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BEGIN:VEVENT
SUMMARY:Renming Song (University of Illinois Urbana-Champaign)
DTSTART:20200623T130000Z
DTEND:20200623T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/1
DESCRIPTION:Title: Factorizations and estimates of Dirichlet heat kernels fo
r non-local operators with critical killings\nby Renming Song (Univers
ity of Illinois Urbana-Champaign) as part of Non-local operators\, probabi
lity and singularities\n\n\nAbstract\nIn this talk I will discuss heat ker
nel estimates for critical perturbations \nof non-local operators. To be m
ore precise\, let $X$ be the reflected \n$\\alpha$-stable process in the c
losure of a smooth open set $D$\, and \n$X^D$ the process killed upon exit
ing $D$. We consider potentials of the \nform $\\kappa(x)=C\\delta_D(x)^{-
\\alpha}$ with positive $C$ and the \ncorresponding Feynman-Kac semigroups
. Such potentials do not belong \nto the Kato class. We obtain sharp two-s
ided estimates for the heat \nkernel of the perturbed semigroups. The inte
rior estimates of the \nheat kernels have the usual $\\alpha$-stable form\
, while the boundary \ndecay is of the form $\\delta_D(x)^p$ with non-nega
tive \n$p\\in [\\alpha-1\, \\alpha)$ depending on the precise value of the
\nconstant $C$. Our result recovers the heat kernel estimates of both \nt
he censored and the killed stable process in $D$. Analogous \nestimates ar
e obtained for the heat kernel of the Feynman-Kac \nsemigroup of the $\\al
pha$-stable process in \n${\\mathbf R}^d\\setminus \\{0\\}$ through the po
tential $C|x|^{-\\alpha}$. \n\nAll estimates are derived from a more gener
al result described as follows: \nLet $X$ be a Hunt process on a locally c
ompact separable metric space in \na strong duality with $\\widehat{X}$. A
ssume that transition densities of \n$X$ and $\\widehat{X}$ are comparabl
e to the function $\\widetilde{q}(t\,x\,y)$ \ndefined in terms of the volu
me of balls and a certain scaling function. \nFor an open set $D$ consider
the killed process $X^D$\, and a critical \nsmooth measure on $D$ with th
e corresponding positive additive functional \n$(A_t)$. We show that the
heat kernel of the the Feynman-Kac semigroup \nof $X^D$ through the multip
licative functional $\\exp(-A_t)$ admits the \nfactorization of the form \
n${\\mathbf P}_x(\\zeta >t)\\widehat{\\mathbf P}_y(\\widehat{\\zeta}>t)\\w
idetilde{q}(t\,x\,y)$.\n\nThis talk is based on a joint paper with Soobin
Cho\, Panki Kim and Zoran Vondracek.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Kulik (Wrocław University of Science and Technology)
DTSTART:20200630T130000Z
DTEND:20200630T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/2
DESCRIPTION:Title: Moment bounds for dissipative semimartingales with heavy
jumps\nby Alexei Kulik (Wrocław University of Science and Technology)
as part of Non-local operators\, probability and singularities\n\n\nAbstr
act\nThe talk is based on a joint research with Ilya Pavlyukevich. We show
that if the jumps of an Ito-semimartingale $X$ admit a finite $p$-moment\
, $p>0$\,\nthe radial part of its drift is dominated at $\\infty$ by $-|X|
^\\kappa$ for some $\\kappa\\geq -1$\, and the balance condition $p+\\kapp
a>1$ holds true\, then\nunder some further minor technical assumptions\n$\
\sup_{t\\geq 0} \\mathbb{E} |X_t|^{p_X}<\\infty$ for each $p_X\\in(0\,p+\\
kappa-1)$. The upper bound $p+\\kappa-1$ is generically optimal.\nThe proo
f is based on the extension of the method of Lyapunov functions to the sem
imartingale framework.\n\nOur study of the uniform-in-time moment estimate
s is strongly motivated by needs of the Stochastic Averaging/Homogenizatio
n theory for Levy driven multi-scale systems\, which will be discussed in
the talk\, as well.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomasz Jakubowski (Wrocław University of Science and Technology)
DTSTART:20200707T130000Z
DTEND:20200707T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/3
DESCRIPTION:Title: Critical Schrödinger perturbations of fractional Laplaci
an\nby Tomasz Jakubowski (Wrocław University of Science and Technolog
y) as part of Non-local operators\, probability and singularities\n\n\nAbs
tract\nLet $p(t\,x\,y)$ be the fundamental solution of the equation $\\par
tial_t u(t\,x) = \\Delta^{\\alpha/2} u(t\,x)$.\nI will consider the integr
al equation\n$$\n\\tilde{p}(t\,x\,y) = p(t\,x\,y) + \\int_0^t \\int_{\\mat
hbb{R}^d} p(t-s\,x\,z) q(z) \\tilde{p}(s\,z\,y) dz ds\,\n$$\nwhere $q(z) =
\\frac{\\kappa}{|z|^{\\alpha}}$ and $\\kappa$ is some constant. The funct
ion $\\tilde{p}$ solving this equation will be called the Schrödinger per
turbations of the function $p$ by $q$. I will present the results concer
ning the estimates of the function $\\tilde{p}$ in both cases $\\kappa>0$
and $\\kappa<0$.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Franziska Kühn (Technical University of Dresden)
DTSTART:20200714T130000Z
DTEND:20200714T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/4
DESCRIPTION:Title: A maximal inequality for martingale problems and applicat
ions\nby Franziska Kühn (Technical University of Dresden) as part of
Non-local operators\, probability and singularities\n\n\nAbstract\nMarting
ale problems aim to characterize stochastic processes by their martingale
properties. A famous example is Lévy's characterization theorem which cha
racterizes Brownian motion by its first two conditional moments. More gene
rally\, a wide class of Markov processes and stochastic differential equat
ions can be described using martingale problems.\n\nIn this talk\, we stud
y martingale problems associated with Lévy-type operators. We present a m
aximal inequality\, which goes back to R. Schilling\, and discuss some var
iants of it. We show that the maximal inequality has many useful applicati
ons in the study of distributional and path properties of the correspondin
g stochastic process\, e.g. criteria for non-explosion in finite time\, ex
istence of moments\, ...\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Panki Kim (Seoul National University)
DTSTART:20200721T130000Z
DTEND:20200721T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/5
DESCRIPTION:Title: Estimates on transition densities of subordinators with j
umping density decaying in mixed polynomial orders\nby Panki Kim (Seou
l National University) as part of Non-local operators\, probability and si
ngularities\n\n\nAbstract\nIn this talk\, we discuss the sharp two-sided e
stimates on the transition densities for subordinators whose Lévy measure
s are absolutely continuous and decaying in mixed polynomial orders. Under
a weaker assumption on Lévy measures\, we also discuss a precise asympto
tic behaviors of the transition densities at infinity. Our results cover g
eometric stable subordinators\, Gamma subordinators and much more. This is
a joint work with Soobin Cho.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krzysztof Bogdan (Wrocław University of Science and Technology)
DTSTART:20200728T130000Z
DTEND:20200728T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/6
DESCRIPTION:Title: Nonlinear nonlocal Douglas identity\nby Krzysztof Bog
dan (Wrocław University of Science and Technology) as part of Non-local o
perators\, probability and singularities\n\n\nAbstract\nI will present res
ults from the joint work with Tomasz Grzywny\, Katarzyna Pietruska-Pałuba
\, Artur Rutkowski with the same title (available at https://arxiv.org/abs
/2006.01932 ). We give Hardy-Stein and Douglas identities for specific non
linear nonlocal Sobolev-Bregman integral forms with unimodal Lévy measure
s. We prove that the corresponding Poisson integral defines an extension o
perator for the Sobolev-Bregman spaces. The results generalizes to the set
ting of $L^p$ spaces the earlier results of the authors\, obtained for the
(quadratic) Dirichlet forms and $L^2$ spaces.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victoria Knopova
DTSTART:20200811T130000Z
DTEND:20200811T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/7
DESCRIPTION:Title: Construction and heat kernel estimates of general stable-
like Markov processes\nby Victoria Knopova as part of Non-local operat
ors\, probability and singularities\n\n\nAbstract\nStarting with a non-sy
mmetric $\\alpha$-stable- like pseudo-differential operator $L$ defined
on the test functions\, we show that the corresponding martingale problem
is well-posed\, and its solution is a strong Markov process which admits
a transition probability density. We investigate the structure of this d
ensity in the vicinity of the starting point. In particular\, we show th
at due to the non-symmetry the respective density is not necessarily bound
ed\, and one needs additional assumptions on the Lévy-type kernel of th
e operator in order to get a point-wise upper bound on the transition pro
bability density.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xicheng Zhang (Wuhan)
DTSTART:20200915T130000Z
DTEND:20200915T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/8
DESCRIPTION:Title: Singular HJB equations with applications to KPZ on the re
al line\nby Xicheng Zhang (Wuhan) as part of Non-local operators\, pro
bability and singularities\n\n\nAbstract\nI will talk about the Hamilton-J
acobi-Bellman equations with distribution-valued coefficients\, which is
not well-defined in the classical sense and shall be understood by using p
aracontrolled distribution method introduced by Gubinelli-Imkeller-Perkows
ki. By a new characterization of weighted Hölder space and Zvonkin's tran
sformation we prove some new a priori estimates\, and therefore\, establis
h the global well-posedness for singular HJB equations. As an application\
, the global well-posedness for KPZ equations on the real line in polynomi
al weighted Hölder spaces is obtained without using Cole-Hopf's transform
ation. This is a joint work with Rongchan Zhu and Xiangchan Zhu.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zoran Vondraček (Zagreb)
DTSTART:20200922T130000Z
DTEND:20200922T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/9
DESCRIPTION:Title: On the potential theory of Markov processes with jump ker
nels decaying at the boundary\nby Zoran Vondraček (Zagreb) as part of
Non-local operators\, probability and singularities\n\n\nAbstract\nIn thi
s talk\, I will consider some potential theory of the process $Y$ on an op
en set $D\\subset \\mathbb{R}^d$ associated with a pure jump Dirichlet for
m whose jump kernel has the form $J(x\,y)=B(x\,y)|x-y|^{-d-\\alpha}$\, $0<
\\alpha<2$. Here $B(x\,y)$ -- the boundary term -- depends on $\\delta_D(x
)\, \\delta_D(y)$ and $|x-y|$\, and is allowed to approach 0 at the bounda
ry. This is in contrast with previous works where $B(x\,y)$ is assumed to
be bounded between two positive constants\, which can be viewed as a unifo
rm ellipticity condition for non-local operators. The conditions and the f
orm of the boundary term $B(x\,y)$ are motivated by jump kernels of some s
ubordinate killed Lévy processes.\n\nWe prove that non-negative harmonic
functions of the process satisfy the Harnack inequality and Carleson's est
imate. Furthermore\, in case when $D$ is the half-space we investigate whe
n the boundary Harnack principle holds. This is joint work with Panki Kim
(Seoul National University) and Renming Song (University of Illinois).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Angkana Rüland (Heidelberg)
DTSTART:20200929T130000Z
DTEND:20200929T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/10
DESCRIPTION:Title: Uniqueness\, stability and single measurement recovery f
or the fractional Calderón problem\nby Angkana Rüland (Heidelberg) a
s part of Non-local operators\, probability and singularities\n\n\nAbstrac
t\nIn this talk I discuss a nonlocal inverse problem\, the\nfractional Cal
derón problem. This is an inverse problem for a\nfractional Schrödinger
equation in which one seeks to recover\ninformation on an unknown potentia
l by exterior measurements. In the\ntalk\, I prove uniqueness and stabilit
y of the "infinite data problem"\nand then address the recovery question.
This also yields surprising\ninsights on the uniqueness properties of the
inverse problem\, in that it\nturns out that a single measurement suffice
s to uniquely recover the\npotential.\n\nThese properties are based on the
very strong unique continuation and\napproximation properties of fraction
al Schrödinger operators\, which are\nof independent interest and which I
also discuss in the talk.\n\nThis is based on joint work with T. Ghosh\,
M. Salo and G. Uhlmann.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hongjie Dong (Brown)
DTSTART:20201006T130000Z
DTEND:20201006T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/11
DESCRIPTION:Title: Evolutionary equations with nonlocal time derivatives\nby Hongjie Dong (Brown) as part of Non-local operators\, probability an
d singularities\n\n\nAbstract\nI will present some recent results about fr
actional parabolic and wave equations with nonlocal Caputo time derivative
s. Under various vanishing mean oscillation (VMO) conditions on the leadin
g coefficients\, we obtained weighted and mixed-norm Sobolev estimates in
the whole space\, half space\, or domains.\n\nThis is based on joint work
with Doyoon Kim (Korea University) and Yanze Liu (Brown University).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daesung Kim (Illinois Urbana-Champaign)
DTSTART:20201013T130000Z
DTEND:20201013T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/12
DESCRIPTION:Title: Quantitative isoperimetric inequalities arising from sto
chastic processes\nby Daesung Kim (Illinois Urbana-Champaign) as part
of Non-local operators\, probability and singularities\n\n\nAbstract\nIt i
s well known that isoperimetric type inequalities hold for a large class o
f quantities arising from Brownian motion. Banuelos and Mendez-Hernandez s
howed that such inequalities can be extended to a wide class of Levy proce
sses. A stability question is if the inequality will be about to achieving
the equality when a given domain is close to being a ball. This question
can be answered by quantitative improvement of such inequalities in terms
of the asymmetry. In this talk\, we discuss the quantitative isoperimetric
inequalities for the expected lifetime of Brownian motion and $\\alpha$-s
table processes\, and some related open problems.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Verbitsky (Missouri)
DTSTART:20201020T140000Z
DTEND:20201020T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/13
DESCRIPTION:Title: Pointwise estimates of positive solutions to linear and
semilinear equations with nonlocal operators\nby Igor Verbitsky (Misso
uri) as part of Non-local operators\, probability and singularities\n\n\nA
bstract\nRecent results will be presented involving sharp global pointwise
estimates of positive solutions to some linear and semilinear partial di
fferential equations and inequalities with nonlocal operators satisfying
various forms of the maximum principle or domination principle. In particu
lar\, equations of the type\n\\[\n(-\\Delta)^{\\frac{\\alpha}{2}} u = g(u)
\\sigma +\\mu \\quad \\text{in} \\\, \\\,\n\\Omega\, \\quad u=0 \\\, \\\,
\\\, \\text{in} \\\, \\\, \\Omega^c\,\n\\]\nwith measure coefficients $\\
sigma$\, $\\mu$\, where $g(u)=u^q$ and $0< \\alpha < n$ in certain domains
$\\Omega \\subseteq {\\mathbb{R}}^n$\, or Riemannian manifolds\, with pos
itive Green's function will be discussed.\n\nJoint work with Alexander Gri
gor'yan.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jie-Ming Wang (Beijing)
DTSTART:20201103T140000Z
DTEND:20201103T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/14
DESCRIPTION:Title: Boundary Harnack Principle for Diffusion with Jumps\
nby Jie-Ming Wang (Beijing) as part of Non-local operators\, probability a
nd singularities\n\n\nAbstract\nFor $d\\geq 3$\, consider the operator ${\
\mathcal L}^{\\bf b}={\\mathcal L}^0+b_1\\cdot \\nabla+{\\mathcal S}^{b_2}
$\,\nwhere ${\\mathcal L}^0$ is a second order elliptic operator in non-
divergence form\,\nthe function $b_1$ belongs to some Kato class and\n$$\
n{\\mathcal S}^{b_2} f(x):=\n\\int_{{\\mathbb R}^d} \\left( f(x+z)-f(x)- \
\nabla f(x) \\cdot\nz\\\, {\\mathbb 1}_{{|z|\\leq 1}} \\right) b_2(x\, z)
J_0(z) \\\,dz\, \\quad f\\in C_b^2({\\mathbb R}^d)\,\n$$\nwhere $J_0(z)$ s
atisfies that there exist positive constants $c_1\, c_2$ and $0<\\beta_1\\
leq \\beta_2 <2$ such that\n $$c_1 (|z_2|/|z_1|)^{d+\\beta_1} \\leq J_0(z
_1)/J_0(z_2)\n\\leq c_2 (|z_2|/|z_1|)^{d+\\beta_2}\n\\quad {f\\!or\\\, any
}\\quad z_1\, z_2\\in {\\mathbb R}^d \\quad{with}\\quad 0<|z_1|\\leq |z_2|
\,$$\n $b_2(x\, z)$ is a real-valued bounded function\non ${\\mathbb R}^
d\\times {\\mathbb R}^d$ satisfying for each $x\\in {\\mathbb R}^d$\,\n$
b_2(x\, \\cdot )\\geq 0$ a.e. on ${\\mathbb R}^d$\, and\n$$\n1_{\\beta_
2=1} \\int_{r<|z|\\leq R}z b_2(x\, z) J_0(z)\\\,dz=0 \\quad {f\\!or\\\, ev
ery}\\quad x\\in {\\mathbb R}^d\n\\quad {and}\\quad 0< r < R < \\infty.\n$
$\nUnder the uniformly ellipticity condition and Hölder condition on the
diffusion coefficient $a_{ij}\,$\nthere exists a conservative Feller proce
ss $X^{\\bf b}$ with its infinitesimal generator ${\\mathcal L}^{\\bf b}$.
\nWe give the two-sided Green function estimates of $X^{\\bf b}$ on a boun
ded $C^{1\,1}$ domain $D$ and further establish the Martin integral repres
entation of harmonic function with respect to $X^{\\bf b}$ on the domain
$D$.\nUsing the Green function estimates and the Martin integral formula
in $D$\, the Harnack principle and the boundary Harnack principle with exp
licit boundary decay rate for the operator ${\\mathcal L}^{\\bf b}$ under
some mild conditions\nare established.\nThis talk is based on a joint wor
k with Professor Z.-Q. Chen.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jian Wang (Fujian)
DTSTART:20201110T140000Z
DTEND:20201110T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/15
DESCRIPTION:Title: Heat kernel upper bounds for symmetric Markov semigroups
\nby Jian Wang (Fujian) as part of Non-local operators\, probability a
nd singularities\n\n\nAbstract\nIt is well known that Nash-type inequaliti
es for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel
upper bounds for associated symmetric Markov semigroups. In this talk\, w
e show that both imply (and hence are equivalent to) off-diagonal heat ker
nel upper bounds under some mild assumptions. Our approach is based on a
new generalized Davies's method. Our results extend that by Carlen-Kusuoka
-Stroock for Nash-type inequalities with power order considerably and also
extend that by Grigor'yan for second order differential operators on a co
mplete non-compact manifold.\n\nThe talk is based on a joint work with Z.-
Q. Chen (Seattle)\, P. Kim (Seoul) and T. Kumagai (Kyoto).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tadeusz Kulczycki (Wroclaw)
DTSTART:20201117T140000Z
DTEND:20201117T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/16
DESCRIPTION:Title: On weak solution of SDE driven by inhomogeneous singular
Lévy noise.\nby Tadeusz Kulczycki (Wroclaw) as part of Non-local ope
rators\, probability and singularities\n\n\nAbstract\nWe study the stochas
tic differential equation\n$dX_t = A_t(X_{t-}) \\\, dZ_t$\, $ X_0 = x$\,\n
where $Z_t = (Z_t^{(1)}\,\\ldots\,Z_t^{(d)})^T$ and for each $i \\in \\{1\
,\\ldots\,d\\}$ $Z_t^{(i)}$ is a one-dimensional\, symmetric $\\alpha_i$-s
table process\, where $\\alpha_i \\in (0\,2)$. Under appropriate condition
s on $\\alpha_1\,\\ldots\,\\alpha_d$ and on matrices $A_t$ we prove exist
ence and uniqueness of the weak solution of the above SDE\, which will be
shown to be a time-inhomogeneous Markov process. We also provide a represe
ntation of the transition probability density of this process as a sum of
explicitly given ‘principal part’\, and a ‘residual part’ subject
to a set of estimates showing that this part is negligible in a short time
. The talk is based on a joint work with A. Kulik and M. Ryznar.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Longmin Wang (Nankai)
DTSTART:20201201T140000Z
DTEND:20201201T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/17
DESCRIPTION:Title: Branching Random Walks on Hyperbolic Spaces\nby Long
min Wang (Nankai) as part of Non-local operators\, probability and singula
rities\n\n\nAbstract\nThe branching Brownian motion on the hyperbolic plan
e with binary\nfission at rate $\\lambda > 0$ exhibits a phase transition
in\n$\\lambda$: For $\\lambda \\leq 1/8$ the number of particles in any\nc
ompact region is eventually $0$\, w.p.1\, but for $\\lambda > 1/8$\nthe nu
mber of particles in any open region grows to $\\infty$\nw.p.1. Lalley and
Sellke (1987) proved that in the subcritical and\ncritical case ($\\lambd
a \\leq 1/8$) the set $\\Lambda$ of all limit\npoints in the boundary circ
le at $\\infty$ consisting of particle\ntrails is a Cantor set\, while in
the supercritical case ($\\lambda\n>1/8$) the set $\\Lambda$ has full Lebe
sgue measure. For $\\lambda\n\\leq 1/8$ the Hausdorff dimension of $\\Lamb
da$ is at most $1/2$\nand has critical exponent $1/2$ near the critical v
alue $\\lambda =\n1/8$. In this talk we will prove the same type of phase
transition\noccurs for branching random walks on hyperbolic spaces.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mateusz Kwaśnicki (Wroclaw)
DTSTART:20201124T140000Z
DTEND:20201124T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/19
DESCRIPTION:Title: Harmonic extensions\, operators with completely monotone
kernels\, and traces of 2-D diffusions\nby Mateusz Kwaśnicki (Wrocla
w) as part of Non-local operators\, probability and singularities\n\nAbstr
act: TBA\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jerome Goldstein (Memphis)
DTSTART:20201208T150000Z
DTEND:20201208T160000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/20
DESCRIPTION:Title: The Boderline between Some Good Problems and the Corresp
onding Bad Problems\nby Jerome Goldstein (Memphis) as part of Non-loca
l operators\, probability and singularities\n\n\nAbstract\nWe will discuss
three problems in PDE for which existence or nonexistence of\ncertain kin
ds of equations is a delicate issue. Many coauthors are involved\, and the
\nproblems are related to each other.\n\nThe first problem involves work f
rom the 1970s about uniqueness for certain\nill posed problems involving t
he Euler-Poisson-Darboux equation. The number of\ninitial conditions requi
red for uniqueness involves the size of negative parameter in\nthe singula
r term and the definition of solution.\n\nThe second problem involves the
Schrödinger operator with the inverse square\npotential multiplied by a c
onstant c. The spectrum of this operator on $L^2(\\mathbb{R}^n)$ is\neithe
r $\\mathbb{R}$ or $\\mathbb{R}^+$\, depending on the choice of c: In the
1980s\, it was proved that the corresponding heat equation has instantaneo
us blow up and no positive solutions\nwhen the spectrum is $\\mathbb{R}$.
The corresponding result is true when Euclidean space is\nreplaced by the
Heisenberg group\; this was proved in 2020.\n\nThe final problem is nonlin
ear and involves the parabolic problem for the fast\ndiffusion equation or
the p-Laplacian heat equation\, perturbed in various ways\, on\nEuclidean
space or on a Riemannaian manifold. In some cases one can show the\nabsen
ce of nonnegative solutions (except for the zero function).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Moritz Kassmann (Bielefeld)
DTSTART:20210112T140000Z
DTEND:20210112T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/21
DESCRIPTION:Title: Heat kernel estimates for mixed systems of diffusions an
d jump processes\nby Moritz Kassmann (Bielefeld) as part of Non-local
operators\, probability and singularities\n\n\nAbstract\nWe prove sharp he
at kernel estimates for symmetric Markov processes that are independent co
pies of one-dimensional jump or diffusion processes. The result can be se
en as a robustness result for heat kernels like the one of Aronson (1968)
for diffusions or the one of Chen/Kumagai (2003) for isotropic jump proces
ses. The talk is based on a joint work together with Jaehoon Kang (KAIST).
\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guohuan Zhao (Bielefeld)
DTSTART:20201215T140000Z
DTEND:20201215T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/22
DESCRIPTION:Title: Regularity properties of jump diffusions with irregular
coefficients\nby Guohuan Zhao (Bielefeld) as part of Non-local operato
rs\, probability and singularities\n\n\nAbstract\nIn this talk\, I plan to
present some results about the regularity properties of strong solutions
to SDEs driven by Lévy processes with irregular drift coefficients. In sh
ort\, I will show the Malliavin differentiability of the unique strong sol
utions as well as the differentiability of the stochastic flows with respe
ct to the spatial variable. Meanwhile\, I will also talk about the Schaude
r's estimate for the resolvent equations corresponding to the SDEs.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carina Geldhauser (Lund)
DTSTART:20210119T140000Z
DTEND:20210119T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/23
DESCRIPTION:Title: The fractional Green function in atmospheric turbulence
models\nby Carina Geldhauser (Lund) as part of Non-local operators\, p
robability and singularities\n\n\nAbstract\nIn this talk we discuss a fami
ly of discrete models for atmospheric turbulence\, often called point vort
ex models.\n\nWe state some of it basic properties and show how we can der
ive an effective PDE\, the so-called mean field limit\, from the discrete
Hamiltonian system\, by using a variational principle. Furthermore\, we d
iscuss the usage and interpretation of these models in statistical physics
.\n\nThe content of this talk is based joint work with Marco Romito (Uni P
isa).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damir Kinzebulatov (Quebec)
DTSTART:20210126T140000Z
DTEND:20210126T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/25
DESCRIPTION:Title: Fractional Kolmogorov operator and desingularizing weigh
ts\nby Damir Kinzebulatov (Quebec) as part of Non-local operators\, pr
obability and singularities\n\n\nAbstract\nThe subject of this talk are sh
arp two-sided bounds on the heat kernel of the fractional Laplacian pertur
bed by a Hardy-type drift\, which we establish by transferring the operato
r to an appropriate weighted space with singular weight. The talk is based
on joint papers with Yu.A.Semenov and K.Szczypkowski.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vanja Wagner (Zagreb)
DTSTART:20210202T140000Z
DTEND:20210202T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/26
DESCRIPTION:Title: Semilinear equations for non-local operators: beyond the
fractional Laplacian\nby Vanja Wagner (Zagreb) as part of Non-local o
perators\, probability and singularities\n\n\nAbstract\nWe study semilinea
r problems in general bounded open sets for non-local operators with exter
ior and boundary conditions\, where the operators are more general than th
e fractional Laplacian. We also give results in case of bounded $C^{1\,1}$
open sets. The talk is based on joint work with Ivan Biočić and Zoran V
ondraček.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Silvestre (The University of Chicago)
DTSTART:20210316T140000Z
DTEND:20210316T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/27
DESCRIPTION:Title: Regularity estimates for the Boltzmann equation without
cutoff\nby Luis Silvestre (The University of Chicago) as part of Non-l
ocal operators\, probability and singularities\n\n\nAbstract\nWe study the
regularization effect of the inhomogeneous Boltzmann equation without cut
off. We obtain a priori estimates for all derivatives of the solution depe
nding only on bounds of its hydrodynamic quantities: mass density\, energy
density and entropy density. As a consequence\, a classical solution to t
he equation may fail to exist after a certain time T only if at least one
of these hydrodynamic quantities blows up. Our analysis applies to the cas
e of moderately soft and hard potentials. We use methods that originated i
n the study of nonlocal elliptic and parabolic equations: a weak Harnack i
nequality in the style of De Giorgi\, and a Schauder-type estimate.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gisèle Goldstein (The University of Memphis)
DTSTART:20210323T140000Z
DTEND:20210323T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/28
DESCRIPTION:Title: On Thomas-Fermi Theory and Extensions\nby Gisèle Go
ldstein (The University of Memphis) as part of Non-local operators\, proba
bility and singularities\n\n\nAbstract\nOf concern to quantum chemists and
solid state physicists is the approximate numerical computation of the gr
ound state wave function\, and the ground state energy and density for mol
ecular and other quantum mechanical systems. Since the number of molecules
in bulk matter is of the order of 1026\, direct computation is too cumber
some or impossible in many situations. In 1927\, L. Thomas and E. Fermi pr
oposed replacing the ground state wave function by the ground state densit
y\, which is a function of only three variables. Independently\, each foun
d an approximate expansion for the energy associated with a density. (The
wave function uniquely determines the density\, but not conversely.)\n\nA
computationally better approximate expansion was found in the 1960’s by
W. Kohn and his collaborators\; for this work Kohn got the Nobel Prize in
Chemistry in 1998. A successful attempt to put Thomas-Fermi theory into a
rigorous mathematical framework was begun in the 1970’s by E. Lieb and B
. Simon and was continued and expanded by Ph. Benilan\, H. Brezis and othe
rs. Very little rigorous mathematics supporting Kohn density functional th
eory is known. In this talk I will present a survey of rigorous Thomas-Fer
mi theory\, including recent developments and open problems (in the\ncalcu
lus of variations and semilinear elliptic systems).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazuhiro Kuwae (Fukuoka University)
DTSTART:20210330T130000Z
DTEND:20210330T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/29
DESCRIPTION:Title: Lp-Kato class measures for symmetric Markov processes un
der heat kernel estimates\nby Kazuhiro Kuwae (Fukuoka University) as p
art of Non-local operators\, probability and singularities\n\n\nAbstract\n
I will talk on the coincidence of two classes of $L^p$-Kato class measures
\nin the framework of symmetric Markov processes admitting upper and lower
estimates of heat kernel under mild conditions. One class of $L^p$-Kato c
lass measures is defined by the $p$-th power of positive order resolvent k
ernel\, another is defined in terms of the $p$-th power of Green kernel de
pending on some exponents related to the heat kernel estimates. We also pr
ove that $q$-th integrable functions on balls with radius $1$ having unifo
rmity of its norm with respect to centers are of $L^p$-Kato class if $q$ i
s greater than a constant related to $p$ and the constants appeared in the
upper and lower estimates of the heat kernel. These are complete extensio
ns of some results\nby Aizenman-Simon and the recent results by the second
named author in the framework of Brownian motions on Euclidean space. We
further give necessary and sufficient conditions\nfor a Radon measure with
Ahlfors regularity to belong to $L^p$-Kato class. Our results can be appl
icable to many examples\, for instance\, symmetric (relativistic) stable p
rocesses\, jump processes on $d$-sets\, Brownian motions on Riemannian man
ifolds\, diffusions on fractals and so on.\nJoint work with Takahiro Mori.
\n\nThe details can be seen in https://arxiv.org/abs/2008.10934\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qi Zhang (University of California\, Riverside)
DTSTART:20210504T130000Z
DTEND:20210504T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/31
DESCRIPTION:Title: Time analyticity and reversibility of some parabolic equ
ations\nby Qi Zhang (University of California\, Riverside) as part of
Non-local operators\, probability and singularities\n\n\nAbstract\nWe desc
ribe a concise way to prove time analyticity for solutions of parabolic eq
uations including the heat and Navier Stokes equations. In some cases\, re
sults under sharp conditions are obtained. An application is a necessary a
nd sufficient condition for the solvability of the backward heat equation
which is ill-posed\, helping to remove an old obstacle in control theory.\
n\nPart of the work is joint with Hongjie Dong\, which is related to earli
er joint work with F. H. Lin.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolai Krylov (University of Minnesota)
DTSTART:20210416T140000Z
DTEND:20210416T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/32
DESCRIPTION:Title: (joint with Montreal-Quebec Analsyis Seminar)\nby Ni
colai Krylov (University of Minnesota) as part of Non-local operators\, pr
obability and singularities\n\n\nAbstract\nFind out more details:\n\nhttps
://researchseminars.org/seminar/MathematicalAnalysis\n\nhttps://www.math.m
cgill.ca/jakobson/analysish/seminar.html\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jamil Chaker (Bielefeld University)
DTSTART:20210420T130000Z
DTEND:20210420T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/33
DESCRIPTION:Title: On nonlocal operators with anisotropic kernels\nby J
amil Chaker (Bielefeld University) as part of Non-local operators\, probab
ility and singularities\n\n\nAbstract\nIn this talk we study a class of (l
inear and nonlinear) integro-differential operators with anisotropic and s
ingular kernels. We present local robust regularity estimates for weak sol
utions in the general framework of bounded measurable coefficients. \nThe
results in this talk are based on joint works with Moritz Kassmann\, Minhy
un Kim and Marvin Weidner.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomasz Grzywny (Wroclaw University of Science and Technology)
DTSTART:20210615T130000Z
DTEND:20210615T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/35
DESCRIPTION:Title: Subordinated Markov processes: estimates for heat kernel
s and Green functions\nby Tomasz Grzywny (Wroclaw University of Scienc
e and Technology) as part of Non-local operators\, probability and singula
rities\n\n\nAbstract\nLet (M\, d) be a metric space and μ a Radon measure
on M. Assume that {S(t)}_{t\\in T} is a Markov process on M such that its
transition function is absolutely continuous with μ\, where T is a set o
f non-negative integers or a set of non-negative real numbers. By A we den
ote the semigroup generator associated with the transition function of {S(
t)}. For a Bernstein function f we define a new semigroup with generator
-f(-A) that is a semigroup for the Markov process {S(K_t)}\, where {K_t} i
s a subordinator on T associated with the function f. During the talk\, t
here will be discussed estimates of the haet kernel/transition function a
nd Green function of {S(K_t)}. The proofs are elementary and do not use es
timates for transition probability of the subordinator. The talk is based
on joint work with Bartosz Trojan.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Timur Yastrzhembskiy (Brown University)
DTSTART:20210601T130000Z
DTEND:20210601T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/37
DESCRIPTION:Title: Global $L_p$-estimates for kinetic Kolmogorov-Fokker-Pla
nck equation\nby Timur Yastrzhembskiy (Brown University) as part of No
n-local operators\, probability and singularities\n\n\nAbstract\nWe study
the degenerate Kolmogorov equation (also known as kinetic Fokker-Planck
equation) in both nondivergence and divergence forms:\n$$\n \\partial_t u
- v \\cdot D_x u - a^{i j} (z) D_{v_i v_j} u + \\lambda u = f\,\n$$\n$$\n
\\partial_t u - v \\cdot D_x u - D_{v_i} (a^{i j} (z) D_{v_j} u) + \\lambd
a u = D_{v_i} f_i + f_0.\n$$\nThe leading coefficients are merely measurab
le in $t$ and satisfy the VMO condition in $x\, v$ with respect to some
quasi-metric. We also assume boundedness and uniform nondegeneracy of $a$.
We prove global a priori estimates in some weighted mixed-norm Lebesgue s
paces and solvability results. Our proof does not rely on kernel estimates
.\n\nThe talk is based on a joint work with Hongjie Dong\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stjepan Šebek (University of Zagreb)
DTSTART:20210608T130000Z
DTEND:20210608T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/38
DESCRIPTION:Title: Limit theorems for a stable sausage\nby Stjepan Šeb
ek (University of Zagreb) as part of Non-local operators\, probability and
singularities\n\n\nAbstract\nIn this talk\, we study fluctuations of the
volume of a stable sausage defined via a d-dimensional rotationally invari
ant alpha-stable process. As the main results\, we establish a central lim
it theorem and functional central limit theorem (in the case when d/alpha
> 3/2) with a standard one-dimensional Brownian motion in the limit\, and
Khintchine’s and Chung’s laws of the iterated logarithm (in the case w
hen d/alpha > 9/5).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mario Maurelli (Università degli Studi di Milano)
DTSTART:20210427T130000Z
DTEND:20210427T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/39
DESCRIPTION:Title: Regularization by noise for transport PDEs: two results<
/a>\nby Mario Maurelli (Università degli Studi di Milano) as part of Non-
local operators\, probability and singularities\n\n\nAbstract\nWe say that
a regularization by noise phenomenon occurs for a possibly ill-posed diff
erential equation if this equation becomes well-posed (in a pathwise sense
) under addition of a suitable noise term. The long-term aim of regulariza
tion by noise is to show this phenomenon for PDEs coming from physics\, es
pecially fluid dynamics.\n\nIn this talk we consider regularization by noi
se for transport-type PDEs. A transport-type PDE is a prototype for many P
DEs from physics and takes the form\n$$\n\\partial_t u(t\,x) +b(t\,x\,u)\\
cdot\\nabla u(t\,x) =0\,\\quad t>0\,x\\in\\mathbb{R}^d\,u(t\,x)\\in\\mathb
b{R}\,\\qquad (1)\n$$\nwith $b$ given vector field. We focus on two cases\
,\n$$\nb(t\,x\,u) = \\tilde b(t\,x) \\text{ and } b(t\,x\,u)=\\tilde b(t\,
x)u\,\\qquad (2)\n$$\ncorresponding respectively to the linear transport e
quation and a scalar conservation law. For irregular vector fields $\\tild
e{b}$\, the corresponding deterministic transport equations (1) are in gen
eral ill-posed. We add to the transport equation the so-called transport n
oise\, namely we consider the stochastic PDE\n$$\ndu(t\,x\,\\omega) +b\\cd
ot\\nabla u\\\,dt +\\nabla u\\circ dW =0\,\\\,t>0\,x\\in\\mathbb{R}^d\,u\\
in\\mathbb{R}\, \\qquad (3)\n$$\nwhere $W$ is a $d$-dimensional Brownian m
otion and $\\circ$ stands for Stratonovich integration. We show that\, for
$b$ of the form (2) for some classes of irregular vector fields $\\tilde{
b}$\, the corresponding stochastic transport PDEs (3) are well-posed.\n\nT
he proofs are based on a combination of the renormalization argument by Di
Perna-Lions and some parabolic bounds.\n\nThis talk is based on the works
Attanasio-Flandoli 2011 and Gess-M. 2018.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katarzyna Pietruska-Pałuba (University of Warsaw)
DTSTART:20210511T130000Z
DTEND:20210511T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/40
DESCRIPTION:Title: Properties of the integrated density of states for rando
m Schrödinger operators driven by nonlocal processes\nby Katarzyna Pi
etruska-Pałuba (University of Warsaw) as part of Non-local operators\, pr
obability and singularities\n\n\nAbstract\nWe will discuss asymptotic prop
erties of the integrated density of states for random systems whose hamilt
onians are driven by nonlocal processes\, and the random field is either o
f Poissonian or alloy type. In both cases the IDS exhibits unusually fast
decay near the bottom of the spectrum (called the Lifschitz singularity).
If time permits\, we will also discuss the relation of the asymptotic beha
viour of the IDS to the long-time behaviour of solutions of the nonlocal p
arabolic Anderson problem.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomasz Klimsiak (Institute of Mathematics Polish Academy of Scienc
es)
DTSTART:20210525T130000Z
DTEND:20210525T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/41
DESCRIPTION:Title: Schrödinger equations with smooth measure potential and
general measure data\nby Tomasz Klimsiak (Institute of Mathematics Po
lish Academy of Sciences) as part of Non-local operators\, probability and
singularities\n\n\nAbstract\nWe study equations driven by Schrödinger op
erators consisting of a self-adjoint Dirichlet operator and a singular pot
ential\, which belongs to a class of positive Borel measures absolutely co
ntinuous with respect to a capacity generated by the operator. In particul
ar\, we cover positive potentials exploding on a set of capacity zero. The
right-hand side of equations is allowed to be a general bounded Borel mea
sure. The class of self-adjoint Dirichlet operators is quite large. Exampl
es include integro-differential operators with the local part of divergenc
e form. \n\nWe give a necessary and sufficient condition for the existence
of a solution\, and prove some regularity and stability results.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Minhyun Kim (Bielefeld University)
DTSTART:20210622T130000Z
DTEND:20210622T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/42
DESCRIPTION:Title: The concentration-compactness principle for the nonlocal
anisotropic $p$-Laplacian of mixed order\nby Minhyun Kim (Bielefeld U
niversity) as part of Non-local operators\, probability and singularities\
n\n\nAbstract\nIn this talk\, we introduce a new class of operators with a
n orthotropic structure having different exponents of integrability and di
fferent orders of differentiability. We prove a robust Sobolev-type inequa
lity and establish the existence of minimizers of the Sobolev quotient. Th
e method for the existence of minimizers is based on the concentration-com
pactness principle which we extend to this class of operators. One consequ
ence of the main result is the existence of a nontrivial nonnegative solut
ion to the corresponding critical problem.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikola Sandrić (University of Zagreb)
DTSTART:20210629T130000Z
DTEND:20210629T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/43
DESCRIPTION:Title: Periodic homogenization of linear degenerate PDEs\nb
y Nikola Sandrić (University of Zagreb) as part of Non-local operators\,
probability and singularities\n\n\nAbstract\nIn this talk\, we discuss fun
ctional CLT for a class of degenerate diffusion processes with periodic co
efficients\, thus generalizing the already classical results in the contex
t of uniformly elliptic diffusions. As an application\, we obtain periodic
homogenization of a class of linear degenerate elliptic and parabolic PDE
s. The talk is based on joint work with Ivana Valentić.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wolfgang Arendt (Ulm University)
DTSTART:20210706T130000Z
DTEND:20210706T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/44
DESCRIPTION:Title: Variational Methods for the Dirichlet-to-Neumann Operato
r and Fractional Powers\nby Wolfgang Arendt (Ulm University) as part o
f Non-local operators\, probability and singularities\n\n\nAbstract\nUsual
ly the DtN operator is considered as a pseudo-differential operator on C-i
nfinity domains.\nWe will present a variational approach which works for L
ipschitz domains and even domains with finite surface. The same approach a
llows one to give a functional framework of the Caffarelli-Silvestre exten
sion leading to the fractional laplacian. We will realize the fractional p
ower of any sectorial operator on a Hilbert space (in the sense of Kato) a
s a DtN operator with precise description of the domains.\n\nReferences:\n
\nW. Arendt\, A.F.M. ter Elst: The Dirichlet-to-Neumann operator on C. Ann
. Sc. Norm. Super. Pisa Cl. Sci. 20 (2020) 1169-1196\n\nW. Arendt\, A.F.M.
ter Elst\, M. Warma: Fractional powers of sectorial operators via the Dir
ichlet-to-Neumann operator.\nComm. PDE 43 (2018) 1-24\n\nJ. Galé\, P. Mia
na\, P.R. Stinga: Extension problem and fractional operators: semigroups a
nd wave equation. J. Evol. Eqn. 13 (2013) 343-368.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yehuda Pinchover (Technion – Israel Institute of Technology)
DTSTART:20210713T130000Z
DTEND:20210713T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/45
DESCRIPTION:Title: Optimal Hardy inequalities for Schrodinger operators on
graphs\nby Yehuda Pinchover (Technion – Israel Institute of Technolo
gy) as part of Non-local operators\, probability and singularities\n\n\nAb
stract\nFor a given subcritical discrete Schrödinger operator $H$ on a we
ighted infinite graph $X$\, we construct a Hardy-weight $w$ which is optim
al in the following sense. The operator $H − \\lambda w$ is subcritical
in $X$ for all $\\lambda < 1$\, null-critical in $X$ for $\\lambda = 1$\,
and supercritical near any neighborhood of infinity in $X$ for any $\\lamb
da>1$. Our results rely on a criticality theory for Schrödinger operators
on general weighted graphs.\n\nThis is a joint work with Matthias Keller
and Felix Pogorzelski.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yana Butko (Universität des Saarlandes)
DTSTART:20210720T130000Z
DTEND:20210720T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/46
DESCRIPTION:Title: Stochastic solutions of generalized time-fractional evol
ution equations\nby Yana Butko (Universität des Saarlandes) as part o
f Non-local operators\, probability and singularities\n\n\nAbstract\nThis
is a joint work with Christian Bender\, Saarland University. We consider
a general class of integro-differential evolution equations which includes
the governing equation of the generalized grey Brownian motion and the ti
me- and space-fractional heat equation:\n\n$$u(t\,x) = u_0(x) + \\int_0^t
k(t\,s)Lu(s\,x)ds\, \\qquad t>0\,\\quad x\\in\\mathbb{R}^d\, \\qquad (1)$$
\n\nwhere $L$ is a pseudo-differential operator associated to a Lévy proc
ess and \n$k(t\,s)$\, $0 < s < t < \\infty$\, is a general memory kernel.
Such equations arise in models of anomalous diffusion.\n\nWe present a gen
eral relation between the parameters of the equation and the distribution
of any stochastic process\, which provides a stochastic solution of Feynm
an-Kac type. More precisely\, we derive a series representation in terms
of the time kernel $k$ and the symbol $-\\psi$ of the pseudodifferential o
perator $L$ for the characteristic function of the one-dimensional margina
ls of any stochastic solution. We explain how this series simplifies in th
e important case of homogeneous kernels which includes the kernel $k(t\,s)
=(t-s)^{\\beta-1}/\\Gamma(\\beta)$ for time-fractional evolution equations
and\, more generally\, kernels corresponding to Saigo-Maeda fractional d
iffintegration operators. The connection between Saigo-Maeda fractional di
ffintegration operators and positive random variables with Laplace transfo
rm given by Prabhakar's three parameter generalization of the Mittag-Leffl
er function is established. These results yield a stochastic representat
ion for (1) with a Saigo-Maeda kernel in terms of a randomly slowed down L
évy process $(Y_{At^\\beta})_{t\\geq 0}$\, where $Y$ is a Lévy process w
ith infinitesimal generator $L$\, $A$ is an independent random variable wi
th Laplace transform given by the three-parameter Mittag-Leffler function\
, and $\\beta$ corresponds to the degree of homogeneity of the kernel. If
$Y$ has a stable distribution (e.g.\, in the case of a symmetric fractiona
l Laplacian in space)\, the randomly slowed down Lévy process can be repl
aced by a randomly scaled linear fractional stable motion\, providing a s
tochastic solution in terms of a self-similar process with stationary incr
ements. Similar results hold also in the case of equation (1) with more g
eneral operator $L$ (it is enough to assume that $L$ generates a strongly
continuous semigroup on some Banach space) and lead to Feynman-Kac formula
e for such equations (if $L:=L_0+V$ where $L_0$ generates a Markov process
and $V$ is a suitable potential). \n\n[1] Ch. Bender\, Ya.A. Butko. Stoc
hastic solutions of generalized time-fractional evolution equations// arXi
v:2102.00117 (2021)\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Grzegorz Karch (University of Wrocław)
DTSTART:20210928T130000Z
DTEND:20210928T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/47
DESCRIPTION:Title: Concentration phenomena in a model of chemotaxis\nby
Grzegorz Karch (University of Wrocław) as part of Non-local operators\,
probability and singularities\n\n\nAbstract\nIn this talk\, I shall discus
s results obtained in the papers:\n\nBiler\, Piotr\; Karch\, Grzegorz\; Zi
enkiewicz\, Jacek\, Large\nglobal-in-time solutions to a nonlocal model of
chemotaxis. Adv. Math.\n330 (2018)\, 834–875.\n\nBiler\, Piotr\; Karch\
, Grzegorz\; Pilarczyk\, Dominika\, Global radial\nsolutions in classical
Keller-Segel model of chemotaxis. J.\nDifferential Equations 267 (2019)\,
no. 11\, 6352–6369.\n\nthe both on solutions to certain parabolic-ellipt
ic models of\nchemotaxis. In these papers\, criteria for existence and non
existence\nof global-in-time solutions have been obtained.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathav Murugan (University of British Columbia)
DTSTART:20211026T140000Z
DTEND:20211026T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/48
DESCRIPTION:Title: On the comparison between jump processes and subordinate
d diffusions\nby Mathav Murugan (University of British Columbia) as pa
rt of Non-local operators\, probability and singularities\n\n\nAbstract\nA
well known method to obtain heat kernel estimates and Harnack inequalitie
s for jump processes is to compare the given jump process with a subordina
ted diffusion process. On any space that admits a diffusion which satisfie
s sub-Gaussian heat kernel bounds\, we show that a large family of jump pr
ocesses have a jump kernel comparable to that of a subordinated diffusion
process. If time permits\, I will also discuss another recent result that
the parabolic Harnack inequality implies that the jump kernel admits a den
sity. This is joint work with Guanhua Liu (Tsinghua University).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Marinelli (University College London)
DTSTART:20211102T140000Z
DTEND:20211102T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/49
DESCRIPTION:Title: On the Malliavin differentiability of solutions to react
ion-diffusion equations with multiplicative noise\nby Carlo Marinelli
(University College London) as part of Non-local operators\, probability a
nd singularities\n\n\nAbstract\nWe discuss some recent results about exist
ence and regularity of the Malliavin derivative of the solution\, evaluate
d at fixed points in time and space\, to a parabolic dissipative stochasti
c PDE on $L^2(G)$\, where $G$ is an open bounded domain in $\\mathbb{R}^d$
with smooth boundary. The equation is driven by a multiplicative Wiener n
oise and the nonlinear drift term is the superposition operator associated
to a real function which is assumed to be monotone\, locally Lipschitz co
ntinuous\, and growing not faster than a polynomial. The arguments are bas
ed on the well-posedness theory in the mild sense for stochastic evolution
equations in Banach spaces\, as well as on monotonicity techniques and on
a maximum principle for stochastic evolution equations.\n\nPartly based o
n joint work with Ll. Quer-Sardanyons.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Nowak (Bielefeld University)
DTSTART:20211012T130000Z
DTEND:20211012T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/50
DESCRIPTION:Title: Regularity theory for nonlocal equations with VMO coeffi
cients\nby Simon Nowak (Bielefeld University) as part of Non-local ope
rators\, probability and singularities\n\n\nAbstract\nWe present some high
er regularity results for nonlocal equations with possibly discontinuous c
oefficients of VMO-type in fractional Sobolev spaces. While for correspond
ing local elliptic equations with VMO coefficients it is only possible to
obtain higher integrability\, in our nonlocal setting we are able to also
prove a substantial amount of higher differentiability. Therefore\, our re
sults are in some sense of purely nonlocal type\, following the recent tre
nd of such results in the literature.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Röckner (Bielefeld University)
DTSTART:20211109T140000Z
DTEND:20211109T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/51
DESCRIPTION:Title: Strong dissipativity of generalized time-fractional deri
vatives and quasi-linear (stochstic) partial differential equations\nb
y Michael Röckner (Bielefeld University) as part of Non-local operators\,
probability and singularities\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Stollmann (Chemnitz University of Technology)
DTSTART:20211019T130000Z
DTEND:20211019T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/52
DESCRIPTION:Title: On the decomposition principle and a Persson type theore
m for general regular Dirichlet forms\nby Peter Stollmann (Chemnitz Un
iversity of Technology) as part of Non-local operators\, probability and s
ingularities\n\n\nAbstract\nWe present a decomposition principle for gener
al regular Dirichlet forms satisfying a spatial local compactness conditio
n. We use the decomposition principle to derive a Persson type theorem for
the\ncorresponding Dirichlet forms. Our setting covers non-local forms a
s well as local ones.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pascal Auscher (University of Paris-Saclay)
DTSTART:20211116T140000Z
DTEND:20211116T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/53
DESCRIPTION:Title: On regularity of weak solutions to linear parabolic syst
ems with measurable coefficients\nby Pascal Auscher (University of Par
is-Saclay) as part of Non-local operators\, probability and singularities\
n\n\nAbstract\nWe discuss what can be seen as an old and seemingly closed
problem in regularity theory for parabolic systems. Still novel phenomenon
s show up. Namely\, what is the regularity of weak solutions to equations
or systems of the form $\\partial_t u - div A\\\, grad\\\, u = f+ div F$ o
n a parabolic cylinder assuming minimal conditions on A and (parabolic) s
cale invariant integrability on f and F? Of course\, it depends on how one
defines a weak solution. We show\, and it seems that was not noticed unde
r minimal assumptions\, that local square integrability of u and its grad
ient implies local $L^2$ bounds uniformly in time and more. Further\, w
ith further integrability property on f and F\, we also obtain Hölder co
ntinuity in time with values in $L^p$ for some $p>2$\, which is new\, as
well as higher integrability for the gradient which was due to Giaquinta a
nd Struwe. This uses of half-order time derivatives with some non-local e
stimates. \n\nJoint work with S. Bortz\, M. Egert and O. Saari (JMPA\, 201
9).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Veraar (Delft University of Technology)
DTSTART:20211214T140000Z
DTEND:20211214T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/54
DESCRIPTION:Title: Stochastic Navier-Stokes equations with gradient noise i
n critical spaces\nby Mark Veraar (Delft University of Technology) as
part of Non-local operators\, probability and singularities\n\n\nAbstract\
nIn this talk I will present some recent results on the stochastic Navier-
Stokes equations on the d-dimensional torus with gradient noise\, which ar
ises in the study of turbulent flows. Under very weak smoothness assumptio
ns on the data one has local well-posedness. Moreover\, using a new bootst
rap method new regularization results for solution are obtained. New blow-
up criteria are introduced and can be seen as stochastic versions of the S
errin blow-up criteria. The latter is used to prove global well-posedness
with high probability for small initial data in critical spaces in any dim
ensions $d\\geq 2$. Moreover\, for d=2 we obtain new global well-posedness
results and regularization phenomena\, which unify and extend several ear
lier results.\n\nThe talk is based on joint work with Antonio Agresti http
s://arxiv.org/abs/2107.03953\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rupert Frank (University of Munich)
DTSTART:20220118T140000Z
DTEND:20220118T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/55
DESCRIPTION:Title: Sobolev norms involving fractional Hardy operators\n
by Rupert Frank (University of Munich) as part of Non-local operators\, pr
obability and singularities\n\n\nAbstract\nWe consider the fractional Schr
ödinger operator with Hardy potential and critical or subcritical couplin
g constant. This operator generates a natural scale of homogeneous Sobolev
spaces which we compare with the ordinary homogeneous Sobolev spaces. As
a byproduct\, we obtain generalized and reversed Hardy inequalities for th
is operator. Our results extend those obtained recently for ordinary (non-
fractional) Schrödinger operators and have an important application in th
e treatment of large relativistic atoms. \n\nThe talk is based on joint wo
rk with K. Merz and H. Siedentop\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeremy Quastel (University of Toronto)
DTSTART:20220208T140000Z
DTEND:20220208T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/56
DESCRIPTION:Title: Integrable fluctuations in random growth\nby Jeremy
Quastel (University of Toronto) as part of Non-local operators\, probabili
ty and singularities\n\n\nAbstract\nWe survey models in the KPZ universali
ty class and the integrable Markov process which governs their asymptotic
fluctuations.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fritz Gesztesy (Baylor University)
DTSTART:20211130T140000Z
DTEND:20211130T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/57
DESCRIPTION:Title: Continuity properties of the spectral shift function for
massless Dirac operators and an application to the Witten index\nby F
ritz Gesztesy (Baylor University) as part of Non-local operators\, probabi
lity and singularities\n\n\nAbstract\nWe report on recent results regardin
g the limiting absorption principle for multi-dimensional\, massless Dirac
-type operators (implying absence of singularly continuous spectrum) and c
ontinuity properties of the associated spectral shift function.\n\nWe will
motivate our interest in this circle of ideas by briefly describing the c
onnection to index theory for non-Fredholm operators\, particularly\, to t
he notion of the Witten index.\n\nThis is based on various joint work with
A. Carey\, J. Kaad\, G. Levitina\, R. Nichols\, D. Potapov\, F. Sukochev\
, and D. Zanin.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jessica Lin (McGill University)
DTSTART:20220308T140000Z
DTEND:20220308T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/58
DESCRIPTION:Title: Asymmetric and Symmetric Cooperative Motion\nby Jess
ica Lin (McGill University) as part of Non-local operators\, probability a
nd singularities\n\n\nAbstract\nWe prove distributional convergence for a
family of random processes on $\\mathbb{Z}$\, which describe a type of ran
dom walk with dependent delay. The model generalizes the "hipster random w
alks" studied by Addario-Berry et al [Probability Theory and Related Field
s\, '20]. We introduce a novel approach which relies on convergence result
s for finite difference schemes of certainly fully nonlinear PDEs. This ta
lk is based on joint work with Louigi Addario-Berry and Erin Beckman.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stéphane Menozzi (Université d'Évry Val d'Essonne)
DTSTART:20220315T140000Z
DTEND:20220315T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/59
DESCRIPTION:Title: Heat kernel of supercritical SDEs with unbounded drifts<
/a>\nby Stéphane Menozzi (Université d'Évry Val d'Essonne) as part of N
on-local operators\, probability and singularities\n\n\nAbstract\nWe consi
der SDEs driven by isotropic α-stable processes\, 0<α<2\, where:\n- the
coefficients are Hölder continuous in space\n- the "diffusion" coefficien
t is bounded and uniformly elliptic\n- the drift can be unbounded\n\nIf β
is the spatial Hölder regularity index of the coefficients\, we obtain u
nder the condition α+β>1 existence of the density for such SDEs. Further
more this density enjoys sharp two-sided estimates and we derive as well s
harp bounds for its logarithmic derivative.\n\nImportantly\, we cover the
whole supercritical range. The proof relies on ad hoc parametrix expansion
s and probabilistic techniques.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giorgio Metafune (Università del Salento)
DTSTART:20220322T140000Z
DTEND:20220322T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/60
DESCRIPTION:Title: A unified approach to degenerate problems in the half-sp
ace\nby Giorgio Metafune (Università del Salento) as part of Non-loca
l operators\, probability and singularities\n\n\nAbstract\nWe study ellipt
ic and parabolic problems governed by the singular elliptic operators\n\n$
\n\\mathcal L =y^{\\alpha_1}\\Delta_{x} +y^{\\alpha_2}\\left(D_{yy}+\\frac
{c}{y}D_y -\\frac{b}{y^2}\\right)\,\\qquad\\alpha_1\, \\alpha_2 \\in\\R\n$
\n\nin the half-space $\\R^{N+1}_+=\\{(x\,y): x \\in \\R^N\, y>0\\}$. This
is a joint paper with\n L. Negro and C. Spina.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Friesen (Dublin City University)
DTSTART:20220405T130000Z
DTEND:20220405T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/61
DESCRIPTION:Title: Continuous affine Volterra processes\nby Martin Frie
sen (Dublin City University) as part of Non-local operators\, probability
and singularities\n\n\nAbstract\nRecent empirical observations on intra-da
y stock market data suggest that volatilities defined as short-term fluctu
ations of asset prices exhibit a highly rough behavior on smaller time sca
les. While such an effect is neither adequately captured nor predicted by
Markovian models\, recent analysis has shown that their rough counterparts
based on the fractional Brownian motion or on Volterra processes perfectl
y capture these effects.\n\n \nIn this talk\, we focus on the particular c
lass of affine Volterra processes being characterized by the feature that
their characteristic function can be expressed in a semi-explicit form in
terms of a solution of a Volterra Riccati equation.\n\nFor this equation\,
we provide a priori growth bounds\, Sobolev regularity in time\, continuo
us dependence on parameters\, and differentiability in the initial state.
Based on these findings\, we derive the existence of limiting distribution
s for a large class of affine Volterra processes. To each of these distrib
utions\, we construct the associated stationary non-Markovian Volterra pro
cess and finally analyze the regularity of its law.\n\nThis is joint work
with P. Jin (BNU-HKBU)\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peixue Wu (University of Illinois at Urbana-Champaign)
DTSTART:20220412T130000Z
DTEND:20220412T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/62
DESCRIPTION:Title: Heat kernel estimates for non-local operators with multi
-singular killing potential.\nby Peixue Wu (University of Illinois at
Urbana-Champaign) as part of Non-local operators\, probability and singula
rities\n\n\nAbstract\nWe study the heat kernel estimates for non-local ope
rators with multi-singular killing potential. To be specific\, given an op
en set $D$ with boundary $\\partial D = \\cup_{k=1}^d \\cup_{j=1}^{m_k} \\
Gamma_{k\,j}$\, where for any $1\\leq k$\, $1 \\leq j \\leq m_k$\, $\\Gamm
a_{k\,j}$ is a $C^{1\,\\beta}$ submanifold without boundary of codimension
$1\\le k\\le d$ and $\\{\\Gamma_{k\,j}\\}_{1\\le k\\le d\, 1\\le j \\le m
_k}$ are disjoint. We show that the heat kernel $p^D(t\,x\,y)$ of the foll
owing non-local operator with multi-singular critical killing potential \n
\n\n$$\n\\big( (\\Delta|_D)^{\\alpha/2} - \\kappa\\big)(f)(x):= p.v. {A}_{
d\,-\\alpha} \\int_D \\frac{f(y)-f(x)}{|y-x|^{d+\\alpha}}dy - \\sum_{k=1}^
d \\sum_{j=1}^{m_k} \\lambda_{k\,j} \\delta_{\\Gamma_{k\,j}}(x)^{-\\alpha}
\,\n$$\n\nwhere $ \\lambda_{k\,j}>0\, \\alpha \\in (0\,2)$ has the followi
ng estimates: for any given $T>0$\, \n\n$$\np^D(t\,x\,y) \\asymp p(t\,x\,y
) \\prod_{k=1}^d \\prod_{j=1}^{m_k} (\\frac{\\delta_{\\Gamma_{k\,j}}(x)}{t
^{1/\\alpha}} \\wedge 1)^{p_{k\,j}}(\\frac{\\delta_{\\Gamma_{k\,j}}(y)}{t^
{1/\\alpha}} \\wedge 1)^{p_{k\,j}}\, \\quad \\forall t\\in (0\,T)\, x\,y\\
in D\,\n$$\n\nwhere $p(t\,x\,y)$ is the heat kernel of the $\\alpha$-stabl
e process on $\\mathbb{R}^d$ and $p_{k\,j}$ and $\\lambda_{k\,j}$ are rela
ted through a strictly increasing function $\\lambda = C(k\,\\alpha\,p)$.
Our method is based on the result established in [Cho et al. Journal de Ma
thématiques Pures et Appliquées 143(2020): 208-256] and a detailed analy
sis of $C^{1\,\\beta}$ manifolds. \n\nThis is joint work with Renming Song
and Shukun Wu: arXiv:2203.03891.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soobin Cho (Seoul National University)
DTSTART:20220510T130000Z
DTEND:20220510T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/63
DESCRIPTION:Title: General law of iterated logarithm for Markov processes\nby Soobin Cho (Seoul National University) as part of Non-local operato
rs\, probability and singularities\n\n\nAbstract\nIn this talk\, we discus
s general criteria and forms of both liminf and limsup laws of\niterated l
ogarithm (LIL) for continuous-time Markov processes. We establish LILs und
er local assumptions near zero (near in finity\, respectively) on uniform
bounds of the first exit time from balls in terms of a function $\\phi$ a
nd uniform bounds on the tails of the jumping measure in terms of a functi
on $\\psi$. One of the main results is that a simple ratio test in terms o
f the functions $\\phi$ and $\\psi$ completely determines whether there ex
ists a positive nondecreasing function $R(t)$ such that $limsup|X_t|/R(t)$
is positive and finite a.s.\, or not. Our results cover a large class of
subordinate dffusions\, jump processes with mixed polynomial local growths
and random conductance models with long range jumps.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Bonforte (University of Madrid)
DTSTART:20220426T130000Z
DTEND:20220426T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/64
DESCRIPTION:Title: Nonlinear and Nonlocal Diffusions. Smoothing effects\, G
reen functions and functional inequalities\nby Matteo Bonforte (Univer
sity of Madrid) as part of Non-local operators\, probability and singulari
ties\n\n\nAbstract\nWe will consider the Cauchy problem for Nonlinear Diff
usion equations of porous medium type $u_t=-\\mathcal{L} u^m$\, with $m>1$
and investigate whether or not integrable data produce bounded solutions.
The diffusion operator belongs to a quite general class of nonlocal opera
tors\, and we will see how different assumption on the operator imply (or
not) smoothing properties. We will briefly compare the approach based on M
oser iteration and the approach through Green functions. On one hand\, we
show that if the linear case ($m=1$) enjoys smoothing properties\, also th
e nonlinear will do. On the other hand\, we see that in some cases the non
linear diffusion enjoys the smoothing properties also when the linear coun
terpart does not\, thanks to the convex nonlinearity.\n\nFollowing Nash' i
deas\, we see how smoothing properties are often equivalent to the validit
y of Gagliardo-Nirenberg-Sobolev (and Nash) inequalities: we explore these
implications also in the nonlinear and nonlocal context and the connectio
n with dual inequalities (Hardy-Littlewood-Sobolev) and Green function est
imates. \n\nThis is a work in progress with J. Endal (UAM\, Madrid).\n\nI
f time allows\, we will complete the panorama by showing related results o
n Euclidean bounded domains (joint works with Figalli\, Ros-Oton\, Sire\,
Vazquez) and/or on Riemannian Manifolds (joint works with Berchio\, Gangul
y\, Grillo\, Muratori)\, together with a small detour on the Fast diffusio
n case $m<1$ (joint work with Ibarrondo and Ispizua).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chiara Spina (University of Salento)
DTSTART:20220531T130000Z
DTEND:20220531T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/65
DESCRIPTION:Title: $L^p$ estimates for a class of degenerate operators\nby Chiara Spina (University of Salento) as part of Non-local operators
\, probability and singularities\n\n\nAbstract\nWe prove $L^p$-estimates f
or the operator\n$$\\mathcal L=\\Delta_x+\\Delta_y +c\\frac{y}{|y|^2}\\
cdot\\nabla_y-\\frac{b}{|y|^{2}}=\\Delta_x+L_y\,$$\nwhere $L_y=\\Delta_y +
c\\frac{y}{|y|^2}\\cdot\\nabla_y-\\frac{b}{|y|^{2}}$. The parameters $b\
,\\ c$ are constant real coefficients subject to the condition $ D:=b+\\le
ft(\\frac{M-2+c}{2}\\right)^2> 0$. \n\nWe work in the space $L^p_c:=L^p(\\
R^{N+M}\, |y|^c\\\, dxdy)$\, motivated by the fact that the weight $|y|^c
$ makes the operator symmetric in $L^2_c$ and we assume $M+c>0$\, so that
the measure $d\\mu=|y|^c\\\, dx\\\, dy$ is locally finite on $\\R^{N+M}$.\
n\n\n\n\nThe operators $\\Delta_x$\, $L_y$ commute and the whole operator
$\\mathcal L$ satisfies the scaling property $I_s^{-1}\\mathcal L I_s=s^2
\\mathcal L$\, if $I_s u(x\,y)=u(sx\,sy)$. It is not difficult to see that
$\\mathcal L$ generates a semigroup in $L^p_c$ if and only if $L_y$ gener
ates in $L^p(\\R^M\, |y|^c\\\, dy)$ and this is equivalent to $(M+c)\\\, \
\left|\\frac{1}{2}-\\frac 1 p\\right|<1+\\sqrt D$.\n\n\nWhen $M=1$ and $b=
0$\, $L_y$ is a Bessel operator and both $\\mathcal L=\\Delta_x+B_y$ and
$D_t-\\mathcal L$ play a major role in the investigation of the fractional
powers $(-\\Delta_x)^s$ and $(D_t-\\Delta_x)^s$\, $s=(1-c)/2$\, through
the ``extension procedure" of Caffarelli and Silvestre [1]. \n\n\nWhen $M
=1$\, that is in the half-space $\\R^{N+1}_+$\, all the results of this p
aper\, and much more\, have been proved in [4] by taking advantage of sop
histicated tools from operator valued harmonic analysis. More general\, n
on symmetrizing weights $|y|^m\\\, dx\\\, dy$ are therein considered and b
oth Dirichlet and Neumann boundary conditions. We refer the reader also to
[2]\, [3] for the case $b=0$ and with variable coefficients. \n\n\nHere
we use a different strategy and show that $L^p$-estimates for the pure $x
$-derivatives\, that is the boundedness of the operators $D_{x_ix_j}\\math
cal L^{-1}$\, follow from sub-solution estimates through an interpolation
theorem in absence of kernels in homogeneous spaces due to Z. Shen. Sub-so
lution estimates\, that is improving of integrability for (sub) solutions
of the homogeneous equation $\\mathcal Lu=0$\, are proved by combining Cac
ciopoli estimates\, weighted Sobolev embeddings and Moser iteration.\n\n[1
] L. Caffarelli\,L. Silvestre: An extension problem related to the fractio
nal Laplacean\,\nComm. Partial Differential Equations\, 32 (2007)\, no. 7-
9 1245-1260.\n\n[2] H. Dong\, T. Phan: On parabolic and elliptic equations
with singular or degenerate coeff-\ncients\, arxiv: 2007.04385 2020\n\n[3
] H. Dong\, T. Phan:Weighted mixed-norm Lp estimates for equations in non-
divergence form\nwith singular coeffcients: the Dirichlet problem arxiv: 2
103.08033 2021\n\n[4] G. Metafune\, L. Negro\, C. Spina: Lp estimates for
the Caffarelli-Silvestre extension\noperators\, Journal of Differential Eq
uations Volume 316\, (2022)\, Pages 290-345.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Błażej Wróbel (University of Wroclaw)
DTSTART:20220621T130000Z
DTEND:20220621T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/66
DESCRIPTION:Title: On Lp estimates for positivity-preserving Riesz transfor
ms related to Schrödinger operators\nby Błażej Wróbel (University
of Wroclaw) as part of Non-local operators\, probability and singularities
\n\n\nAbstract\nWe study $L^p$ boundedness properties of positivity preser
ving Riesz transforms related to a \nSchrödinger operator. Using interpol
ation technique we establish $L^p$ boundedness for general non-negative po
tentials. Then we present a counterexample showing that the $L^{\\infty}$
boundedness may fail. Next we give integral type conditions on the potenti
al $V$ that guarantee the boundedness on the endpoints $p=1$ and $p=\\inft
y$. Our conditions depend only on the global properties of V and are resis
tant to small perturbations. In particular $V$ may have a power growth or
an exponential growth. \n\nThe talk is based on a joint work with Maciej K
ucharski.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jie Xiao (Memorial University)
DTSTART:20220628T130000Z
DTEND:20220628T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/67
DESCRIPTION:Title: Energy formulae for fractional Schrodinger-Poisson syste
m\nby Jie Xiao (Memorial University) as part of Non-local operators\,
probability and singularities\n\n\nAbstract\nThrough a geometric-capacity-
theoretic approach\, this talk presents two new formulas for the fractiona
l energy of a quantum particle arising from the fractional Schrodinger-Po
isson system which models the behavior of a quantum particle within an unk
nown electrostatic field.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xue-Mei Li (EPFL and Imperial College London)
DTSTART:20221115T140000Z
DTEND:20221115T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/70
DESCRIPTION:Title: Hessian estimates of the logarithmic heat kernel\nby
Xue-Mei Li (EPFL and Imperial College London) as part of Non-local operat
ors\, probability and singularities\n\n\nAbstract\nHessian estimates on th
e heat kernel and its logarithmic estimates for important first steps towa
rd understanding\ntoe Brownian bridge measure\, a natural measure on loop
spaces. It also find applications in studying the existence of Lipschitz\n
transport maps. I shall discuss some new techniques in these studies and r
esults. This is joint work with Xin Chen and Bo Wu.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Longjie Xie (Jiangsu Normal University)
DTSTART:20221011T130000Z
DTEND:20221011T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/71
DESCRIPTION:Title: Weak and strong well-posedness of critical and supercrit
ical SDEs with singular coefficients\nby Longjie Xie (Jiangsu Normal U
niversity) as part of Non-local operators\, probability and singularities\
n\n\nAbstract\nConsider the following time-dependent stable-like operator
with drift:\n$$\n\\mathscr{L}_t\\varphi(x)=\\int_{\\mathbb{R}^d}\\big[\\va
rphi(x+z)-\\varphi(x)-z^{(\\alpha)}\\cdot\\nabla\\varphi(x)\\big]\\sigma(t
\,x\,z)\\nu_\\alpha(d z)+b(t\,x)\\cdot\\nabla \\varphi(x)\,\n$$\nwhere $d\
\geq 1$\, $\\nu_\\alpha$ is an $\\alpha$-stable type Lévy measure with $\
\alpha\\in(0\,1]$ and $z^{(\\alpha)}=1_{\\alpha=1}1_{|z|\\leq1}z$\, $\\sig
ma$ is a real-valued Borel function on $\\mathbb{R}_+\\times\\mathbb{R}^d\
\times\\mathbb{R}^d$ and $b$ is an $\\mathbb{R}^d$-valued Borel function\n
on $\\mathbb{R}_+\\times\\mathbb{R}^d$. By using the Littlewood-Paley theo
ry\, we establish the well-posedness for the martingale problem associated
with $\\mathscr{L}_t$ under the sharp balance condition $\\alpha+\\beta\\
geq1$\, where $\\beta$ is the Hölder index of $b$ with respect to $x$.\nM
oreover\, we also study a class of stochastic differential equations drive
n by Markov processes with generators of the form $\\mathscr{L}_t$.\nWe pr
ove the pathwise uniqueness of strong solutions for such equations when th
e coefficients are in certain Besov spaces.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:The Anh Bui (Macquarie University)
DTSTART:20221213T140000Z
DTEND:20221213T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/72
DESCRIPTION:Title: On Sobolev norms involving generalized Hardy operators\nby The Anh Bui (Macquarie University) as part of Non-local operators\,
probability and singularities\n\n\nAbstract\nConsider the operator on $L^
{2}(\\mathbb{R}^d)\, d\\ge 1$\n $$\n \\mathcal L_a = (-\\Delta)^{\\alpha
/2}+a|x|^{-\\alpha} \\quad \\text{with} \\quad 0<\\alpha<\\min\\{2\, d\\}.
\n $$\n Under the condition $a\\ge -\\frac{2^\\alpha\\Gamma((d+\\alpha)/
4)^2}{\\Gamma((d-\\alpha)/4)^2}$ the operator is non negative and self-adj
oint.\n We prove that fractional powers {$\\mathcal{L}^{s/2}_a$}\n for $
s\\in(0\,2]$ satisfy the estimates\n $$\n \\|\\mathcal{L}_{a}^{s/2}f\\
|_{L^{p}}\n \\lesssim\\|(-\\Delta)^{\\alpha s/4}f\\|_{L^{p}}\,\n \
\qquad\n \\|(-\\Delta)^{s/2}f\\|_{L^{p}}\n \\lesssim \\|\\mathcal{
L}_{a}^{\\alpha s/4}f\\|_{L^{p}}\n $$\n for suitable ranges of $p$. Our
result fills the remaining gap\n in earlier results. The method of proof
is based on square function estimates for operators whose heat kernel has
a weak decay. This talk is based on joint work with P. D'Ancona.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabrice Baudoin (University of Connecticut)
DTSTART:20221206T140000Z
DTEND:20221206T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/73
DESCRIPTION:Title: Asymptotic windings of the unitary Brownian motion\n
by Fabrice Baudoin (University of Connecticut) as part of Non-local operat
ors\, probability and singularities\n\n\nAbstract\nWe study several matrix
diffusion processes constructed from a unitary Brownian motion. In partic
ular\, we use the Stiefel fibration to lift the Brownian motion of the com
plex Grassmannian to the complex Stiefel manifold and deduce a skew-produc
t decomposition of the Stiefel Brownian motion. As an application\, we pro
ve asymptotic laws for the determinants of the block entries of the unitar
y Brownian motion. This is a joint work with Jing Wang (Purdue University)
.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergio Polidoro (University of Modena and Reggio Emilia)
DTSTART:20221220T140000Z
DTEND:20221220T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/74
DESCRIPTION:Title: Asymptotic bounds for the relativistic Fokker-Planck ope
rator\nby Sergio Polidoro (University of Modena and Reggio Emilia) as
part of Non-local operators\, probability and singularities\n\n\nAbstract\
nWe consider a class of second order degenerate kinetic operators L in the
framework of special relativity. We first describe L as an Hörmander ope
rator which is invariant with respect to Lorentz transformations. Then we
prove a Lorentz-invariant Harnack type inequality\, and we derive accurate
asymptotic lower bounds for positive solutions to Lf=0. As a consequence
we obtain upper and lower bounds for the density of the relativistic stoc
hastic process associated to L.\n\nThis is a joint work with Francesca Anc
eschi and Annalaura Rebucci.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abdelaziz Rhandi (University of Salerno)
DTSTART:20221108T130000Z
DTEND:20221108T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/75
DESCRIPTION:Title: Bounds for the gradient of the transition kernel for ell
iptic operators with unbounded diffusion\, drift and potential terms\n
by Abdelaziz Rhandi (University of Salerno) as part of Non-local operators
\, probability and singularities\n\n\nAbstract\nWe prove global Sobolev re
gularity and pointwise upper bounds for the gradient of transition densiti
es associated with second order differential operators in R^d with unbound
ed diffusion\, drift and potential terms.\n\nThis is joint work with Marku
s Kunze and Marianna Porfido.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaj Nyström (Uppsala University)
DTSTART:20221025T130000Z
DTEND:20221025T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/76
DESCRIPTION:Title: Parabolic uniform rectifiability and caloric measure I:
$A_\\infty$ implies parabolic uniform rectifiability of a parabolic Lipsch
itz graph\nby Kaj Nyström (Uppsala University) as part of Non-local o
perators\, probability and singularities\n\n\nAbstract\nWe prove that if a
parabolic Lipschitz graph domain has the property that its caloric measur
e is a parabolic $A_\\infty$ weight with respect to surface measure\, then
the function defining the graph has a half-order time derivative in the s
pace of (parabolic) bounded mean oscillation. Equivalently\, we prove that
the $A_\\infty$ property of caloric measure implies that the boundary is
parabolic uniformly rectifiable. Consequently\, by combining our result wi
th the work of Lewis and Murray we resolve\, in the setting of parabolic L
ipschitz graph domains\, a longstanding open problem in the field by provi
ng that the $L^p$ solvability (for some $p > 1$) of the Dirichlet problem
for the heat equation is equivalent to parabolic uniformly rectifiability.
This is joint work with S. Bortz\, S. Hofmann\, and J.M. Martell.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-François Jabir (Higher School of Economics)
DTSTART:20230117T140000Z
DTEND:20230117T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/77
DESCRIPTION:Title: Multidimensional stable-driven McKean-Vlasov SDEs with d
istributional interaction kernel\nby Jean-François Jabir (Higher Scho
ol of Economics) as part of Non-local operators\, probability and singular
ities\n\n\nAbstract\nThis talk will be focused on presenting existence and
uniqueness results\, in a weak and a strong sense\, for McKean-Vlasov mod
els driven by alpha-stable Lévy processes and an interaction kernel lying
in a Besov space with non-positive exponent. In this specific setting\,
we exhibit how\, quantitatively\, the McKean non-linearity\, together with
the noise\, provides a regularisation effect allowing to pass beyond clas
sical - or more recently established - characteristic thresholds ensuring
the wellposedness of linear SDEs with singular derive. Application to phys
ically-based McKean-Vlasov models and the propagation of chaos property of
some related particle approximation will be also discussed. \n\n\n(This w
ork is a joint collaboration with P-E. Chaudru de Raynal\, Laboratoire de
Mathématiques Jean Leray\, Nantes\, and S. Menozzi\, LaMME\, Univ. Evry
Val d’Essonne - Paris Saclay.)\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xicheng Zhang (Beijing Institute of Technology)
DTSTART:20230131T140000Z
DTEND:20230131T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/78
DESCRIPTION:Title: Second order fractional mean-field SDEs with singular
kernels and measure initial data\nby Xicheng Zhang (Beijing Institute
of Technology) as part of Non-local operators\, probability and singularit
ies\n\n\nAbstract\nIn this work we establish the local and global well-pos
edness of weak and strong solutions to second order fractional mean-field
SDEs with singular/distribution interaction kernels and measure initial va
lue\, where the kernel can be\nNewton or Coulomb potential\, Riesz potenti
al\, Biot-Savart law\, etc. Moreover\, we also show the stability\, smooth
ness and the short time singularity and large time decay estimates of the
density.\nOur results reveal a phenomenon that for nonlinear mean-field eq
uations\, the regularity of the initial distribution could balance the sin
gularity of the kernel. The precise relationship between the singularity o
f kernels and the regularity of\ninitial value are calculated\, which belo
ngs to the subcritical regime in scaling sense. In particular\, our result
s provide microscopic probability explanation and establish a unified tre
atment for\nmany physical models such as fractional Vlasov-Poisson-Fokker-
Planck system\, the vorticity formulation of 2D-fractal Navier-Stokes equa
tions\, surface quasi-geostrophic models\, fractional porous medium equati
on with viscosity\, etc.\n(This is a joint work with Zimo Ham and Michael
Rockner.)\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lorenzo Marino (Polish Academy of Sciences)
DTSTART:20230124T140000Z
DTEND:20230124T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/79
DESCRIPTION:Title: Weak regularisation by degenerate Lévy noise\nby Lo
renzo Marino (Polish Academy of Sciences) as part of Non-local operators\,
probability and singularities\n\n\nAbstract\nThe current talk presents so
me new results about the regularisation by noise phenomena for multidimens
ional ODEs\, where the random disturbance stands as a Lévy process (with
suitable properties) and it perturbs the dynamics only on some components.
In particular\, we aim to exhibit the minimal Hölder regularity on the d
eterministic drift ensuring the well-posedness\, in a weak probabilistic s
ense\, of the associated SDE. Due to the noise degeneracy\, an hypoellipti
c-type framework is implemented assuming a weak Hörmander condition on th
e drift. As a by-product of our method of proof\, Krylov-type estimates fo
r the canonical solution process are also established. In conclusion\, we
also show through suitable counter-examples that there exists indeed an (a
lmost) sharp threshold on the Hölder regularity exponents of the drift en
suring the weak well-posedness for the SDE.\n\n\nThe work this talk is bas
ed upon is a joint collaboration with S. Menozzi (LaMME\, Université d’
Evry Val d’Essonne).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Renming Song (University of Illinois Urbana-Champaign)
DTSTART:20230207T140000Z
DTEND:20230207T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/80
DESCRIPTION:Title: Potential theory of Dirichlet forms with jump kernels bl
owing up at the boundary\nby Renming Song (University of Illinois Urba
na-Champaign) as part of Non-local operators\, probability and singulariti
es\n\n\nAbstract\nIn this talk\, I will present some recent results on pot
ential theory of Dirichlet forms on the half-space $\\R^d_+$ defined by th
e jump kernel $J(x\,y)=|x-y|^{-d-\\alpha}\\mathcal{B}(x\,y)$\, where $\\al
pha\\in (0\,2)$ and $\\mathcal{B}(x\,y)$ can blow up to infinity at the bo
undary. The main results include boundary Harnack principle and sharp two-
sided Green function estimates.\n\nThis talk is based on a joint paper wit
h Panki Kim and Zoran Vondracek.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krzysztof Bogdan (Wroclaw University of Science and Technology)
DTSTART:20230314T140000Z
DTEND:20230314T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/81
DESCRIPTION:Title: The fractional Laplacian with reflections\nby Krzysz
tof Bogdan (Wroclaw University of Science and Technology) as part of Non-l
ocal operators\, probability and singularities\n\n\nAbstract\nMotivated by
the notion of isotropic $\\alpha$-stable Lévy process confined\, by ''re
flections''\, to a bounded open Lipschitz set\, we study related analytica
l objects. In particular\, we construct the corresponding transition semig
roup\, and prove the exponential speed of convergence of the semigroup to
a unique stationary distribution over a long time. This is a joint work wi
th Markus Kunze.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Butkovsky (Technische Universität Berlin)
DTSTART:20230404T130000Z
DTEND:20230404T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/82
DESCRIPTION:Title: Stochastic equations with singular drift driven by fract
ional Brownian motion\nby Oleg Butkovsky (Technische Universität Berl
in) as part of Non-local operators\, probability and singularities\n\n\nAb
stract\nJoint work with Khoa Le and Leonid Mytnik [1]. We consider stochas
tic differential equation\n$$\nd X_t=b(X_t) dt +d W_t^H\,\n$$\nwhere the d
rift $b$ is either a measure or an integrable function\, and $W^H$ is a $d
$-dimensional fractional Brownian motion with Hurst parameter $H\\in(0\,1)
$\, $d\\in\n$. For the case where $b\\in L_p(\\R^d)$\, $p\\in[1\,\\infty]$
we show weak existence of solutions to this equation under the conditio
n\n$$\n\\frac{d}p<\\frac1H-1\,\n$$\nwhich is an extension of the Krylov-Ro
ckner condition (2005) to the fractional case. We construct a counter-exa
mple showing optimality of this condition. If $b$ is a Radon measure\, par
ticularly the delta measure\, we prove weak existence of solutions to this
equation under the optimal condition $H<\\frac1{d+1}$. We also show str
ong well-posedness of solutions to this equation under certain conditions.
\nTo establish these results\, we utilize the stochastic sewing technique
and develop a new version of the stochastic sewing lemma.\n\n[1] Butkovsky
\, O.\, Lê\, K.\, & Mytnik\, L. (2023). Stochastic equations with singula
r drift driven by fractional Brownian motion. arXiv preprint arXiv:2302.11
937.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincenzo Ambrosio (Università Politecnica delle Marche)
DTSTART:20230627T130000Z
DTEND:20230627T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/83
DESCRIPTION:Title: The nonlinear fractional relativistic Schrödinger equat
ion\nby Vincenzo Ambrosio (Università Politecnica delle Marche) as pa
rt of Non-local operators\, probability and singularities\n\n\nAbstract\nW
e consider the following class of fractional relativistic Schrödinger equ
ations:\n\\[\n\\begin{cases}\n(-\\Delta+m^{2})^{s}u + V(\\varepsilon x) u=
f(u)\,\\quad {\\rm in} \\quad \\mathbb{R}^{N}\,\\\\\nu \\in H^{s}(\\mathb
b{R}^{N})\, \\quad u>0 \\quad {\\rm in} \\quad \\mathbb{R}^{N}\,\n\\end{ca
ses}\n\\]\nwhere $\\varepsilon>0$ is a small parameter\, $s\\in (0\, 1)$\,
$m>0$\, $N> 2s$\, $(-\\Delta+m^{2})^{s}$ is the fractional relativistic S
chrödinger operator\, $V:\\mathbb{R}^{N}\\rightarrow \\mathbb{R}$ is a c
ontinuous potential satisfying a local condition\, and $f:\\mathbb{R}\\rig
htarrow \\mathbb{R}$ is a continuous subcritical nonlinearity. We first sh
ow that\, for $\\varepsilon>0$ small enough\, the above problem has a weak
solution $u_{\\varepsilon}$ (with exponential decay at infinity) which co
ncentrates around a local minimum point of $V$ as $\\varepsilon\\rightarro
w 0$. We also relate the number of positive solutions with the topology of
the set where the potential $V$ attains its minimum value.\n\nThe main re
sults will be established by using a penalization technique\, the generali
zed Nehari manifold method and Ljusternik-Schnirelman theory.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marvin Weidner (Universitat de Barcelona)
DTSTART:20230516T130000Z
DTEND:20230516T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/84
DESCRIPTION:Title: The nonlocal Bernstein technique and the nonlocal obstac
le problem\nby Marvin Weidner (Universitat de Barcelona) as part of No
n-local operators\, probability and singularities\n\n\nAbstract\nThe Berns
tein technique is an elementary but powerful tool in the regularity theory
for elliptic and parabolic equations. It is based on the insight that\, i
f derivatives of a solution are also subsolutions to an equation\, then th
e maximum principle can be used in order to obtain regularity estimates fo
r these solutions.\nIn the first part of this talk\, we explain how the Be
rnstein technique can be extended to a large class of integro-differential
equations driven by nonlocal operators that are comparable to the fractio
nal Laplacian. In the second part\, we discuss several applications of thi
s technique to the regularity theory for the nonlocal obstacle problem in
a bounded domain.\nThis talk is based on a joint work with Xavier Ros-Oton
and Damià Torres-Latorre.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikolay Krylov (University of Minnesota)
DTSTART:20230523T140000Z
DTEND:20230523T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/85
DESCRIPTION:Title: On weak solutions of time inhomogeneous Ito's equations
with VMO diffusion and Morrey drift\nby Nikolay Krylov (University of
Minnesota) as part of Non-local operators\, probability and singularities\
n\n\nAbstract\nWe prove the existence and weak uniqueness of weak solution
s of Ito's stochastic time dependent equations with irregular diffusion an
d drift terms of Morrey class with mixed norms.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Kulik (Wrocław University of Science and Technology)
DTSTART:20230606T130000Z
DTEND:20230606T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/86
DESCRIPTION:Title: Drift reduction and parametrix construction for stochast
ic differential equations driven by cylindrical Lévy noises.\nby Alex
ei Kulik (Wrocław University of Science and Technology) as part of Non-lo
cal operators\, probability and singularities\n\n\nAbstract\nIn the talk\,
we will present an analytic construction for the unique weak solution \no
f an SDE driven by a cylindrical Lévy noise which are spatially inhomogen
eous in the sense that different coordinates of the driving Lévy process
may have different scaling properties. We will discuss how the classical
parametrix method for constructing fundamental solutions to parabolic equ
ations should be adapted in order to handle numerous difficulties which a
rise in this non-local setting\, including essential singularity\, lack of
scaling\, and presence of the drift (gradient) term which may be not orde
r-dominated by the noise.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guohuan Zhao (Chinese Academy of Sciences)
DTSTART:20230711T130000Z
DTEND:20230711T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/87
DESCRIPTION:Title: Lévy-type operators with low singularity kernels: regul
arity estimates and martingale problem\nby Guohuan Zhao (Chinese Acade
my of Sciences) as part of Non-local operators\, probability and singulari
ties\n\n\nAbstract\nThe main focus of this talk is on the linear non-local
operator $L$ defined by\n$$\nL u (x) = \\int_{\\mathbb{R}^d} (u(x+z)-u(x)
) a(x\,z)J(z)~d z.\n$$\nHere $J$ is the jumping kernel of a L\\'evy proces
s\, which exhibits only a low-order singularity near the origin and does n
ot permit standard scaling. To analyze elliptic equations associated with
$L$\, I will introduce generalized Orlicz-Besov spaces that are specifical
ly tailored for this purpose. Moreover\, I will establish certain regulari
ty properties of the solutions to such equations in these spaces. Addition
ally\, I intend to introduce the martingale problem associated with $L$. B
y exploiting analytic results\, we demonstrate the well-posedness of the m
artingale problem under mild conditions\, and establish a new Krylov-type
estimate for the corresponding Markov processes. This is based on joint wo
rk with Eryan Hu from Tianjin University.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Phan Thành Nam (Ludwig-Maximilians-Universität München)
DTSTART:20231024T130000Z
DTEND:20231024T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/88
DESCRIPTION:Title: Cwikel-Lieb-Rozenblum type estimates for Schrödinger op
erators with Hardy potential\nby Phan Thành Nam (Ludwig-Maximilians-U
niversität München) as part of Non-local operators\, probability and sin
gularities\n\n\nAbstract\nThe celebrated Cwikel-Lieb-Rozenblum (CLR) inequ
ality states that the number of negative eigenvalues of the Schrödinger o
perator $-\\Delta-V(x)$ in $L^2(R^d)$ is bounded from above by the integra
tion of |V|^{d/2}. Up to a universal constant factor\, this bound is optim
al for a wide range of fermionic systems\, from one-body systems where it
is equivalent to the standard Sobolev inequality\, to large systems where
it is consistent with Weyl's semiclassical approximation. I will discuss e
xtensions of the CLR bound when $V(x)$ may be as singular as the Hardy pot
ential $(d/2-1)^2 |x|^{-2}$. The critical singularity requires a logarithm
ic correction which has been noticed for the one-body case but seems unkno
wn for the general case. The talk is based on joint work with Giao Ky Duon
g\, Thi Minh Thao Le\, and Phuoc Tai Nguyen.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nguyen H. Lam (Memorial University)
DTSTART:20231121T140000Z
DTEND:20231121T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/89
DESCRIPTION:Title: Hardy-Rellich type inequalities: A new approach and symm
etrization principle\nby Nguyen H. Lam (Memorial University) as part o
f Non-local operators\, probability and singularities\n\n\nAbstract\nWe pr
esent a new way to use the notion of Bessel pair to establish the optimal
Hardy-Rellich type inequalities. We also talk about necessary and sufficie
nt conditions on the weights for the Hardy-Rellich inequalities to hold. S
ymmetry properties of the Rellich type and Hardy-Rellich type inequalities
will also be discussed. The talk is based on joint work with Anh Do\, Guo
zhen Lu\, and Lu Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Russo (ENSTA Paris)
DTSTART:20231205T140000Z
DTEND:20231205T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/90
DESCRIPTION:Title: Weak Dirichlet processes with jumps and applications
\nby Francesco Russo (ENSTA Paris) as part of Non-local operators\, probab
ility and singularities\n\n\nAbstract\nIn this talk we will revisit the no
tion of weak Dirichlet process\nwhich is the natural extension of semimart
ingale with jumps.\nIf $X$ is such a process\, then it is the sum of a loc
al martingale $M$ and a\nmartingale ortogonal process $A$ in the sense\nth
at $[A\,N] = 0$ for every continuous local martingale $N$.\nWe remark that
if $[A] = 0$ then $X$ is a Dirichlet process.\nThe notion of Dirichlet pr
ocess is not very suitable in the\njump case since in this case $A$ is for
ced to be continuous.\n\nThe talk will discuss the following points.\n\n-
To provide a (unique) decomposition which\n is also significant for semim
artingales with jumps.\n\n- To introduce the notion of characteristics\n
(similarly to the case of semimartingales)\n in equivalence with some It\
\^o type chain rules.\n\n- To discuss various examples of such processes\n
arising from path-dependent martingale problems.\n This includes path-d
ependent stochastic differential equations\n with involving a distributio
nal drift and with jumps. \n\n\nThe talk is based on a joint paper with E
. Bandini (Bologna).\n\nhttp://uma.ensta-paristech.fr/$\\sim$russo\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elena Issoglio (University of Torino)
DTSTART:20231003T130000Z
DTEND:20231003T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/91
DESCRIPTION:Title: McKean SDEs with singular coefficients\nby Elena Iss
oglio (University of Torino) as part of Non-local operators\, probability
and singularities\n\n\nAbstract\nIn this talk we consider a class of SDEs
with drift depending on the law density of the solution\, known as McKean
SDEs. The novelty here is that the drift is singular in the sense that it
is `multiplied' by a generalised function (element of a negative fractiona
l Sobolev space). Those equations are interpreted in the sense of a suitab
le singular martingale problem\, thus a key tool is the study of the corre
sponding singular Fokker-Planck equation. We define the notion of solution
to the singular McKean equation and show its existence and uniqueness. Th
is is based on a joint work with F. Russo (ENSTA).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhen-Qing Chen (University of Washington)
DTSTART:20231114T140000Z
DTEND:20231114T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/92
DESCRIPTION:Title: Boundary Harnack principle for non-local operators\n
by Zhen-Qing Chen (University of Washington) as part of Non-local operator
s\, probability and singularities\n\n\nAbstract\nThe classical boundary Ha
rnack principle asserts that two positive harmonic\nfunctions that vanish
on a portion of the boundary of a smooth domain decay\nat the same rate. I
t is well known that scale invariant boundary Harnack\ninequality holds fo
r Laplacian \\Delta on uniform domains and holds for\nfractional Laplacian
s \\Delta^s on any open sets. It has been an open\nproblem whether the sca
le-invariant boundary Harnack inequality holds on\nbounded Lipschitz domai
ns for Levy processes with Gaussian components such\nas the independent su
m of a Brownian motion and an isotropic stable process\n(which corresponds
to \\Delta + \\Delta^s).\n \nIn this talk\, I will present a necessary an
d sufficient\ncondition for the scale-invariant boundary Harnack inequalit
y to hold for a\nclass of non-local operators on metric measure spaces thr
ough a\nprobabilistic consideration. This result will then be applied to g
ive a\nsufficient geometric condition for the scale-invariant boundary Har
nack\ninequality to hold for subordinate Brownian motions having Gaussian
components \non bounded Lipschitz domains in Euclidean spaces. This condit
ion is almost optimal and \na counterexample will be given showing that t
he scale-invariant BHP may fail\non some bounded Lipschitz domains with la
rge Lipschitz constants.\n\nBased on joint work with Jieming Wang.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Franco Flandoli (Scuola Normale Superiore di Pisa)
DTSTART:20240305T140000Z
DTEND:20240305T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/93
DESCRIPTION:Title: Stochasticity into fluids\nby Franco Flandoli (Scuol
a Normale Superiore di Pisa) as part of Non-local operators\, probability
and singularities\n\n\nAbstract\nFrom the beginning of the advent of Stoch
astic Partial Differential Equations (SPDEs)\, classes of equations relate
d to fluid dynamics were considered. Even earlier\, Landau and Lischitz wr
ote a Navier-Stokes equation perturbed by additive noise\, in their volume
on fluid mechanics. However\, deciding the form of stochasticity realisti
c or interesting for fluid dynamics remains one of the most important ques
tions still debated and stochastic analysis helps a lot to address this pr
oblem. I will try to review part of the present understanding of this mode
ling issue and its consequences.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:José Luis Pérez Garmendia (CIMAT)
DTSTART:20231219T140000Z
DTEND:20231219T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/94
DESCRIPTION:Title: Multi-armed Lévy bandits with periodic decision opportu
nities\nby José Luis Pérez Garmendia (CIMAT) as part of Non-local op
erators\, probability and singularities\n\n\nAbstract\nWe consider a versi
on of the continuous-time multi-armed bandit problem where decision\noppor
tunities arrive at Poisson arrival times and study its Gittins index polic
y. When driven by a Lévy process\, we will show that the Gittins index ca
n be expressed in terms of a Wiener-Hopf factorization of the Lévy proces
s observed at the arrival times of an independent Poisson process.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vassili Kolokoltsov (Moscow State University\, Higher School of Ec
onomics and the University of Warwick)
DTSTART:20240123T140000Z
DTEND:20240123T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/95
DESCRIPTION:Title: Convergence rates for functional central limit theorems
with stable laws and domains of quasi-attraction\nby Vassili Kolokolts
ov (Moscow State University\, Higher School of Economics and the Universit
y of Warwick) as part of Non-local operators\, probability and singulariti
es\n\n\nAbstract\nThe talk will be devoted to the three new directions of
research:\n\n1) Rates of convergence in the functional CLT with stable lim
its\; 2) Domains of quasi-attraction\n\nas distributions\, whose normalise
d sums of $n$ i.i.d terms approach stable laws for large\,\n\nbut not too
large $n$ (full quantitative and qualitative description of this effect in
a functional setting)\;\n\n3) Rates of convergence of CTRWs (continuous t
ime\n\nrandom walks) to fractional evolutions. The ideas of the talk are t
aken from the recent author's papers\n\n(1) The Rates of Convergence for F
unctional Limit Theorems with\n\nStable Subordinators and for CTRW Approxi
mations to\n\nFractional Evolutions. Fractal Fract. (2023)\, 7\, 335.\n\nh
ttps://doi.org/10.3390/fractalfract7040335\n\n(2) Domains of Quasi Attract
ion: Why Stable Processes Are Observed in Reality?\n\nFractal Fract. (2023
)\, 7\, 752.\n\nhttps://doi.org/10.3390/fractalfract7100752\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krzysztof Burdzy (University of Washington)
DTSTART:20240416T150000Z
DTEND:20240416T160000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/96
DESCRIPTION:Title: Simple nonlinear PDEs inspired by billiards\nby Krzy
sztof Burdzy (University of Washington) as part of Non-local operators\, p
robability and singularities\n\n\nAbstract\nHow many times can $n$ billiar
d balls collide\nin the open $d$-dimensional space? I will provide some\ne
stimates. I will explain how the above question leads\nto a ``pinned billi
ard balls'' model. On a large scale\,\nthe model seems to have a hydrodyna
mic limit.\nThe parameters of the conjectured limit should satisfy\nsimple
nonlinear PDEs. While the existence and properties\nof the conjectured hy
drodynamic limit are open questions\,\nI will provide a quite complete ana
lysis of the conjectured PDEs.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alberto Saldaña (National Autonomous University of Mexico)
DTSTART:20240319T140000Z
DTEND:20240319T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/97
DESCRIPTION:Title: The logarithmic Laplacian: a new tool in the analysis of
PDEs with fractional diffusions\nby Alberto Saldaña (National Autono
mous University of Mexico) as part of Non-local operators\, probability an
d singularities\n\n\nAbstract\nIn this talk\, I will give an introduction
to the logarithmic Laplacian\, which is a relatively new pseudodifferentia
l operator that has shown to be a powerful tool in the study of linear and
nonlinear fractional PDEs. I will describe some of its main properties\,
applications\, and some recent regularity results.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikita Simonov (Sorbonne Université)
DTSTART:20240409T130000Z
DTEND:20240409T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/98
DESCRIPTION:Title: Fast diffusion equations\, tails and convergence rates\nby Nikita Simonov (Sorbonne Université) as part of Non-local operator
s\, probability and singularities\n\n\nAbstract\nUnderstanding the interme
diate asymptotic and computing convergence rates towards equilibria are am
ong the major problems in the study of parabolic equations. Convergence ra
tes depend on the tail behaviour of solutions. This observation raised the
following question: how can we understand the tail behaviour of solutions
from the tail behaviour of the initial datum?\n\nIn this talk\, I will di
scuss the asymptotic behaviour of solutions to the fast diffusion equation
. It is well known that non-negative solutions behave for large times as t
he Barenblatt (or fundamental) solution\, which has an explicit expression
. In this setting\, I will introduce the Global Harnack Principle (GHP)\,
precise global pointwise upper and lower estimates of non-negative solutio
ns in terms of the Barenblatt profile. I will characterize the maximal (he
nce optimal) class of initial data such that the GHP holds by means of an
integral tail condition. As a consequence\, I will provide rates of conver
gence towards the Barenblatt profile in entropy and in stronger norms such
as the uniform relative error.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:René Schilling (Technische Universität Dresden)
DTSTART:20240507T130000Z
DTEND:20240507T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/99
DESCRIPTION:Title: The Liouville Theorem for Lévy Generators (and beyond)
and the Unique Continuation Property\nby René Schilling (Technische U
niversität Dresden) as part of Non-local operators\, probability and sing
ularities\n\n\nAbstract\nWe discuss necessary and sufficient criteria for
certain Fourier\nmultiplication operators to satisfy the Liouville propert
y (bounded\nharmonic functions are a.s.\\ constant) and the local continua
tion\nproperty (bounded functions\, that are harmonic and identically zero
on a\ndomain\, are a.s. zero on the whole space). Since the operators gen
erate\nstochastic processes\, there is also a probabilistic interpretation
of\nthese findings.\n\nThis is a joint work with David Berger (TU Dresden
)\,\nEugene Shargorodsky (King's College\, London) and Teo Sharia (Royal\n
Holloway\, London).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zimo Hao (Universität Bielefeld)
DTSTART:20240312T140000Z
DTEND:20240312T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/100
DESCRIPTION:Title: SDEs with supercritical distributional drifts\nby Z
imo Hao (Universität Bielefeld) as part of Non-local operators\, probabil
ity and singularities\n\n\nAbstract\nLet $d\\geq 2$. In this talk\, we in
vestigate the following stochastic\ndifferential equation (SDE) in ${\\ma
thbb R}^d$ driven by Brownian motion\n$$\n{\\rm d} X_t=b(t\,X_t){\\rm d} t
+\\sqrt{2}{\\rm d} W_t\,\n$$\nwhere $b$ belongs to the space ${\\mathbb L}
_T^q \\mathbf{H}_p^\\alpha$\nwith $\\alpha \\in [-1\, 0]$ and $p\,q\\in[2\
, \\infty]$\, which is a\ndistribution-valued and divergence-free vector f
ield.\nIn the subcritical case $\\frac dp+\\frac 2q<1+\\alpha$\, we establ
ish the\nexistence and uniqueness of a weak solution to the integral equat
ion:\n$$\nX_t=X_0+\\lim_{n\\to\\infty}\\int^t_0b_n(s\,X_s){\\rm d} s+\\sqr
t{2} W_t.\n$$\nHere\, $b_n:=b*\\phi_n$ represents the mollifying approxima
tion\, and the\nlimit is taken in the $L^2$-sense.\nIn the critical and su
percritical case $1+\\alpha\\leq\\frac dp+\\frac\n2q<2+\\alpha$\, assuming
the initial distribution has an $L^2$-density\, we\nshow the existence of
weak solutions and associated Markov processes.\nMoreover\, under the add
itional assumption that $b=b_1+b_2+\\div a$\, where\n$b_1\\in {\\mathbb L}
^\\infty_T{\\mathbf B}^{-1}_{\\infty\,2}$\, $b_2\\in\n{\\mathbb L}^2_TL^2$
\,\nand $a$ is a bounded antisymmetric matrix-valued function\, we establi
sh\nthe convergence of mollifying approximation solutions without the need
\nto subtract a subsequence.\nTo illustrate our results\, we provide examp
les of Gaussian random fields\nand singular interacting particle systems\,
including the two-dimensional\nvortex models.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Milica Tomasevic (École polytechnique)
DTSTART:20240528T130000Z
DTEND:20240528T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/101
DESCRIPTION:Title: Particle approximation of the doubly parabolic Keller-S
egel equation in the plane\nby Milica Tomasevic (École polytechnique)
as part of Non-local operators\, probability and singularities\n\n\nAbstr
act\nIn this talk\, we study a stochastic system of $N$ particles associat
ed with the parabolic-parabolic Keller-Segel system in the plane. This par
ticle system is singular and non Markovian in that its drift term depends
on the past of the particles. When the sensitivity parameter is sufficient
ly small\, we show that this particle system indeed exists for any $N \\ge
q 2$\, we show tightness in $N$ of its empirical measure\, and that any we
ak limit point of this empirical measure\, as $N\\to \\infty$\, solves som
e nonlinear martingale problem\, which in particular implies that its fami
ly of time-marginals solves the parabolic-parabolic Keller-Segel system in
some weak sense. The main argument of the proof consists of a Markovianiz
ation of the interaction kernel: We show that\, in some loose sense\, the
two-by-two path-dependant interaction can be controlled by a two-by-two Co
ulomb interaction\, as in the parabolic-elliptic case. This is a joint wor
k with N. Fournier (Sorbonne Université).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Huyuan Chen (Jiangxi Normal University)
DTSTART:20240521T130000Z
DTEND:20240521T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/102
DESCRIPTION:Title: The Cauchy problem associated to the logarithmic Laplac
ian\nby Huyuan Chen (Jiangxi Normal University) as part of Non-local o
perators\, probability and singularities\n\n\nAbstract\nIn this talk\, we
study the Cauchy problem \n$$\\partial_tu+ \\mathcal{L}_{\\Delta} u=0 \\
\\ {\\rm in}\\ \\\, (0\,\\frac N2) \\times \\R^N\,\\quad\\quad u(0\,\\c
dot)=0\\ \\ {\\rm in}\\ \\\, \\R^N\\setminus \\{0\\}.$$\nwhere $L_\\Delta
$ is the logarithmic Laplacian operator\, a singular integral operator wi
th symbol $2\\log |\\zeta|$. We apply our results to give a classification
of the solutions of\n$$\n\\begin{cases}\n \\partial_t u+\\mathcal{L}_{\\D
elta} u=0 \\quad \\ &{\\rm in}\\ \\ (0\,T)\\times \\R^N \\\\\n \\phanto
m{ \\ \\\, }\n\\displaystyle u(0\,\\cdot)=f\\quad \\ &{\\rm{in}}\\ \
\ \\R^N\n\\end{cases}\n$$\nand obtain an expression of the fundamental
solution of the associated stationary equation in $\\R^N$\,\nand of the fu
ndamental solution in a bounded domain\, i.e.\n$$\\mathcal{L}_{\\Delta} u
=k\\delta_0\\quad {\\rm in}\\ \\ \\mathcal{D}'(\\Omega)\\quad {\\rm such\
\ that \n }\\quad u=0\\quad {\\rm in}\\ \\ \\R^N\\setminus\\Omega. $$\n\nT
his is a joint work with Laurent Véron.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivan Biočić (University of Zagreb)
DTSTART:20240618T130000Z
DTEND:20240618T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/103
DESCRIPTION:Title: Semilinear equations for subordinate spectral Laplacian
: moderate and large solutions\nby Ivan Biočić (University of Zagreb
) as part of Non-local operators\, probability and singularities\n\n\nAbst
ract\nIn this talk\, we solve semilinear problems in bounded $C^{1\,1}$ do
mains for non-local operators with a non-homogeneous Dirichlet boundary co
ndition\, based on the work [1] and a joint work in progress with Vanja Wa
gner. The operators cover and extend the case of the spectral fractional L
aplacian\, and are modelled using the process called subordinate killed Br
ownian motion. Our focus will be on the potential-probabilistic approach t
o these problems with an emphasis on methods\, intuition\, and calculation
s. This approach is a consequence of recent developments in [2\,3].\n\nWe
present an integral representation of harmonic functions for such non-loca
l operators and give sharp boundary behaviour of Green and Poisson potenti
als. H\\"older regularity of distributional solutions is given as well as
a version of Kato's inequality. We explore moderate (i.e. harmonically bou
nded) solutions and large (i.e. harmonically unbounded) solutions to the s
emilinear problem. Large solutions are obtained by using a Keller-Osserman
-type condition\, by an approximation method.\n\n\n[1] I. Biočić\, Semil
inear Dirichlet problem for subordinate spectral Laplacian\, Communication
s on Pure and Applied Analysis\, 22 (2023)\, 851-898.\n\n[2] I. Biočić\
, Z. Vondraček\, V. Wagner\, Semilinear equations for non-local operators
: Beyond the fractional Laplacian\, Nonlinear Analysis\, 207 (2021)\, 1123
03.\n\n[3] P. Kim\, R. Song\, Z. Vondraček\, Potential theory of subordin
ate killed Brownian motion\, Transactions of the American mathematical soc
iety\, 371 (2019)\, 3917-3969.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleksandar Mijatović (University of Warwick)
DTSTART:20240702T130000Z
DTEND:20240702T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/105
DESCRIPTION:Title: Subexponential lower bounds for $f$-ergodic Markov proc
esses\nby Aleksandar Mijatović (University of Warwick) as part of Non
-local operators\, probability and singularities\n\n\nAbstract\nIn this ta
lk I will describe a criterion for establishing lower bounds on the rate
of convergence in $f$-variation of a continuous-time ergodic Markov proces
s to its invariant measure. The criterion consists of novel super- and sub
martingale conditions for certain functionals of the Markov process. It pr
ovides a general approach for proving lower bounds on the tails of the inv
ariant measure and the rate of convergence in $f$-variation of a Markov pr
ocess\, analogous to the widely used Lyapunov drift conditions for upper
bounds. Our key innovation\, which will be discussed in the talk\, produce
s lower bounds on the tails of the heights and durations of the excursions
from bounded sets of a continuous-time Markov process using path-wise arg
uments. \n\nI will present applications of our theory to elliptic diffusi
ons and Levy-driven stochastic differential equations with known polynomi
al/stretched exponential upper bounds on their rates of convergence. Our l
ower bounds match asymptotically the known upper bounds for these classes
of models\, thus establishing their rate of convergence to stationarity. T
he generality of our approach suggests that\, analogous to the Lyapunov dr
ift conditions for upper bounds\, our methods can be expected to find app
lications in many other settings. This is joint work with Miha Brešar at
Warwick. The paper is available on https://arxiv.org/abs/2403.14826\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergio Andraus (Tsukuba Gakuin University)
DTSTART:20240924T130000Z
DTEND:20240924T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/106
DESCRIPTION:Title: Collision times of multivariate Bessel processes with t
heir Weyl chambers' boundaries and their Hausdorff dimension\nby Sergi
o Andraus (Tsukuba Gakuin University) as part of Non-local operators\, pro
bability and singularities\n\n\nAbstract\nIn this talk\, I consider multiv
ariate Bessel processes\, which are\nmultivariate generalizations of the w
ell-known Bessel processes and\nwhich depend on a choice of root system. T
hese processes are confined\nto a subset of N-dimensional space\, the Weyl
chamber\, which depends on\nthis root system\, and they are an active top
ic of research in\nmathematical physics as well as probability theory. The
ir confinement\nto the Weyl chamber is a consequence of repulsive drifts t
hat drive\nthe processes away from the boundaries\, and the strength of th
ese\ndrifts depends on a set of parameters\, called multiplicities.\n\nGiv
en the root system $R$\, the multiplicities $k(\\alpha)>0\,\\ \\alpha\\in\
nR$\, and a standard\, $N$-dimensional Brownian motion\n$\\{B(t)\\}_{t\\ge
q0}$\, the multivariate Bessel process' evolution\,\n$\\{X(t)\\}_{t\\geq0}
$\, is given by the SDE\n\\[\n\\textrm{d}X(t)=\\textrm{d}B(t)+\\sum_{\\alp
ha\\in\nR}\\frac{k(\\alpha)}2\\frac{\\alpha}{\\langle\\alpha\,X(t)\\rangle
}\\\,\\textrm{d}\nt\,\\ X(0)=x_0.\n\\]\nDemni showed that in spite of the
singular drift\, $X(t)$ hits the Weyl\nchamber's boundary in finite time a
lmost surely whenever a\nmultiplicity is less than $1/2$. The main objecti
ve of the talk is to\nshow that the set of hitting times at the Weyl chamb
er's boundary has\na fractal structure given by the following result: the
Hausdorff\ndimension of collision times with the boundary is given by\n\\[
\n\\dim[X^{-1}(\\partial W)]=\\max\\Big\\{0\,\\frac12-\\min_{\\alpha\\in R
}k(\\alpha)\\Big\\}.\n\\]\n\nI start by giving a quick overview of one par
ticular case\, namely\nDyson's Brownian motion\, for which one can perform
a direct\ncalculation of the Hausdorff dimension based on the transition
density\nof the process and an asymptotic formula by Graczyk and Sawyer. T
hen\,\nI move on to the general case where the root system is reduced and\
ngive an outline of the proof of our result based on an observation by\nJ.
Małecki: a particular polynomial of the process given by the\nalternatin
g polynomial defined by the root system has an SDE which is\nnothing but a
time-changed squared Bessel process SDE.\n\nThis is joint work with N. Hu
fnagel at Düsseldorf\, and the paper can\nbe found here: https://arxiv.or
g/abs/2312.05420\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomasz Komorowski (IMPAN - Institute of Mathematics Polish Academy
of Sciences)
DTSTART:20241029T140000Z
DTEND:20241029T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/107
DESCRIPTION:Title: Diffusive and superdiffusive limits for a kinetic equat
ion with a transmitting-reflecting-absorbing boundary condition\nby T
omasz Komorowski (IMPAN - Institute of Mathematics Polish Academy of Scien
ces) as part of Non-local operators\, probability and singularities\n\n\nA
bstract\nWe consider the limit of a linear kinetic equation with a degen
erate scattering kernel and a reflection-transmission-absorption condition
at an interface. An equation of this type arises from the kinetic limit
of a microscopic harmonic chain of oscillators whose dynamics is perturbed
by a stochastic term\, conserving energy and momentum. The chain is in
contact\, via one oscillator\, with a heat bath\, which\, in the limit\, g
enerates the boundary condition at the interface. \n\nIt is known that
in the absence of the interface\, the solution of the kinetic equation ex
hibits either\n\nsuperdiffusive\, or diffusive behavior\, in the proper lo
ng time - large scale limit\, depending on the dispersion relation of the
harmonic chain. We discuss how the presence of the interface influences t
he boundary condition for the limiting diffusion\, or anomalous diffusion.
\n\nThe presented results have been obtained in collaboration with G. Bas
ile (Univ. Roma I)\, \n\nA. Bobrowski (Lublin Univ. of Techn.)\, K. Bogdan
(Wrocław Univ. of Sci. and Techn.)\, L. Arino (Ensta\, Paris)\, S. Olla
(Univ. Paris-Dauphine and GSSI\, L’Aquila)\, L. Ryzhik (Stanford Univ.)\
, H. Spohn (TU\, Munich).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aníbal Rodríguez Bernal (Universidad Complutense de Madrid)
DTSTART:20241008T130000Z
DTEND:20241008T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/108
DESCRIPTION:Title: Homogeneous spaces\, operators and semigroups: optimal
estimates and selfsimilarity\nby Aníbal Rodríguez Bernal (Universida
d Complutense de Madrid) as part of Non-local operators\, probability and
singularities\n\n\nAbstract\nWe present some results on evolution problems
governed by homogeneous\noperators. We show that many relevant features o
f these problems do\nnot really depend from specifics of the equations but
stem from\nhomogeneity. These include sharp estimates for solutions in\nh
omogeneous spaces and several selfsimilarity results. These\nproperties ho
ld in particular for higer order equations and fractional\ndiffusion ones.
\n\nThis is a joint work with J. Cholewa (U. Silesia)\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Bobrowski (Lublin University of Technology)
DTSTART:20241112T140000Z
DTEND:20241112T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/109
DESCRIPTION:Title: A couple of recent approximations of skew Brownian moti
on\nby Adam Bobrowski (Lublin University of Technology) as part of Non
-local operators\, probability and singularities\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katarzyna Pietruska-Pałuba (University of Warsaw)
DTSTART:20241203T140000Z
DTEND:20241203T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/110
DESCRIPTION:Title: Functional identities related to nonlocal Levy process
es: Douglas\, Hardy-Stein\nby Katarzyna Pietruska-Pałuba (University
of Warsaw) as part of Non-local operators\, probability and singularities\
n\n\nAbstract\nWe will present developments concerned with a class of\nfun
ctional inequalities related to nonlocal Levy processes and their\ngenerat
ors. The identities considered are of Douglas and Hardy-Stein type.\nThey
have been obtained in collaboration with Krzysztof Bogdan\, Tomasz\nGrzywn
y\, Michal Gutowski and Artur Rutkowski.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoan Tardy (École polytechnique)
DTSTART:20250107T140000Z
DTEND:20250107T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/111
DESCRIPTION:Title: Collisions of the supercritical Keller-Segel particle s
ystem\nby Yoan Tardy (École polytechnique) as part of Non-local opera
tors\, probability and singularities\n\n\nAbstract\nWe study a particle sy
stem naturally associated to the 2-dimensional Keller-Segel equation. It c
onsists of N Brownian particles in the plane\, interacting through a binar
y attraction in θ/(Nr)\, where r stands for the distance between two part
icles. When the intensity θ of this attraction is greater than 2\, this p
article system explodes in finite time. We assume that N>3θ and study in
details what happens near explosion. There are two slightly different scen
arios\, depending on the values of N and θ\, here is one: at explosion\,
a cluster consisting of precisely k0 particles emerges\, for some determin
istic k0≥7 depending on N and θ. Just before explosion\, there are infi
nitely many (k0−1)-ary collisions. There are also infinitely many (k0−
2)-ary collisions before each (k0−1)-ary collision. And there are infini
tely many binary collisions before each (k0−2)-ary collision. Finally\,
collisions of subsets of 3\,…\,k0−3 particles never occur. The other s
cenario is similar except that there are no (k0−2)-ary collisions.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giorgio Metafune (Università del Salento)
DTSTART:20250211T140000Z
DTEND:20250211T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/112
DESCRIPTION:Title: The spectrum of the Ornstein-Uhlenbeck operator\nby
Giorgio Metafune (Università del Salento) as part of Non-local operators
\, probability and singularities\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Enrico Valdinoci (The University of Western Australia)
DTSTART:20241217T120000Z
DTEND:20241217T130000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/113
DESCRIPTION:Title: Nonlocal minimal surfaces: a tricky question about the
strict maximum principle\nby Enrico Valdinoci (The University of Weste
rn Australia) as part of Non-local operators\, probability and singulariti
es\n\n\nAbstract\nSuppose that two nonlocal minimal surfaces are included
one into the other and touch at a point. Then\, they must coincide. But th
is is perhaps less obvious than what it seems at first glance.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cédric Bernardin (Higher School of Economics)
DTSTART:20250422T130000Z
DTEND:20250422T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/114
DESCRIPTION:Title: Microscopic Fluctuations Theory for systems with long-r
ange interactions\nby Cédric Bernardin (Higher School of Economics) a
s part of Non-local operators\, probability and singularities\n\n\nAbstrac
t\nMicroscopic Fluctuations Theory is the cornerstone of modern non-equil
ibrium statistical mechanics. Developed for bounary driven diffusive sytem
s during the last 25 years\, it remains mainly limited to interacting part
icles systems with short range interactions. In this talk we will explain
how to generalise this theory for microscopic models with long range inter
actions which are macrocopically described by fractional diffusion equatio
n.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federica Gregorio (Università di Salerno)
DTSTART:20250311T140000Z
DTEND:20250311T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/115
DESCRIPTION:Title: Fourth-order operators with polynomially growing coeff
icients\nby Federica Gregorio (Università di Salerno) as part of Non-
local operators\, probability and singularities\n\n\nAbstract\nIn this tal
k we will prove generation results in $L^p$ spaces as well as domain chara
cterization for some fourth-order operators with polynomially growing coef
ficients under suitable growth conditions on the coefficients.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Haojie Hou\, Xicheng Zhang (School of Mathematics and Statistics\,
Beijing)
DTSTART:20250121T140000Z
DTEND:20250121T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/116
DESCRIPTION:Title: Heat kernel estimates for nonlocal kinetic operators\nby Haojie Hou\, Xicheng Zhang (School of Mathematics and Statistics\, B
eijing) as part of Non-local operators\, probability and singularities\n\n
\nAbstract\nIn this paper\, we employ probabilistic techniques to derive s
harp\, explicit two-sided estimates for the heat kernel of the nonlocal ki
netic operator\n $$\n \\Delta^{\\alpha/2}_v + v \\cdot \\nabla_x\, \\qua
d \\alpha \\in (0\, 2)\,\\ (x\,v)\\in{\\mathbb R}^{d}\\times{\\mathbb R}^d
\,\n $$\nwhere $ \\Delta^{\\alpha/2}_v $ represents the fractional Laplac
ian acting on the velocity variable $ v $. Additionally\, we establish log
arithmic gradient estimates with respect to both the spatial ariable $ x $
and the velocity variable $v$. In fact\, the estimates are developed for
more general non-symmetric stable-like operators\, demonstrating explicit
dependence on the lower and upper bounds of the kernel functions. These r
esults\, in particular\, provide a solution to a fundamental problem in th
e study of nonlocal kinetic operators. This is a joint work with Haojie Ho
u.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eryan Hu (Tianjin University)
DTSTART:20250218T140000Z
DTEND:20250218T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/117
DESCRIPTION:Title: Dirichlet heat kernel estimates for rectilinear stable
processes\nby Eryan Hu (Tianjin University) as part of Non-local opera
tors\, probability and singularities\n\n\nAbstract\nLet $d \\ge 2$\, $\\al
pha \\in (0\,2)$\, and $X$ be the rectilinear $\\alpha$-stable process on
$\\mathbb{R}^d$. We first present a geometric characterization of open sub
set $D\\subset \\mathbb{R}^d$ so that the part process $X^D$ of $X$ in $D
$ is irreducible. We then study the properties of the transition density f
unctions of $X^D$\, including the strict positivity property as well as th
eir sharp two-sided bounds in $C^{1\,1}$ domains in $\\mathbb{R}^d$. Our b
ounds are shown to be sharp for a class of $C^{1\,1}$ domains.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucio Galeati (University of L'Aquila)
DTSTART:20250408T130000Z
DTEND:20250408T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/118
DESCRIPTION:Title: Regularity of the conditional densities for singular fr
actional SDEs\nby Lucio Galeati (University of L'Aquila) as part of No
n-local operators\, probability and singularities\n\n\nAbstract\nWe consid
er multidimensional SDEs with singular drift\, driven by additive fraction
al Brownian motion (fBm). Under appropriate regularity assumptions\, such
equations are known to be solvable in a strong sense\, thanks to modern to
ols like the Stochastic Sewing Lemma (SSL). However\, due to the singulari
ty of the drift and the non-Markovian nature of the noise\, many standard
methods to estimate the density of the law of the solution are not availab
le anymore\; conditional estimates are even harder to attain. In this talk
I will present several results in this direction\, based on a combination
of duality arguments\, sewing techniques\, Romito's lemma and Girsanov tr
ansform. As a consequence\, we provide novel existence and uniqueness resu
lts for McKean-Vlasov equations driven by fBm with convolutional drift\, t
hanks to a regularity bootstrapping procedure.\n\nBased on an ongoing join
t work with Lukas Anzeletti\, Alexandre Richard and Etienne Tanré.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandre Veretennikov (Moscow State University)
DTSTART:20250513T130000Z
DTEND:20250513T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/119
DESCRIPTION:Title: On some recent news about SDEs and McKean - Vlasov equa
tions with irregular drift\nby Alexandre Veretennikov (Moscow State Un
iversity) as part of Non-local operators\, probability and singularities\n
\n\nAbstract\nSome recent news about MV SDEs concern solutions\nwith irreg
ular drift\, in particular\, for degenerate SDEs\,\nas well as some moment
conditions on the initial\ndistributions. For Ito SDEs there is a new var
iation of\none of Yamada - Watanabe theorems on pathwise uniqueness.\nPart
ly\, the results in both branches are obtained jointly\nwith some my colle
agues and some my students.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Perkowski (Freie Universität Berlin)
DTSTART:20250401T130000Z
DTEND:20250401T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/120
DESCRIPTION:Title: Energy solutions of singular SPDEs on Hilbert spaces\nby Nicolas Perkowski (Freie Universität Berlin) as part of Non-local o
perators\, probability and singularities\n\n\nAbstract\nI will discuss a u
nified setting for well-posedness of a class of nonlinear stochastic (part
ial) differential equations with singular noise\, such as the KPZ/Burgers
equation\, stochastic Navier-Stokes equations or finite-dimensional diffus
ions with distributional drift. Under structural assumptions we develop a
probabilistic theory\, in duality with potentially infinite-dimensional pa
rtial differential equations\, which allows to easily include boundary con
ditions in SPDEs. This is motivated by the goal of deriving (weakly) unive
rsal fluctuations of interacting particle systems with boundary effects. T
he talk is based on joint work with Lukas Gräfner\, Shyam Popat and partl
y on ongoing work with Adrian Martini.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolai Krylov (University of Minnesota)
DTSTART:20250520T140000Z
DTEND:20250520T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/121
DESCRIPTION:Title: Morrey-Sobolev spaces and second-order elliptic and par
abolic PDEs with singular first-order coefficients\nby Nicolai Krylov
(University of Minnesota) as part of Non-local operators\, probability and
singularities\n\n\nAbstract\nIn recent years we witness growing interest\
nin using Real Analysis methods and results\nin the theory of nondiverge
nce form partial differential equations (PDEs)\nand the goal of this lectu
re is to give a brief account of \napplications of several results in Rea
l Analysis to the theory of\nelliptic and parabolic equations in Sobolev a
nd Sobolev-Morrey spaces.\nIn particular\, we concentrate on some results\
nobtained by using Hardy-Littlewood maximal function\ntheorem\, Fefferman-
Stein theorem\,\ntheory of Muckenhoupt weights\, and Rubio de Francia\next
rapolation theorem and their role in Sobolev or Morrey-Sobolev space\ntheo
ry of parabolic equations with mixed norms.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikolaj Sierzega (University of Warsaw & George Mason University)
DTSTART:20251104T140000Z
DTEND:20251104T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/122
DESCRIPTION:Title: Differential Harnack bounds for fractional heat equatio
ns\nby Mikolaj Sierzega (University of Warsaw & George Mason Universit
y) as part of Non-local operators\, probability and singularities\n\n\nAbs
tract\nHarnack-type estimates lie at the very heart of the regularity theo
ry for partial differential equations. One way to obtain such bounds is by
integrating differential Harnack inequalities\; for instance\, in the cas
e of the standard heat equation\, integrating the remarkable Li–Yau ineq
uality yields the classical Gaussian bound\, also known as the Hadamard–
Pini inequality. In this talk\, I will describe how a seemingly straightfo
rward attempt to extend this framework to fractional heat flows requires r
ecasting the Li–Yau technique to accommodate a broader class of equatio
ns and leads to an interesting reformulation of classical Harnack bounds.\
n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yu-Ting Chen (University of Victoria)
DTSTART:20251014T140000Z
DTEND:20251014T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/123
DESCRIPTION:Title: Stochastic path integrals in the two-dimensional stocha
stic heat equation\nby Yu-Ting Chen (University of Victoria) as part o
f Non-local operators\, probability and singularities\n\n\nAbstract\nThe t
wo-dimensional stochastic heat equation (SHE) at criticality was\nintroduc
ed around the end of the ’90s. It arises from problems of statistical ph
ysics via\nseveral stochastic models of surface growth dynamics and from t
he disordered system\nof a directed polymer in a random medium. The delta-
Bose gas in quantum physics is\nalso involved. This talk will introduce st
ochastic path integrals in the two-dimensional\nSHE at criticality and emp
hasize those at the level of annealed expectations. These\nemphasized stoc
hastic path integrals have the interesting feature of being governed by\nS
DEs with supercritical singular drift and taking a special “sum form”
that contrasts\nthe known “product form” in the one-dimensional case\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rohan Sarkar (Binghamton University SUNY)
DTSTART:20251007T130000Z
DTEND:20251007T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/124
DESCRIPTION:Title: Spectrum of Lévy-Ornstein-Uhlenbeck semigroups on R^d<
/a>\nby Rohan Sarkar (Binghamton University SUNY) as part of Non-local ope
rators\, probability and singularities\n\n\nAbstract\nWe investigate spect
ral properties of Markov semigroups associated with Ornstein-Uhlenbeck (OU
) processes driven by Lévy processes. These semigroups are generated by n
on-local\, non-self-adjoint operators. In the special case where the drivi
ng Lévy process is Brownian motion\, one recovers the classical diffusion
OU semigroup\, whose spectral properties have been extensively studied ov
er past few decades. Our main results show that\, under suitable condition
s on the Lévy process\, the spectrum of the Lévy-OU semigroup in the $L^
p$-space weighted with the invariant distribution coincides with that of t
he diffusion OU semigroup. Furthermore\, when the drift matrix $B$ is diag
onalizable with real eigenvalues\, we derive explicit formulas for eigenfu
nctions and co-eigenfunctions. A key ingredient in our approach is intertw
ining relationship: we prove that every Lévy-OU semigroup is intertwined
with a diffusion OU semigroup\, thereby preserving the spectral properties
.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gerald Trutnau (Seoul National University)
DTSTART:20251125T140000Z
DTEND:20251125T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/125
DESCRIPTION:Title: Existence and uniqueness of (infinitesimally) invariant
measures for second order partial differential operators on Euclidean spa
ce\nby Gerald Trutnau (Seoul National University) as part of Non-local
operators\, probability and singularities\n\n\nAbstract\nWe consider a lo
cally uniformly strictly elliptic second order partial differential operat
or in Euclidean space with dimension greater or equal to two\, with low re
gularity assumptions on its coefficients\, as well as an associated Hunt p
rocess and semigroup. The Hunt process is known to solve a corresponding s
tochastic differential equation that is pathwise unique. In this situation
\, we study the relation of invariance\, infinitesimal invariance\, recurr
ence\, transience\, conservativeness and L^r-uniqueness. Our main result i
s that recurrence implies uniqueness of infinitesimally invariant measures
\, as well as existence and uniqueness of invariant measures. We can hence
make in particular use of various explicit analytic criteria for recurren
ce that have been previously developed in the context of (generalized) Dir
ichlet forms and present diverse examples and counterexamples for uniquene
ss of infinitesimally invariant\, as well as invariant measures and an exa
mple where L^1-uniqueness fails although pathwise uniqueness holds. This i
s joint work with Haesung Lee.\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soobin Cho (University of Illinois Urbana-Champaign)
DTSTART:20251202T140000Z
DTEND:20251202T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/126
DESCRIPTION:Title: Approximate factorizations for non-symmetric jump proce
sses\nby Soobin Cho (University of Illinois Urbana-Champaign) as part
of Non-local operators\, probability and singularities\n\n\nAbstract\nIn t
his talk\, we first discuss approximate factorizations of heat kernels and
Green functions for purely discontinuous Markov processes\, and their equ
ivalence. In the second part\, we present applications of these factorizat
ions to obtain two-sided heat kernel estimates for two classes of processe
s: stable-like processes with critical killing in $C^{1\,Dini}$ open sets\
, and non-symmetric stable processes in $C^{1\,2-Dini}$ open sets. In part
icular\, we derive sharp\, explicit two-sided estimates for killed and cen
sored stable processes in $C^{1\,Dini}$ open sets. We also discuss the opt
imality of the $C^{1\,Dini}$ condition for heat kernel estimates of killed
stable processes. This talk is based on joint work with Professor Renming
Song (UIUC).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Bertoin (University of Zurich)
DTSTART:20260113T140000Z
DTEND:20260113T150000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/127
DESCRIPTION:Title: On the elephant random walk and its zeros\nby Jean
Bertoin (University of Zurich) as part of Non-local operators\, probabilit
y and singularities\n\n\nAbstract\nThe so-called elephant random walk is a
simple random process with\nmemory on $\\Z$. After recalling some well-kn
own results about its asymptotic behavior\,\nwe shall turn our interest to
its zero-set. I will notably present recent results obtained by Zheng Fan
g in his (ongoing) PhD about the following related questions:\n- For how l
ong has to be trained the elephant ?\n- For how long does the elephant re
member ?\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Takashi Kumagai (Waseda University)
DTSTART:20260210T130000Z
DTEND:20260210T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/128
DESCRIPTION:Title: Quantitative homogenization on time-dependent random co
nductance models with stable-like jumps\nby Takashi Kumagai (Waseda Un
iversity) as part of Non-local operators\, probability and singularities\n
\n\nAbstract\nIn this talk\, I will present quantitative homogenization re
sults for stable-like long range random walks in time-dependent random con
ductance models\, where the conductances are bounded from above\, but may
be degenerate.\n\nThis talk is based on joint work with X. Chen (Shanghai)
\, Z.-Q. Chen (Seattle) and J. Wang (Fuzhou).\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaella Servadei (Università degli Studi di Urbino Carlo Bo)
DTSTART:20260505T130000Z
DTEND:20260505T140000Z
DTSTAMP:20260404T110645Z
UID:NonLocalOperators/129
DESCRIPTION:by Raffaella Servadei (Università degli Studi di Urbino Carlo
Bo) as part of Non-local operators\, probability and singularities\n\nInt
eractive livestream: https://ulaval.zoom.us/j/69592078369?pwd=8v4uUabn7Jmy
X7zyQBYp34LVuQh9Yb.1\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/NonLocalOperators/129/
URL:https://ulaval.zoom.us/j/69592078369?pwd=8v4uUabn7JmyX7zyQBYp34LVuQh9Y
b.1
END:VEVENT
END:VCALENDAR