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SUMMARY:Serguei Popov (Universidade de Porto)
DTSTART:20210112T140000Z
DTEND:20210112T150000Z
DTSTAMP:20260404T132231Z
UID:PSA/4
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/PSA/4
 /">Conditioned SRW in two dimensions and some of its surprising properties
 </a>\nby Serguei Popov (Universidade de Porto) as part of Probability and 
 Stochastic Analysis at Tecnico Lisboa\n\n\nAbstract\nWe consider the two-d
 imensional simple random walk conditioned on never hitting the origin. Thi
 s process is a Markov chain\, namely it is the Doob $h$-transform of the s
 imple random walk\nwith respect to the potential kernel. It is known to be
  transient and we show that it is "almost recurrent" in the sense that eac
 h infinite set is visited infinitely often\, almost surely. After discussi
 ng some basic properties of this process (in particular\, calculating its 
 Green's function)\, we prove that\, for a "large" set\, the proportion of 
 its sites visited by the conditioned walk is approximately a Uniform$[0\,1
 ]$ random variable. Also\, given a set $G\\subset R^2$ that does not "surr
 ound" the origin\, we prove that a.s. there is an infinite number of $k$'s
  such that $kG\\cap Z^2$ is unvisited. These results suggest that the rang
 e of the conditioned walk has "fractal" behavior. Also\, we obtain estimat
 es on the speed of escape of the walk to infinity\, and prove that\, in sp
 ite of transience\, two independent copies of conditioned walks will a.s. 
 meet infinitely many tim\n
LOCATION:https://stable.researchseminars.org/talk/PSA/4/
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