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SUMMARY:Serguei Popov (Universidade de Porto)
DTSTART:20210112T140000Z
DTEND:20210112T150000Z
DTSTAMP:20260423T004550Z
UID:PSA/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PSA/4/">Cond
 itioned SRW in two dimensions and some of its surprising properties</a>\nb
 y Serguei Popov (Universidade de Porto) as part of Probability and Stochas
 tic Analysis at Tecnico Lisboa\n\n\nAbstract\nWe consider the two-dimensio
 nal simple random walk conditioned on never hitting the origin. This proce
 ss is a Markov chain\, namely it is the Doob $h$-transform of the simple r
 andom walk\nwith respect to the potential kernel. It is known to be transi
 ent and we show that it is "almost recurrent" in the sense that each infin
 ite set is visited infinitely often\, almost surely. After discussing some
  basic properties of this process (in particular\, calculating its Green's
  function)\, we prove that\, for a "large" set\, the proportion of its sit
 es visited by the conditioned walk is approximately a Uniform$[0\,1]$ rand
 om variable. Also\, given a set $G\\subset R^2$ that does not "surround" t
 he origin\, we prove that a.s. there is an infinite number of $k$'s such t
 hat $kG\\cap Z^2$ is unvisited. These results suggest that the range of th
 e conditioned walk has "fractal" behavior. Also\, we obtain estimates on t
 he speed of escape of the walk to infinity\, and prove that\, in spite of 
 transience\, two independent copies of conditioned walks will a.s. meet in
 finitely many tim\n
LOCATION:https://researchseminars.org/talk/PSA/4/
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