Conditioned SRW in two dimensions and some of its surprising properties

Abstract: We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient and we show that it is "almost recurrent" in the sense that each infinite 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 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 "surround" 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 range of the conditioned walk has "fractal" behavior. Also, we obtain estimates on the 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 infinitely many tim

analysis of PDEscombinatoricsprobability

Audience: researchers in the topic

Probability and Stochastic Analysis at Tecnico Lisboa

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