Recurrent Sets

Abstract

As is well known ([7], [12], [13] and [14]) Markov processes may be studied via an appropriate potential theory, which for symmetric random walks is discrete Newtonian potential theory (see [8], [19]). Here we will be concerned with the problem of deciding when a set of lattice points in $s \geqq 3$ dimensions is recurrent, that is, when it is visited infinitely often a.s. by the symmetric random walk. In [14] a necessary and sufficient condition was given to decide this problem. Intuitively this test determines whether or not $\infty$ is a regular point of a lattice set (see [11]). However the test tells one very little about the recurrence properties of an arbitrarily given lattice set. Here we desire more precise information about lattice sets. Namely can one impose regularity conditions on lattice sets such that there exists a weighting $\mu(a)$, such that the divergence of $\sum_{a\epsilon A} \mu(a)$ is necessary and sufficient for the recurrence of the lattice set $A$? As a step in this direction we give a necessary and sufficient condition for recurrence of a set for a general Markov chain in terms of the existence of a non-negative solution to a Wiener-Hopf type equation on the set in question. As a byproduct of this equation for finite lattice sets, we obtain interesting bounds for the probabilistic capacity of finite sets as well as explicit expressions for the probability of leaving a finite set forever. As an application of our criterion for recurrence in $s = 3$ dimensions, regularity conditions are given on lattice sets so that $\mu(a) = |a|^{-1}$, where $|a|$ is the Euclidean distance from the lattice point $a$ to the origin, is the appropriate weighting so that $\sum_{a\varepsilon A} \mu(a) = \infty \iff A$ is recurrent. Further it is shown that the regularity conditions cannot be removed, as we exhibit a set for which the above series is divergent but the set is not recurrent. Further the above regularity conditions are invariant under arbitrary possibly different rotations of each lattice point about the origin and hence recurrent lattice sets satisfying these regularity conditions remain recurrent under arbitrary rotation. Finally a necessary condition and a sufficient condition are given for subsets of the axis in 3 dimensions.