Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities

Abstract

Abstract. The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures, so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model and establish an analogue of the Poisson formula for (random) bounded harmonic functions. Under natural conditions on transition probabilities we identify these boundaries for several classes of groups with hyperbolic properties and prove the boundary triviality (i.e., the absence of non-constant random bounded harmonic functions) for groups of subexponential growth, in particular, for nilpotent groups. IntroductionRandom walks on groups were intensively studied during the last 40 years (see, for instance, [Ka96] and the references therein). Their importance is due to numerous applications, in particular, to the description of boundaries and spaces of harmonic functions and to the study of ergodic properties of group actions. Such random walks are Markov chains which are homogeneous both in time and space and can also be represented as products of independent identically distributed (i.i.d.) group elements. In the important special case of products of random matrices additional tools such as Lyapunov exponents can be employed.2000 Mathematics Subject Classification. 60J50, 37A30, 60B99.

Ergodic theory on stationary random graphs

“…A stationary random graph (G, ρ) is a random rooted graph whose distribution is invariant under rerooting along a simple random walk started at the root ρ (see Section 1.1 for a precise definition). The entropy technique and characterization of the Liouville property for groups, homogeneous graphs or random walk in random environment [24,25,26,27,29,30] are adapted to this context. In particular we have Theorem 1.1.…”

Hyperbolic and Parabolic Unimodular Random Maps

“…that every invariantly nonamenable unimodular random rooted graph with finite expected degree and at most exponential growth has positive speed. Meanwhile, it is a result of Benjamini and Curien [9], generalizing the work of Kaimanovich, Vershik, and others [54,53,52,51,50,49], that every non-Liouville unimodular random rooted graph with finite expected degree and at most exponential growth has positive speed. In general, however, there do exist invariantly nonamenable, non-Liouville, unimodular random rooted graphs with finite expected degree such that the random walk has zero speed almost surely.…”

Hyperbolic and Parabolic Unimodular Random Maps

“…This can be seen in several ways: it is an easy consequence of a theorem of Benjamini, Lyons, and Schramm [14, Theorem 3.2] (see also [2,Theorem 8.13] and [4, Theorem 3.2]) that every invariantly nonamenable unimodular random rooted graph with finite expected degree and at most exponential growth has positive speed. Meanwhile, it is a result of Benjamini and Curien [9], generalizing the work of Kaimanovich, Vershik, and others [54,53,52,51,50,49], that every non-Liouville unimodular random rooted graph with finite expected degree and at most exponential growth has positive speed. In general, however, there do exist invariantly nonamenable, non-Liouville, unimodular random rooted graphs with finite expected degree such that the random walk has zero speed almost surely.…”