Ben-Zion A. Rubshtein

Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities

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.

Limit theorems for random walks with dynamical random transitions

SummaryLet θ be an ergodic and conservative non-singular transformation of (Ω, Σ,m) (thedynamic environment), let μw be a random probability on a locally compact second countable groupG, and define

Lacunary isomorphism of decreasing sequences of measurable partitions

v_{_w }^{(n)} = \mu _{\theta ^n } - 1_w *\mu _{\theta ^n } - 2_w *...*\mu _{\theta w} *\mu _w

Intersections and Sums of Orlicz Spaces

Conditions for the convergence limn→∞∥vω(n)*(f−δt*f)∥1=0 for a.e. ω and everyf∈L1(G) andt∈G, are given, whenG is Abelian or compact.

Comparison of Orlicz Spaces

AbstractLetG be a compact group andM1(G) be the convolution semigroup of all Borel probability measures onG with the weak topology. We consider a stationary sequence {μn}n=−∞+∞ of random measures μn=μn(ω) inM1(G) and the convolutions

Embeddings L1 ∩ L∞ ⊆ X ⊆ L1 + L∞ ⊆ L0

v_{m,n} (\omega ) = \mu _m (\omega )* \cdots *\mu _{n - 1} (\omega ), m< n