Bilateral random walks on compact semigroups

Abstract:

Let $\mu$ be a regular Borel probability measure with support $F$ on a compact semigroup $S$. Let ${X_0},{X_{ \pm 1}},{X_{ \pm 2}}, \cdots$ be a sequence of independent random variables with values in $S$, having identical distribution $P({X_n} \in B) = \mu (B)$. The random walk ${W_n} = {X_{ - n}} \cdots {X_{ - 1}}{X_0}{X_1} \cdots {X_n}$ is studied in this paper. Let $D$ be the closed semigroup generated by $F$ and let $K$ be the kernel of $D$. An element $x \in D$ is called recurrent iff ${P_x}({W_n} \in {N_x}{\text {i}}{\text {.o}}{\text {.}}) = 1$ for every open neighborhood ${N_x}$ of $x$. We prove: $x$ is essential for ${W_n}$ if and only if $x \in K$ if and only if $x$ is recurrent if and only if $\Sigma {P_x}({W_n} \in {N_x}) = \Sigma {\mu ^n} \ast [{\mu ^n}({x^{ - 1}} \cdot )]({N_x}) = \infty$ for every ${N_x}$. Moreover all states in $K$ are recurrent positive. These results extend results of the authors for the unilateral random walks (using different methods) and recent results of Larisse for the discrete case.