On recurrent random walks on semigroups

Abstract:

Let $\mu$ be a regular Borel probability measure on a locally compact semigroup S and consider the right (resp. left) random walk on $D = \overline {{\text {U}}{F^n}} ,F = {\text {Supp}}\;\mu$, with transition function ${P^n}(x,B) \equiv {\mu ^n}({x^{ - 1}}B)\;({\text {resp}}.\;{\mu ^n}(B{x^{ - 1}}))$. These Markov chains can be represented as ${Z_n} = {X_1}{X_2} \cdots {X_n}\;({\text {resp}}.\;{S_n} = {X_n}{X_{n - 1}} \cdots {X_1}),\;{X_i}$’s independent $\mu$-distributed with values in S defined on an infinite-sequence space $(\Pi _1^\infty {S_i},P),{S_i} = S$ for all i. Let ${R_r}\;({\text {resp}}.\;{R_t}) = \{ x \in D;{P_x}({Z_n}({S_n}) \in {N_x}\;{\text {i.o.}}) = 1$ for all neighborhoods ${N_x}$ of x} and ${R’_r}({R’_t}) = \{ x \in D;P({Z_n}({S_n}) \in {N_x}\;{\text {i.o.}}) = 1$ for all ${N_x}$ of x}. Let S be completely simple ($= E \times G \times F$, usual Rees product) in the results (1), (2), (3), (4), (5) below: (1) $x \in {R_r}\;iff\;\Sigma \;{\mu ^n}({x^{ - 1}}{N_x}) = \infty$ for all neighborhoods ${N_x}$ of $x\;iff\;\Sigma \;{\mu ^n}({N_x}) = \infty$ for all ${N_x}$ of x. (2) Either ${R_r} = {R_t} = \emptyset$ or ${R_r} = {R_t} = D =$ also completely simple. (3) If the group factor G is compact, then there are recurrent values and we have ${R_r} = {R_t} = D =$ completely simple. (4) ${R’_r} \ne \emptyset$ implies ${R’_r} = {R_r} = {R_t} = D =$ a right subgroup of S (but ${R’_t}$ may be $\emptyset$). (5) S can support a recurrent random walk (i.e., a r. walk with ${R_r} \ne \emptyset$) iff G (= the group factor) can support a recurrent random walk. Finally (6) if S is compact abelian, then always $R’ = R = K =$ the kernel of S. These results extend previously known results of Chung and Fuchs and Loynes.