Bilateral random walks on compact semigroups

Let /i be a regular Borel probability measure with support F on a compact semigroup S. Let XO, X+1, X+2-.. be a sequence of independent random variables with values in S, having identical distribution P(X E B)= 8t(B). The random walk W = X *-. X XXI *-XN n n -n 101 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 E D is called recurrent iff Px(Wn e Nx i.o.) = 1 for every open neighborhood Nx of x. We prove: x is essential for W if and only if x E K if and only if x is recurrent if n and only if ZP x(W e ) =UJn* [1z(x1 * )](N )oo for every N . 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. 1. Let S be a compact semigroup, i.e., a compact Hausdorff space which is algebraically a semigroup with jointly continuous multiplication and let S have a countable basis. (See Concluding remarks on the blanket assumption of 2nd countability and how it can be removed.) Let yt be a regular probability measure defined on the Borel subsets-of S whose support is denoted by F. Let D = UF , the closed subsemigroup generated by F, and let K be the kernel of D. The measure ji induces various random walks on D among which the principal ones are the right, left and bilateral (symmetric) random walks induced by the probability transition functions (respectively) (prn(x, B)-n(x 1 B(,,)n(B), rP , B) it ~x1 B) n Pn(x, B) = Pn(x, B) jn(Bx1) (Ln) (B tPn(x, B) ,n x jnl(a, b); axb E B, a, b 6 Fn} On * X(Un)(B) for n = 1, 2,...,2 where It denotes the nth convolution (*) power of ji. For Received by the editors May 17, 1973 and, in revised form, November 5, 1973. AMS (MOS) subject classifications (1970). Primary 60G50, 60J15, 43A05.