Existence of finite invariant measures for Markov processes

Abstract

Let xt be a Markov process on a locally compact metric space (8, p) with a countable base. Let 63 be the o-algebra generated by the open sets, and for ACG6, t>O, set P(t, x, A)=Pr{xtCAjxo=x}. Khasminskii [1 ] has related, in a natural way, the existence of a finite invariant measure for xt with the mean time to hit a compact set. Our object is to relate the existence of such a measure to properties of the measures P(t, x, * ), t > O, xEl. We assume that for fixed t and A, P(t, x, A) is continuous in x, that for an e-neighborhood Sf(x) about x, P(t, x, S,(x))-l as t--O uniformly on compacta, and that P(t, x, 8) = 1. Let C+ be the strictly positive cone of the Banach space ca(&, 63) consisting of the countably additive set functions on 63 with variation norm. Let Ut, t _ 0, be the semigroup defined for M&ca(&, G3) by the formula