On the existence of Markovian measures on continuous paths

Abstract

Let $\eta$ be a positive Radon measure on the space of continuous paths from R into a locally compact Polish space $Y$, and assume that $\eta$ admits an invariant measure. We explicit conditions on $\eta$ such that successive Markovianisation of $\eta$ over any dense and countable set of times have all limit points (in the weak-star topology on Radon measures) satisfying the strong Markov property. We show that if Y is a locally compact Polish group, and $\eta$ is left or right translation invariant, then $\eta$ satisfies such conditions. Our proof uses the Zermalo-Fraenkel and Axiom of Dependant Choice axiomatisation of set theory, in which countable products of compacts are compact.