Asymptotic stability for a class of Markov semigroups

Abstract

Let $U\subset K$ be an open and dense subset of a compact metric space and let $\{\Phi_t\}_{t\ge0}$ be a Markov semigroup on the space of bounded Borel measurable functions on $U$ with the strong Feller property. Suppose that for each $x\in\bdu$ there exists a barrier $h\in C(K)$ at $x$ such that $\Phi_t(h)\ge h$ for all $t\ge0$. Suppose also that every real-valued $g\in C(K)$ with $\Phi_t(g)\ge g$ for all $t\ge0$ and which attains its global maximum at a point inside $U$ is constant. Then for each $f\in C(K)$ there exists the uniform limit $F=\lim_{t\to\infty}\Phi_t(f)$. Moreover $F$ is continuous on $K$, agrees with $f$ on $\partial{U}$ and $\Phi_t(F)=F$ for all $t\ge0$.