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 $\{{\Phi }_t{\}}_{t\ge 0}$ 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 $t\ge 0$. Suppose also that every real-valued $g\in C(K)$ with ${\Phi }_t(g)\ge g$ for all $t\ge 0$ 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 $t\ge 0$.

4 pages