Construction of Markov Processes from Hitting Distributions II

Abstract

Let $K$ be a compact metric space, $\Delta$ a fixed point in $K, \mathcal{O}$ a base for the topology of $K$ closed under the formation of finite unions and finite intersections, and $\mathscr{D} = \{(K - U)\cup\Delta\mid U\in\mathcal{O}\}$ (here $\Delta$ stands for $\{\Delta\}$). Let $\{ H_D(x, \cdot)\mid x\in K, D\in\mathscr{D}\}$ be a family of probability measures satisfying the obvious necessary conditions of being the hitting distributions (as suggested in the notation) of a Hunt process on $K$ with $\Delta$ as the death point and the following conditions: (a) if $x\not\in D$ there exists $D'$ such that $\sup_{\gamma\in D'-\Delta}\int H_D(y, dz)H_D'(z, D' - \Delta) < 1$; (b) if $D_n\downarrow\Delta$ and $D - \Delta$ is compact $\int H_{D_n}(x, dy)H_D(y, D - \Delta)\rightarrow 0$ uniformly on compact subsets of $K - \Delta$; (c) there is a subclass $\mathscr{D}'$ of $\mathscr{D}$ such that the sets $K - D, D\in\mathscr{D}'$, have compact closure in $K - \Delta$ and form a base for the topology of $K - \Delta$, and for $D\in\mathscr{D}'$ and real continuous $f$ on $K \int H_D(x, dy)f(y)$ is continuous on $K - D$. Then a Hunt process is constructed from the prescribed hitting distributions $H_D(x, \cdot)$. This improves an earlier result of the author in that the smoothness condition (c) is much weaker than before; in fact the smoothness condition we actually assume is somewhat weaker than (c).