The Annals of Mathematical Statistics

Construction of Markov Processes from Hitting Distributions II

C. T. Shih

Full-text: Open access

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).

Article information

Source
Ann. Math. Statist., Volume 42, Number 1 (1971), 97-114.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177693498

Digital Object Identifier
doi:10.1214/aoms/1177693498

Mathematical Reviews number (MathSciNet)
MR279893

Zentralblatt MATH identifier
0278.60049

JSTOR
links.jstor.org

Citation

Shih, C. T. Construction of Markov Processes from Hitting Distributions II. Ann. Math. Statist. 42 (1971), no. 1, 97--114. doi:10.1214/aoms/1177693498. https://projecteuclid.org/euclid.aoms/1177693498


Export citation