A reverse ergodic theorem for inhomogeneous killed Markov chains and application to a new uniqueness result for reflecting diffusions

TL;DR

This paper establishes a reverse ergodic theorem for inhomogeneous killed Markov chains and applies it to prove a new uniqueness result for reflecting diffusions with oblique reflection in domains with singular points.

Contribution

It introduces a novel reverse ergodic theorem for inhomogeneous transition functions and uses it to demonstrate existence and uniqueness of certain reflecting diffusions with oblique reflection.

Findings

Proves a reverse ergodic theorem for inhomogeneous sub-probability transition functions.

Establishes existence and uniqueness of semimartingale diffusions with oblique reflection.

Shows that if reflecting Brownian motion in a smooth cone is a semimartingale, then the parameter α is less than 1.

Abstract

Bass and Pardoux (1987) deduce from the Krein-Rutman theorem a reverse ergodic theorem for a sub-probability transition function, which turns out to be a key tool in proving uniqueness of reflecting Brownian Motion in cones in Kwon and Williams (1991) and Taylor and Williams (1993). By a different approach, we are able to prove an analogous reverse ergodic theorem for a family of inhomogeneous sub-probability transition functions. This allows us to prove existence and uniqueness for a semimartingale diffusion process with varying, oblique direction of reflection, in a domain with one singular point that can be approximated, near the singular point, by a smooth cone, under natural, easily verifiable geometric conditions. Along the way we also show that if the reflecting Brownian motion in a smooth cone is a semimartingale then the parameter $\alpha$ of Kwon and Williams (1991) is strictly less than 1, thus partially extending the results of Williams (1985) to higher dimension.