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arXiv · 2106.07208

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

Cristina Costantini·Thomas G. Kurtz
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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 $α$ of Kwon and Williams (1991) is strictly less than 1, thus partially extending the results of Williams (1985) to higher dimension.

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Cristina Costantini, Thomas G. Kurtz. 2024-08-14. A reverse ergodic theorem for inhomogeneous killed Markov chains and application to a new uniqueness result for reflecting diffusions. https://doi.org/10.1214/23-aap2047

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