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

Abstract

Bass and Pardoux (Probab. Theory Related Fields (1987) 76 557–572) deduce from the Krein–Rutman theorem a reverse ergodic theorem for a subprobability transition function, which turns out to be a key tool in proving uniqueness of reflecting Brownian motion in cones in Kwon and Williams (Trans. Amer. Math. Soc (1991) 32 739–780) and Taylor and Williams (Probab. Theory Related Fields (1993) 96 283–317). By a different approach, we are able to prove an analogous reverse ergodic theorem for a family of inhomogeneous subprobability 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, under our conditions, the parameter α of Kwon and Williams (1991) is strictly less than 1, thus partially extending the results of Williams (Z. Warsch. Verw. Gebiete (1985) 69 161–176) to higher dimension.