A REVERSE ERGODIC THEOREM FOR INHOMOGENEOUS KILLED MARKOV CHAINS AND APPLICATION TO A NEW UNIQUENESS RESULT FOR REFLECTING DIFFUSIONS

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 of Kwon andWilliams (1991) is strictly less than 1, thus partially extending the results of Williams (1985) to higher dimension.