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

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.

References in corpus (7)

• Fluid model for a network operating under a fair bandwidth-sharing policy
• State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy
• Switched networks with maximum weight policies: Fluid approximation and multiplicative state space collapse
• Viscosity methods giving uniqueness for martingale problems
• Existence and uniqueness of reflecting diffusions in cusps
• Markov selection for constrained martingale problems
• Localization for constrained martingale problems and optimal conditions for uniqueness of reflecting diffusions in 2-dimensional domains

Cited by in corpus (2)

• Localization for constrained martingale problems and optimal conditions for uniqueness of reflecting diffusions in 2-dimensional domains
• Obliquely reflecting Brownian motion in nonpolyhedral, piecewise smooth cones, with an example of application to diffusion approximation of bandwidth sharing queues