  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 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.
