An inhomogeneous reverse ergodic theorem 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 in 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 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.

[1]  Ruth J. Williams.  Reflected brownian motion in a wedge: Semimartingale property , 1985 .

[2]  Ruth J. Williams.  Brownian motion in a wedge with oblique reflection at the boundary , 1985 .

[3]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[4]  Ruth J. Williams,et al.  Existence and Uniqueness of Semimartingale Reflecting Brownian Motions in Convex Polyhedrons , 1996 .

[5]  R. Bass,et al.  Uniqueness for diffusions with piecewise constant coefficients , 1987 .

[6]  Xiongzhi Chen.  Brownian Motion and Stochastic Calculus , 2008 .

[7]  R. J. Williams,et al.  Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant , 1993 .

[8]  Thomas G. Kurtz,et al.  Martingale Problems for Constrained Markov Problems , 1990 .

[9]  P. Donnelly.  MARKOV PROCESSES Characterization and Convergence (Wiley Series in Probability and Mathematical Statistics) , 1987 .

[10]  Markov selection for constrained martingale problems , 2019, Electronic Journal of Probability.

[11]  T. Kurtz,et al.  Existence and uniqueness of reflecting diffusions in cusps , 2017, 1710.06281.

[12]  Ruth J. Williams,et al.  Reflected Brownian motion in a cone with radially homogeneous reflection field , 1991 .

[13]  P. Protter.  Stochastic integration and differential equations , 1990 .

[14]  P. Dupuis,et al.  SDEs with Oblique Reflection on Nonsmooth Domains , 2008 .

[15]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[16]  M. Kreĭn,et al.  Linear operators leaving invariant a cone in a Banach space , 1950 .

[17]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .