A Dirichlet process characterization of a class of reflected diffusions

Abstract

For a class of stochastic differential equations with reflection for which a certain ${\mathbb{L}}^{p}$ continuity condition holds with p>1, it is shown that any weak solution that is a strong Markov process can be decomposed into the sum of a local martingale and a continuous, adapted process of zero p-variation. When p=2, this implies that the reflected diffusion is a Dirichlet process. Two examples are provided to motivate such a characterization. The first example is a class of multidimensional reflected diffusions in polyhedral conical domains that arise as approximations of certain stochastic networks, and the second example is a family of two-dimensional reflected diffusions in curved domains. In both cases, the reflected diffusions are shown to be Dirichlet processes, but not semimartingales.