On the stochastic regularity of diffusion processes associated with (non-symmetric) Dirichlet forms

Abstract

First for any starting point in Rd we identify the stochastic differential equation of distorted Brownian motion with respect to a certain discontinuous Muckenhoupt A2-weight under the assumption of Fukushimas absolute continuity condition. We then systematically develop general tools to apply the absolute continuity condition. These tools comprise methods to obtain a Hunt process on a locally compact separable metric state space whose transition function has a density w.r.t. the reference measure and methods to estimate drift potentials comfortably. Our results are applied to distorted Brownian motions and construct weak solutions to singular stochastic differential equations, i.e. equations with possibly unbounded and discontinuous drift and reflection terms which may be the sum of countably many local times. The solutions can start from any point of the explicitly specified state space. We consider different kinds of weights, like Muckenhoupt A2 weights and weights with moderate growth at singularities as well as different kind of (multiple) boundary conditions. We also apply the general schemes to degenerate elliptic forms and show solutions to the corresponding stochastic differential equations. Finally we extend the results of symmetric distorted Brownian motions to non-symmetric ones. Using elliptic regularity results in weighted spaces, stochastic calculus and the theory of non-symmetric Dirichlet forms, we first show weak existence of non-symmetric distorted Brownian motion for any starting point in some domain E of Rd, where E is explicitly given as the points of strict positivity of the unique continuous version of the density to its invariant measure. Once having shown weak existence, we obtain from a result of [43] that the constructed weak solution is indeed strong as well as pathwise unique up to its explosion time. As a consequence of our approach, we can use the theory of Dirichlet forms to prove further properties of the solutions. More precisely, we obtain new non-explosion criteria for them.