Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and
 non-explosion results

Abstract

Using elliptic regularity results in weighted spaces, stochastic calculus and the theory of non-symmetric Dirichlet forms, we first showweak existence of non-symmetric distorted Brownian motion for any starting point in some domain $E$ of $\mathbb{R}^d$, where $E$ is explicitly given as the points of strictpositivity of the unique continuous version of the density to its invariant measure. This non-symmetric distorted Brownian motion is also proved to be strong Feller. Non-symmetric distorted Brownian motion is a singular diffusion, i.e. a diffusion that typically hasan unbounded and discontinuous drift. Once having shown weak existence, we obtain from a result of [13] that the constructedweak solution is indeed strong and weakly as well as pathwise unique up to its explosion time. As a consequence of our approach, we can use the theory of Dirichlet formsto prove further properties of the solutions. For example, we obtain new non-explosion criteria for them.We finally present concrete existence and non-explosion results for non-symmetric distorted Brownian motion relatedto a class of Muckenhoupt weights and corresponding divergence free perturbations.