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

TL;DR

This paper establishes the existence, uniqueness, and non-explosion criteria for non-symmetric distorted Brownian motions using advanced stochastic calculus, elliptic regularity, and Dirichlet form theory, including concrete examples.

Contribution

It provides the first comprehensive analysis of strong solutions and non-explosion criteria for non-symmetric distorted Brownian motions with explicit conditions.

Findings

Weak existence of non-symmetric distorted Brownian motion in specified domains.

Strong and pathwise unique solutions up to explosion time.

New non-explosion criteria based on Dirichlet form theory.

Abstract

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 $\mathbb{R}^d$, where $E$ is explicitly given as the points of strict positivity of the unique continuous version of the density to its invariant measure. Non-symmetric distorted Brownian motion is a singular diffusion, i.e. a diffusion that typically has an unbounded and discontinuous drift. Once having shown weak existence, we obtain from a result of \cite{KR} that the constructed weak 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 forms to prove further properties of the solutions. More precisely, we obtain new non-explosion criteria for them. We finally present concrete existence and non-explosion results for non-symmetric distorted Brownian motion related to a class of Muckenhoupt weights and corresponding divergence free perturbations.