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

Michael  Röckner; Jiyong  Shin; Gerald  Trutnau

doi:10.3934/dcdsb.2016095

Article Contents

2016, Volume 21, Issue 9: 3219-3237. Doi: 10.3934/dcdsb.2016095

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

1.
Fakultät für Mathematik, Universität Bielefeld, Universitätsstrasse 25, 33615 Bielefeld
2.
School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro Dongdaemun-gu, Seoul 02445
3.
Department of Mathematical Sciences and Research Institute of Mathematics of Seoul National University, 1, Gwanak-Ro, Gwanak-Gu, Seoul 08826

Received: August 31, 2015

Revised: February 29, 2016

Published: October 2016

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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. 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 has an unbounded and discontinuous drift. Once having shown weak existence, we obtain from a result of [13] 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. 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 related to a class of Muckenhoupt weights and corresponding divergence free perturbations.

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Mathematics Subject Classification: Primary: 31C25, 60J60, 47D07; Secondary: 31C15, 60J35, 60H20.

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