Elliptic regularity results: n-regularized Liouville Brownian motion and non-symmetric diffusions associated with degenerate forms

TL;DR

This paper utilizes advanced elliptic regularity results to establish the existence of solutions for certain degenerate and non-symmetric stochastic differential equations, including the n-regularized Liouville Brownian motion, through Dirichlet form theory.

Contribution

It introduces new elliptic regularity techniques to prove weak existence of singular SDEs associated with degenerate and non-symmetric Dirichlet forms, including the Liouville Brownian motion.

Findings

Weak existence of singular SDEs for degenerate and non-symmetric forms

Existence of n-regularized Liouville Brownian motion from all points in R^2

Application of elliptic regularity to stochastic calculus in degenerate settings

Abstract

We apply improved elliptic regularity results to a concrete symmetric Dirichlet form and various non-symmetric Dirichlet forms with possibly degenerate symmetric diffusion matrix. Given the (non)-symmetric Dirichlet form, using elliptic regularity results and stochastic calculus we show weak existence of the corresponding singular stochastic differential equation for any starting point in some subset E of R^d. As an application of our approach we can show the existence of n-regularized Liouville Brownian motion only via Dirichlet form theory starting from all points in R^2.