Construction of $\mathcal L^p$-strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions

TL;DR

This paper develops a method to construct $ abla^p$-strong Feller processes using Dirichlet forms, enabling the creation of elliptic diffusions with irregular coefficients on complex domains.

Contribution

It introduces a general scheme for constructing $ abla^p$-strong Feller processes from Dirichlet forms, applicable to elliptic diffusions with singular coefficients.

Findings

Constructed processes solve martingale problems from regularity assumptions.

Applied to elliptic diffusions with Lipschitz coefficients and singular drifts.

Established regularity conditions via elliptic regularity results.

Abstract

We provide a general construction scheme for $\mathcal L^p$-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvents of kernels having the $\mathcal L^p$-strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients and singular drifts on general open sets with absorption at the boundary. In this application elliptic regularity results imply the desired regularity assumptions.