Skorokhod decomposition for a reflected $\mathcal L^p$-strong Feller diffusions with singular drift

TL;DR

This paper develops a method to decompose reflected diffusions with singular drifts using Skorokhod decomposition, enabling analysis of complex stochastic systems with physically relevant singular potentials.

Contribution

It introduces a Skorokhod decomposition framework for singular drift diffusions with reflecting boundaries, utilizing Dirichlet form theory and Sobolev regularity techniques.

Findings

Constructed Skorokhod decompositions for singular drift diffusions.

Proved regularity properties of potentials related to surface measures.

Enabled modeling of particle systems with highly singular interactions.

Abstract

We construct Skorokhod decompositions for diffusions with singular drift and reflecting boundary behavior on open subsets of $\mathbb R^d$ with $C^2$-smooth boundary except for a sufficiently small set. This decomposition holds almost surely under the path measures of the process for every starting point from an explicitly known set. This set is characterized by the boundary smoothness and the singularities of the drift term. We apply modern methods of Dirichlet form theory and $\mathcal L^p$-strong Feller processes. These tools have been approved as useful for the pointwise analysis of stochastic processes with singular drift and various boundary conditions. Furthermore, we apply Sobolev space theorems and elliptic regularity results to prove regularity properties of potentials related to surface measures. These are important ingredients for the pointwise construction of the boundary local time of the diffusions under consideration. As an application we construct stochastic dynamics for particle systems with hydrodynamic and pair interaction. Our approach allows highly singular potentials like Lennard-Jones potentials and position-dependent diffusion coefficients and thus the treatment of physically reasonable models.