Strong Feller property of sticky reflected distorted Brownian motion

TL;DR

This paper constructs a multidimensional sticky reflected distorted Brownian motion using Girsanov transformations and proves its transition semigroup has the strong Feller property, with applications to wetting models.

Contribution

It introduces a new multidimensional diffusion process with sticky reflection and establishes its strong Feller property using Dirichlet form and probabilistic methods.

Findings

Constructed a conservative diffusion on $[0, Infty)^n$ with sticky reflection.

Proved the strong Feller property for the transition semigroup.

Characterized the process via Dirichlet form methods.

Abstract

Using Girsanov transformations we construct from sticky reflected Brownian motion on $[0,\infty)$ a conservative diffusion on $E:=[0,\infty)^n$, $n \in \mathbb{N}$, and prove that its transition semigroup possesses the strong Feller property for a specified general class of drift functions. By identifying the Dirichlet form of the constructed process, we characterize it as sticky reflected distorted Brownian motion. In particular, the relations of the underlying analytic Dirichlet form methods to the probabilistic methods of random time changes and Girsanov transformations are presented. Our studies are motivated by its applications to the dynamical wetting model with $\delta$-pinning and repulsion.