Construction and analysis of a sticky reflected distorted Brownian motion

TL;DR

This paper constructs a sticky reflected distorted Brownian motion in non-negative space using Dirichlet forms, analyzes its boundary behavior, and applies it to a dynamical wetting model across various dimensions.

Contribution

It introduces a Dirichlet form approach for constructing and analyzing sticky boundary behavior in distorted Brownian motion, and applies this to a wetting model in multiple dimensions.

Findings

The process exhibits sticky boundary behavior with positive occupation time.

The constructed process solves the associated stochastic differential equation weakly.

Application to a dynamical wetting model in all dimensions d .

Abstract

We give a Dirichlet form approach for the construction of a distorted Brownian motion in $E:=[0,\infty)^n$, $n\in\mathbb{N}$, where the behavior on the boundary is determined by the competing effects of reflection from and pinning at the boundary (sticky boundary behavior). In providing a Skorokhod decomposition of the constructed process we are able to justify that the stochastic process is solving the underlying stochastic differential equation weakly for quasi every starting point with respect to the associated Dirichlet form. That the boundary behavior of the constructed process indeed is sticky, we obtain by proving ergodicity of the constructed process. Therefore, we are able to show that the occupation time on specified parts of the boundary is positive. In particular, our considerations enable us to construct a dynamical wetting model (also known as Ginzburg--Landau dynamics) on a bounded set $D_{\scriptscriptstyle{N}}\subset \mathbb{Z}^d$ under mild assumptions on the underlying pair interaction potential in all dimensions $d\in\mathbb{N}$. In dimension $d=2$ this model describes the motion of an interface resulting from wetting of a solid surface by a fluid.