Construction of distorted Brownian motion with permeable sticky behaviour on sets with Lebesgue measure zero

TL;DR

This paper constructs a distorted Brownian motion with permeable sticky behavior on measure-zero sets, providing explicit generator representations and analyzing its recurrence, irreducibility, and ergodic properties.

Contribution

It introduces a new class of stochastic processes with permeable sticky behavior on measure-zero sets, including explicit generator formulas and detailed process properties.

Findings

Explicit generator representation for the process.

Process is irreducible and recurrent.

Positive sojourn time on the set A.

Abstract

The starting point is a gradient Dirichlet form with respect to $\varrho\lambda^d$ on the space $L^2({\mathbb{R}}^d, \varrho\mu)$. Here $\lambda^d$ is the Lebesgue measure on ${\mathbb R}^d$, $\varrho$ a strictly positive density and $\mu$ puts weight on a set $A\subset {\mathbb R}^d$ with Lebesgue measure zero. We show that the Dirichlet form admits an associated stochastic process $X$. We derive an explicit representation of the corresponding generator if $A$ is a Lipschitz boundary. This representation together with the Fukushima decomposition identifies $X$ as a distorted Brownian motion with drift given by the logarithmic derivative of $\varrho$ in ${\mathbb R}^d \setminus A$. Furthermore, we prove $X$ to be irreducible and recurrent. Finally, via ergodicity we prove positive s\'ejour time of $X$ on $A$. Hence we obtain a stochastic process $X$ with permeable sticky behaviour on $A$.