Construction of distorted Brownian motion with permeable sticky behaviour on sets with Lebesgue measure zero
Torben Fattler, Martin Grothaus, Nathalie Steil
TL;DR
This work constructs a diffusion on $\mathbb{R}^d$ that exhibits permeable sticky interaction with a Lebesgue-null set $A$ by coupling a gradient Dirichlet form with a speed measure $\varrho\mu$, where $\mu=\lambda^d+\mathcal{S}$ concentrates on $A$. By deriving an explicit generator under Lipschitz boundary conditions for $A$ and applying the Fukushima decomposition, the authors show the process outside $A$ is a distorted Brownian motion with drift $\nabla\log\varrho$, while the interaction with $A$ induces stickiness. They establish existence of a diffusion $\mathbf{M}^{\varrho}$ associated to $(\mathcal{E}^{\varrho},D(\mathcal{E}^{\varrho}))$, prove irreducibility and recurrence, and demonstrate ergodicity, yielding positive séjour time on $A$ and permeability of $A$ when $\varrho\mu(\mathbb{R}^d)<\infty$. The paper provides explicit generator formulas, an SDE representation outside $A$, and a rigorous framework for modeling diffusions with sticky interfaces on measure-zero sets, with potential applications to permeable boundary phenomena in high dimensions.
Abstract
The starting point is a gradient Dirichlet form with respect to $\varrhoλ^d$ on the space $L^2({\mathbb{R}}^d, \varrhoμ)$. Here $λ^d$ is the Lebesgue measure on ${\mathbb R}^d$, $\varrho$ a strictly positive density and $μ$ 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 séjour time of $X$ on $A$. Hence we obtain a stochastic process $X$ with permeable sticky behaviour on $A$.