Uniqueness of diffusion on domains with rough boundaries

TL;DR

This paper characterizes when a diffusion process on domains with rough, fractal boundaries has a unique Markov extension, based on boundary regularity and a parameter related to the boundary's Hausdorff dimension.

Contribution

It establishes a precise condition for Markov uniqueness of diffusion forms on irregular domains with fractal boundaries, extending previous results to more complex geometries.

Findings

Markov uniqueness holds if and only if elta (s-(d-1))

Results apply to Lipschitz and fractal boundaries like the von Koch snowflake

Provides a criterion linking boundary regularity and diffusion uniqueness

Abstract

Let $\Omega$ be a domain in $\mathbf R^d$ and $h(\varphi)=\sum^d_{k,l=1}(\partial_k\varphi, c_{kl}\partial_l\varphi)$ a quadratic form on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where the $c_{kl}$ are real symmetric $L_\infty(\Omega)$-functions with $C(x)=(c_{kl}(x))>0$ for almost all $x\in \Omega$. Further assume there are $a, \delta>0$ such that $a^{-1}d_\Gamma^{\delta}\,I\le C\le a\,d_\Gamma^{\delta}\,I$ for $d_\Gamma\le 1$ where $d_\Gamma$ is the Euclidean distance to the boundary $\Gamma$ of $\Omega$. We assume that $\Gamma$ is Ahlfors $s$-regular and if $s$, the Hausdorff dimension of $\Gamma$, is larger or equal to $d-1$ we also assume a mild uniformity property for $\Omega$ in the neighbourhood of one $z\in\Gamma$. Then we establish that $h$ is Markov unique, i.e. it has a unique Dirichlet form extension, if and only if $\delta\ge 1+(s-(d-1))$. The result applies to forms on Lipschitz domains or on a wide class of domains with $\Gamma$ a self-similar fractal. In particular it applies to the interior or exterior of the von Koch snowflake curve in $\mathbf R^2$ or the complement of a uniformly disconnected set in $\mathbf R^d$.