Dirichlet forms for singular diffusion in higher dimensions

TL;DR

This paper develops a mathematical framework using Dirichlet forms to model singular diffusion processes in higher-dimensional bounded regions, combining classical diffusion with jump behavior across different parts of the domain.

Contribution

It introduces a novel approach to describe singular diffusion in higher dimensions via form methods and characterizes the associated operators and semigroups.

Findings

Positivity and contractivity of the semigroup established

Operator characterization for singular diffusion in higher dimensions

Description of stochastic processes with mixed diffusion and jump behavior

Abstract

We describe singular diffusion in bounded subsets $\Omega$ of $\mathbb{R}^n$ by form methods and characterize the associated operator. We also prove positivity and contractivity of the corresponding semigroup. This results in a description of a stochastic process moving according to classical diffusion in one part of $\Omega$, where jumps are allowed through the rest of $\Omega$.