Nonlocal quadratic forms with visibility constraint

TL;DR

This paper investigates nonlocal quadratic forms constrained by visibility within a domain, establishing regularity, existence of associated jump processes, and Poincaré inequalities, especially in complex geometries like dumbbell-shaped domains.

Contribution

It introduces a new class of nonlocal forms with visibility constraints, analyzes their regularity, and derives Poincaré inequalities with specific scaling properties.

Findings

Existence of jump processes with visibility constraints.

Poincaré inequalities hold with diffusive scaling in certain domains.

Results relate to eigenvalue convergence in perturbed domains.

Abstract

Given a subset $D$ of the Euclidean space, we study nonlocal quadratic forms that take into account tuples $(x,y) \in D \times D$ if and only if the line segment between $x$ and $y$ is contained in $D$. We discuss regularity of the corresponding Dirichlet form leading to the existence of a jump process with visibility constraint. Our main aim is to investigate corresponding Poincar\'{e} inequalities and their scaling properties. For dumbbell shaped domains we show that the forms satisfy a Poincar\'{e} inequality with diffusive scaling. This relates to the rate of convergence of eigenvalues in singularly perturbed domains.