Nonlocal quadratic forms with visibility constraint

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 Poincaré inequalities and their scaling properties. For dumbbell shaped domains we show that the forms satisfy a Poincaré inequality with diffusive scaling. This relates to the rate of convergence of eigenvalues in singularly perturbed domains.