PUB - Publikationen an der Universität Bielefeld

Given a subset D of the Euclidean space, we study nonlocal quadratic forms that take into account tuples (x, y) is an element of D x 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 Poincare inequalities and their scaling properties. For dumbbell shaped domains we show that the forms satisfy a Poincare inequality with diffusive scaling. This relates to the rate of convergence of eigenvalues in singularly perturbed domains.