Torsional rigidity in random walk spaces

Abstract

In this paper we study the (nonlocal) torsional rigidity in the ambient space of random walk spaces. We get the relation of the (nonlocal) torsional rigidity of a set $\Omega$ with the spectral $m$-heat content of $\Omega$, what gives rise to a complete description of the nonlocal torsional rigidity of $\Omega$ by using uniquely probability terms involving the set $\Omega$; and recover the first eigenvalue of the nonlocal Laplacian with homogeneous Dirichlet boundary conditions by a limit formula using these probability term. For the random walk in $\R^N$ associated with a non singular kernel, we get a nonlocal version of the Saint-Venant inequality, and, under rescaling we recover the classical Saint-Venant inequality. We study the nonlocal $p$-torsional rigidity and its relation with the nonlocal Cheeger constants. We also get a nonlocal version of the P\'{o}lya-Makai-type inequalities. We relate the torsional rigidity given here for weighted graphs with the torsional rigidity on metric graphs.