Torsional Rigidity in Random Walk Spaces

.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 \(Ω\) with the spectral \(m\)-heat content of \(Ω\), which gives rise to a complete description of the nonlocal torsional rigidity of \(Ω\) by using uniquely probability terms involving the set \(Ω\), and we recover the first eigenvalue of the nonlocal Laplacian with homogeneous Dirichlet boundary conditions by a limit formula using these probability terms. For the random walk in \( R^N\) associated with a nonsingular kernel, we get a nonlocal version of the Saint-Venant inequality, and, under rescaling, we recover the classical Saint-Venant inequality. We also study the nonlocal \(p\)-torsional rigidity and its relation with the nonlocal Cheeger constants, and we prove a nonlocal version of the Pólya–Makai-type inequalities. We relate the torsional rigidity given here for weighted graphs with the torsional rigidity on metric graphs.Keywordstorsion rigidityrandom walksweighted graphsSaint-Venant inequalityFaber–Krahn inequalityMSC codes35K5547H0647J35