The Heat Flow on Metric Random Walk Spaces

Abstract

In this paper we study the Heat Flow on Metric Random Walk Spaces, which unifies into a broad framework the heat flow on locally finite weighted connected graphs, the heat flow determined by finite Markov chains and some nonlocal evolution problems. We give different characterizations of the ergodicity and prove that a metric random walk space with positive Ollivier-Ricci curvature is ergodic. Furthermore, we prove a Cheeger inequality and, as a consequence, we show that a Poincaré inequality holds if, and only if, an isoperimetric inequality holds. We also study the Bakry-Émery curvature-dimension condition and its relation with functional inequalities like the Poincaré inequality and the transport-information inequalities.

References in corpus (4)

• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Transportation cost-information inequalities and applications to random dynamical systems and diffusions
• Ricci curvature of finite Markov chains via convexity of the entropy
• Non-Markov property of certain eigenvalue processes analogous to Dyson's model

Cited by in corpus (4)

• The Total Variation Flow in Metric Random Walk Spaces
• Evolution problems of Leray-Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces
• $(BV,L^p)$-decomposition, $p=1,2$, of Functions in Metric Random Walk Spaces
• Nonlocal doubly nonlinear diffusion problems with nonlinear boundary conditions