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 we also 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 isoperimetrical 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. µ * m(A) := X m x (A)dµ(x), for all Borel sets A ⊂ X,

Least gradient functions in metric random walk spaces

“…] is a metric random walk space and it easy to see ( [17]) that ν Ω is reversible for m Ω . In the case that Ω is a closed bounded subset of R N , we obtain in this way the metric random walk…”

Gradient flows in metric random walk spaces

“…Specifically, together with the existence and uniqueness of solutions to the aforementioned problems, a wide variety of their properties have been studied (some of which are listed in the contents section), as well as the nonlocal diffusions operators involved in them. This survey is mainly based on the results that we have obtained in [46,47] (see also the more recent work [55]) and [48]. Related to the above problems, we have also studied (see [49]) the (BV , L p )-decomposition, p = 1 and p = 2, of functions in metric random walk spaces.…”

Evolution problems of Leray–Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces

“…Here, the homogeneous Neumann boundary conditions are understood in the sense that the jumps of the Markov chain are restricted to staying in Ω (which is consistent with what happens in the classical local model). See also [20,Example 2.3] for the linear case, i.e., p = 2, in metric random walk spaces. The linear case with nonhomogeneous boundary conditions has been addressed by different authors.…”

“…When dealing with a metric random walk space [X, d, m], we will assume that there exists an invariant and reversible measure for the random walk, which we will always denote by ν, such that m x ≪ ν for all x ∈ X. Moreover, we will assume that the metric random walk space with the measure ν is m-connected (see [20]). Important examples of metric random walk spaces are the following:…”