In this paper we study the (BV, L p )-decomposition, p = 1, 2, of functions in metric random walk spaces, developing a general theory that can be applied to weighted graphs and nonlocal models in Image Processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case p = 1 we also study the associated geometric problem and the thresholding parameters. m J x (A) := A J(x − y)dL N (y) for every Borel set A ⊂ R N , where J : R N → [0, +∞[ is a measurable, nonnegative and radially symmetric function verifying J =1. Furthermore, given a metric measure space (X, d, µ) we can obtain a metric random walk space Date: July 26, 2019.

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,

In this paper we study the {(\mathrm{BV},L^{p})}-decomposition, {p=1,2}, of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case {p=1} we also study the associated geometric problem and the thresholding parameters describing the behavior of its solutions.

In this paper we study evolution problems of Leray-Lions type with nonhomogeneous Neumann boundary conditions in the framework of metric random walk spaces. This includes as particular cases evolution problems with nonhomogeneous Neumann boundary conditions for the p-Laplacian operator in weighted discrete graphs and for nonlocal operators with nonsingular kernel in R N . A J(x − y)dL N (y) for every Borel set A ⊂ R N , where J : R N → [0, +∞[ is a measurable, nonnegative and radially symmetric function with J = 1. See Section 1.1 for more details.The aim of this paper is to study p-Laplacian type evolution problems like the one given by the following reference model:|u(y) − u(x)| p−2 (u(y) − u(x))dm x (y), x ∈ Ω, 0 < t < T,(1.1)

Recently, motivated by problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes, there has been an increasing interest in the research of nonlocal partial differential equations. In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X, d) together with a probability measure assigned to each $$x\in X$$ x ∈ X , which encode the jumps of a Markov process. In this way, we have unified into a broad framework the study of partial differential equations in weighted discrete graphs and in other nonlocal models of interest. Our aim here is to provide a summary of the results that we have obtained for the heat flow and the total variational flow in metric random walk spaces. Moreover, some of our results on other problems related to the diffusion operators involved in such processes are also included, like the ones for evolution problems of p-Laplacian type with nonhomogeneous Neumann boundary conditions.