Gradient flows in metric random walk spaces

• Published in: SeMA journal Vol. 79; no. 1; pp. 3 - 35
• Main Authors: Mazón, José M., Solera, Marcos, Toledo, Julián
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Boundary conditions; Gradient flow; Heat transmission; Image processing; Markov processes; Mathematics; Mathematics and Statistics; Metric space; Operators (mathematics); Partial differential equations; Random walk; 05C21; Calibrable sets; Functions of bounded variation; 26A45; Random walk; Sets of finite perimeter; 45C99; Total variation flow; 35R02; 05C80; Cheeger sets; Nonlocal operators
• ISSN: 2254-3902; 2281-7875
• DOI: 10.1007/s40324-021-00272-z

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 ∈ 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.