Gradient flows in random walk spaces

  1. Solera Diana, Marcos
Supervised by:
  1. José Manuel Mazón Ruiz Director
  2. José Julián Toledo Melero Co-director

Defence university: Universitat de València

Fecha de defensa: 24 September 2021

Committee:
  1. Noureddine Igbida Chair
  2. Salvador Moll Secretary
  3. María del Mar González Nogueras Committee member
Department:
  1. MATHEMATICAL A

Type: Thesis

Abstract

The aim of this thesis is to unify into a broad framework the study of a wide range of nonlocal diffusion processes that have recently arisen in various scientific fields, as well as the study of partial differential equations on graphs. In order to do so, we note that there is a strong relation between some of these problems and probability theory, and it is in this field in which we find the appropriate spaces in which to develop this unifying study. Indeed, this possibility is provided by random walk spaces. We study some gradient flows in the general framework of a random walk space. In particular, we study the heat flow, the total variation flow, and evolutions problems of Leray-Lions type with different types of nonhomogeneous boundary conditions. Specifically, together with the existence and uniqueness of solutions to these problems and the asymptotic behavior of its solutions, a wide variety of their properties are studied, as well as the nonlocal diffusion operators involved in them.