Gradient flows in random walk spaces

Resumo

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.