Stochastic models associated to a Nonlocal Porous Medium Equation

TL;DR

This paper explores the connection between nonlocal porous medium equations involving fractional operators and stochastic processes like random flights, providing insights into their probabilistic interpretations and solutions.

Contribution

It establishes a link between space-fractional porous medium equations and isotropic transport processes, offering a probabilistic interpretation of their solutions.

Findings

Explicit nonnegative, compactly supported solutions representing probability densities.

Interpretation of weak solutions via random flights.

Analysis of the relation between stochastic processes and nonlocal diffusion.

Abstract

The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative, compactly supported weak solution representing a probability density function. In this paper we analyze the link between isotropic transport processes, or random flights, and the nonlocal porous medium equation. In particular, we focus our attention on the interpretation of the weak solution of the nonlinear diffusion equation by means of random flights.