Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case

TL;DR

This paper develops a probabilistic framework for solving a possibly degenerate porous media equation with discontinuous coefficients, motivated by complex critical systems, by approximating it through non-degenerate equations.

Contribution

It introduces a novel probabilistic representation for solutions of degenerate porous media equations with discontinuous coefficients, including new analytical properties derived from approximation methods.

Findings

Established a probabilistic representation for the degenerate equation

Proved analytical properties of solutions via approximation

Addressed singular behaviors in complex systems

Abstract

We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. The main idea consists in approximating the equation by equations with monotone non-degenerate coefficients and deriving some new analytical properties of the solution.