Probabilistic representation for solutions of an irregular porous media type equation

TL;DR

This paper develops a probabilistic framework for solving a one-dimensional porous media equation with discontinuous coefficients, motivated by complex critical systems, and introduces a new uniqueness result for related linear PDEs.

Contribution

It provides a novel probabilistic representation for solutions of irregular porous media equations and establishes a new uniqueness theorem for linear PDEs with non-continuous coefficients.

Findings

Probabilistic representation of solutions for irregular porous media equations.

New uniqueness result for linear PDEs with discontinuous coefficients.

Application to complex self-organized critical systems.

Abstract

We consider a 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. One of the main analytic ingredients of the proof, is a new result on uniqueness of distributional solutions of a linear PDE on $\R^1$ with non-continuous coefficients.