Probabilistic representation for solutions of a porous media type equation with Neumann boundary condition: the case of the half-line

TL;DR

This paper introduces a probabilistic framework for solving a porous media equation with Neumann boundary conditions on a half-line, using reflected stochastic differential equations to represent solutions.

Contribution

It develops a generalized solution concept and establishes a probabilistic representation via reflected stochastic differential equations for the porous media equation.

Findings

Existence of a solution in law for the reflected SDE.

Unique probabilistic representation of the porous media solution.

Connection between microscopic diffusion and macroscopic PDE solution.

Abstract

The purpose of this paper consists in proposing a generalized solution for a porous media type equation on a half-line with Neumann boundary condition and prove a probabilistic representation of this solution in terms of an associated microscopic diffusion. The main idea is to construct a stochastic differential equation with reflection which has a solution in law and whose marginal law densities provide the unique solution of the porous media type equation.