A probabilistic algorithm approximating solutions of a singular PDE of porous media type

Abstract

The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient $β$, which is well-posed as an evolution problem in $L^1(\mathbb{R})$. In some recent papers of Blanchard et alia and Barbu et alia, the solution was represented by the solution of a non-linear stochastic differential equation in law if the initial condition is a bounded integrable function. We first extend this result, at least when $β$ is continuous and the initial condition is only integrable with some supplementary technical assumption. The main purpose of the article consists in introducing and implementing a stochastic particle algorithm to approach the solution to (PDE) which also fits in the case when $β$ is possibly irregular, to predict some long-time behavior of the solution and in comparing with some recent numerical deterministic techniques.