Abstract

We consider a porous media type equation (PME) over the real line with monotone discontinuous coefficient and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion.

We will distinguish between two different situations: the so-called {f non-degenerate} and {f degenerate} cases. In the first case we show existence and uniqueness, however in the second one for which we only show existence. One of the main analytic ingredients of the proof (in the non-degerate case) is a new result on uniqueness of distributional solutions of a linear PDE on $R^1$ with non-continuous coefficients. In the degenerate case, the proofs require a careful analysis of the deterministic (PME) equation. Some comments about an associated stochastic PDE with multiplicative noise will be provided.

This talk is based partly on two joint papers: the first with Ph. Blanchard and M. R”ockner, the second one with V. Barbu and M. R”ockner}.