{Probabilistic representation for solutions of an irregular porous media type equation.

Philippe Blanchard (1) , Michael Röckner (2) , Francesco Russo (3)

1 Fakultät für Physik
2 SFB 705 - Fakultät für Mathematik [Bielefeld]
3 LAGA - Laboratoire Analyse, Géométrie et Applications

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.

Mots clés: self-organized criticality (SOC); singular porous media type equation; probabilistic representation; self-organized criticality (SOC).

Domaines: Probabilités [math.PR]

Philippe Blanchard, Michael Röckner, Francesco Russo. {Probabilistic representation for solutions of an irregular porous media type equation.. 2008. ⟨hal-00279975v1⟩