hal-00723821, version 1

## Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation.

Nadia Belaribi () a12, François Cuvelier () a2, Francesco Russo (, ) 1

Résumé : The object of this paper is a multi-dimensional generalized porous media equation (PDE) with not smooth and possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R}^d)$. This work continues the study related to the one-dimensional case by the same authors. One expects that a solution of the mentioned PDE can be represented through the solution (in law) of a non-linear stochastic differential equation (NLSDE). A classical tool for doing this is a uniqueness argument for some Fokker-Planck type equations with measurable coefficients. When $\beta$ is possibly discontinuous, this is often possible in dimension $d = 1$. If $d > 1$, this problem is more complex than for $d = 1$. However, it is possible to exhibit natural candidates for the probabilistic representation and to use them for approximating the solution of (PDE) through a stochastic particle algorithm. We compare it with some numerical deterministic techniques that we have implemented adapting the method of a paper of Cavalli et al. whose convergence was established when $\beta$ is Lipschitz. Special emphasis is also devoted to the case when the initial condition is radially symmetric. On the other hand assuming that $\beta$ is continuous (even though not smooth), one provides existence results for a mollified version of the (NLSDE) and a related partial integro-differential equation, even if the initial condition is a general probability measure.

• Collaboration : EPFL Lausanne, SFB 701 Universitaet Bielefeld
• Domaine : Mathématiques/Probabilités
• Mots-clés : Stochastic particle algorithm – porous media equation – monotonicity – stochastic differential equations – non-parametric density estimation – kernel estimator

• hal-00723821, version 1
• oai:hal.inria.fr:hal-00723821
• Contributeur :
• Soumis le : Lundi 20 Août 2012, 10:44:26
• Dernière modification le : Lundi 20 Août 2012, 11:06:05