A probabilistic algorithm approximating solutions of a singular PDE of porous media type. (arXiv:1011.3107v1 [math.PR])

A probabilistic algorithm approximating solutions of a singular PDE of porous media type. (arXiv:1011.3107v1 [math.PR]): "

The object of this paper is a one-dimensional generalized porous media
equation (PDE) with possibly discontinuous coefficient $\beta$, 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 $\beta$ 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 $\beta$ is
possibly irregular, to predict some long-time behavior of the solution and in
comparing with some recent numerical deterministic techniques.

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