In this paper, we are interested in the one-dimensional porous medium equation when the initial condition is the distribution function of a probability measure. We associate a nonlinear martingale problem with it. After proving uniqueness for the martingale problem, we show existence owing to a propagation of chaos result for a system of weakly interacting di usion processes. The particle system obtained by increasing reordering from these di usions is proved to solve a stochastic di erential equation with normal re ection. Last, we obtain propagation of chaos for the reordered particles to a probability measure which does not solve the martingale problem but is also linked to the porous medium equation. c 2000 Elsevier Science B.V. All rights reserved. The ow of an ideal gas in a homogeneous porous medium is governed by the following three relations between the density , the pressure p and the speed v: where p 0 ¿ 0; ¿1; ¿ 0 is the porosity of the medium (i.e. the fraction of the volume available to the gas), ¿ 0 is the viscosity of the gas and ¿ 0 the permeability of the medium. Eliminating the pressure and the speed and getting rid of the constants by scaling, we obtain the porous medium equation satisÿed by the scaled density u: This equation also arises in many other applications: population dynamics (Gurtin and Mac Camy, 1977) , theory of ionized gases at high temperature. For a survey of its mathematical properties, see Aronson (1986) . Writing the Laplacian in divergence form,
Jourdain REF studies a system related to a Cauchy problem for the porous medium equation where the initial datum is the distribution function of a probability measure. But this approach is limited to dimension d = 1 and does not cover the interesting case where the initial datum is a probability measure.
Probabilistic approximation for a porous medium equation
Stochastic Processes and their Applications
Mathematics