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for a given real number α > 1, where Δ is the d -dimensional Laplacian. This equation was introduced by Muskat as an (empirical) equation of the density u of a gas flowing through a homogeneous porous medium and is called a porous medium equation ([!]). Analogously to Kac's approach to a Boltzmann equation [10] we introduce a Markov system of many particles as a simple model of the gas. The porous medium equation (1) is derived from the equation for the empirical density of the number of particles. We prove that a macroscopic limit of the empirical density is a solution of (1). We also prove Kac-McKean's propagation of chaos for the system as follows. Let Sh = {(hzl9 •••, hzd): z l 5 •••, zdεZ} be a d -dimensional lattice of the width h > 0, and τ > 0 be a unit time. We define a system of N-particles on Sh with the following stochastic interaction. For each integer n > 0, let

Derivation of a porous medium equation from many Markovian particles and the propagation of chaos