# Probabilistic Solution of the Porous Medium Equation

It is well known that the transition density for standard Brownian motion $B_t$ in $\mathbb{R}^d$ yields a solution to the global Cauchy problem for the heat equation $$u_t = \Delta u$$ with initial condition given by the Dirac distribution $\delta_0$.

Unlike the heat equation, the porous medium equation $$u_t = \Delta(u^m)$$ with exponent $m>1$ has finite speed of propagation.

• When is the infinitesimal generator of a stochastic process linear?

• Is there a probabilistic solution of this non-linear diffusion equation?

• Would anyone who has looked at any of the material related to Hunt and Ewing's "Percolation Theory for Flow in Porous Media" care to weigh in here? – Alexander Moll Jul 13 '11 at 14:15
• André's response below does answer the original question. Still, I'd like to know - is there some aspect of say bond percolation on the lattice $\mathbb{Z}^d$ that satisfies the discrete version of the equation above? I'd guess that $m>1$ would entail $p < p_c$... – Alexander Moll Jul 15 '11 at 22:23

A probabilisitc solution is given by MR1469575 a nonlinear diffusion $$Y_t = Y_0 + \int_0^t u^{\frac{m-1}{2}}(s,Y_s)\; \mathrm{d}W_s , \qquad \mathrm{law}(Y_0) = u(0,\cdot)$$ then $$\mathrm{law}(Y_t) = u(t,\cdot) .$$ This is true for a general class of nonlinear diffusion equations. The best references I've found are MR1775228 and MR2722788.
$dX^i = -\sum_{j\neq i} \nabla W_\epsilon(X^i-X^j) \, dt + \delta \, dB^i.$
The parameter $\epsilon$ is the spatial range of $W$, and in the limit $\epsilon\to0$ the interaction becomes purely local, and leads to a nonlinear diffusion term. If one also lets $\delta\to0$, then the purely Brownian contribution also vanishes. Only the nonlinear diffusion is then left.