Probabilistic solution of the porous medium equation

Question

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?

Answer 1

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.

update 22.05.2023: A rescaled zero-range process for the porous medium equation

In the recent preprint arXiv:2304.11189, Gess and Heydecker show that the nearest neighbor zero-range process with jump rates $k^\alpha$ on the torus of size $N$ converges after some suitable rescaling of particle size and time to the solution of the PME $\partial_t u = \frac{1}{2} \Delta u^\alpha$.

Answer 2

The processes described by Andre work by having the interaction act at the level of the mobility of the particles.

There are other ways, too. One is in the work of Philipowski (see also Figalli & Philipowski). Here the idea is to take interactions of potential type, i.e. for instance

$$ dX^i = -\sum_{j\neq i} \nabla W_\epsilon(X^i-X^j) \, dt + \delta \, dB^i. $$

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.