A class of stochastic evolutions that scale to the porous medium equation
A class of reversible Markov jump processes on a periodic lattice is described. The process can be viewed as a randomly interacting configuration of sticks that evolves through exchanges of stick pieces between nearest neighbours. Namely, after an exponentially distributed time the stick \(x_k\) at a site \(k\in Z_N^d\), where \(Z_N^d\) is a finite-dimensional cube lattice, breaks and transfers a randomly long piece \(u\) to a stick at a randomly chosen neighbouring site \(\ell\) (the distribution depends on a parameter \(\alpha>1\)). Let \(\mu_t^N(d\theta)\) be a random measure induced by the above described random evolution. Then the main result of the paper states that \(\mu_{N^2t}^N(d\theta)\) converges in probability to a weak solution \(u(t,\theta)\) of the porous medium equation which is the nonlinear heat equation \(\partial_t u = \kappa \Delta(u^\alpha)\) on the \(d\)-dimensional torus.
scientific article; zbMATH DE number 1279723