Transport and localization in random media: theory and applications

The random heat equation in dimensions three and higher

Lenya Ryzhik

Stanford University


We consider the heat equation with a multiplicative Gaussian potential in dimensions three and higher. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards--Wilkinson model, that is, the stochastic heat equation (SHE) with an additive white noise, with an effective diffusion and an effective variance. This is a joint work with Alex Dunlap, Yu Gu and Ofer Zeitouni.