Renormalization flow for the 2D nonlinear stochastic heat equation: pointwise statistics and universality

Abstract

We consider a two-dimensional stochastic heat equation with noise correlated at scale $\ddot{I}\ll 1$ and of strength $|\log \ddot{I}{|}^{-1/2}\ddot{I}ƒ(v)$ depending nonlinearly on the solution $v$. Under certain conditions, the first author and Gu have shown that the one-point statistics of $v$ converge in law as $\ddot{I}\to 0$ to the terminal value of an associated forward-backward SDE. Here, we show that the 2D stochastic heat equation is stable under renormalization with a new effective nonlinearity tied to the decoupling function of the forward-backward SDE. This allows us to extend the pointwise results to a much broader class of nonlinearities. We also show that these limiting pointwise statistics are insensitive to the fine details of the noise, and thus universal.

91 pages; 2 figures. v3: expositional improvements