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

TL;DR

This paper studies the 2D stochastic heat equation with correlated noise and nonlinearities, demonstrating stability under renormalization and establishing universality of pointwise statistics in the small-scale limit.

Contribution

It introduces a renormalization framework for the 2D stochastic heat equation, extending pointwise statistical results to broader nonlinearities and proving universality.

Findings

Pointwise statistics converge to a limit described by a forward-backward SDE.

The equation is stable under renormalization with an effective nonlinearity.

Limiting statistics are universal, insensitive to noise details.

Abstract

We consider a two-dimensional stochastic heat equation with noise correlated at scale $\rho \ll 1$ and of strength $|\log\rho|^{-1/2}\sigma(v)$ 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 $\rho\to 0$ 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.