A forward-backward SDE from the 2D nonlinear stochastic heat equation

TL;DR

This paper studies the asymptotic behavior of solutions to a 2D nonlinear stochastic heat equation with weak noise, showing convergence of one-point distributions to a limit described by a forward-backward SDE, and extends known results even in the linear case.

Contribution

It introduces a novel approach to characterize the limiting distribution of the solution via a forward-backward SDE in the weak noise regime for the 2D nonlinear stochastic heat equation.

Findings

One-point distribution converges as noise vanishes.

Limiting multipoint statistics are characterized similarly.

Explicit solution of the FBSDE in the linear case recovers previous results.

Abstract

We consider a nonlinear stochastic heat equation in spatial dimension $d=2$, forced by a white-in-time multiplicative Gaussian noise with spatial correlation length $\varepsilon>0$ but divided by a factor of $\sqrt{\log\varepsilon^{-1}}$. We impose a condition on the Lipschitz constant of the nonlinearity so that the problem is in the "weak noise" regime. We show that, as $\varepsilon\downarrow0$, the one-point distribution of the solution converges, with the limit characterized in terms of the solution to a forward-backward stochastic differential equation (FBSDE). We also characterize the limiting multipoint statistics of the solution, when the points are chosen on appropriate scales, in similar terms. Our approach is new even for the linear case, in which the FBSDE can be solved explicitly and we recover results of Caravenna, Sun, and Zygouras (Ann. Appl. Probab. 27(5):3050--3112, 2017).