Gaussian fluctuations for the nonlinear stochastic heat equation with drift

TL;DR

This paper establishes a Quantitative Central Limit Theorem for the spatial average of solutions to a nonlinear stochastic heat equation with drift, using Malliavin calculus and novel heat kernel estimates.

Contribution

It introduces a new CLT for the nonlinear stochastic heat equation with drift, employing advanced Malliavin calculus techniques and heat kernel product estimates.

Findings

Proves a QCLT for the solution's spatial average in dimension 1.

Develops a novel heat kernel product estimate of independent interest.

Provides a functional CLT corresponding to the main result.

Abstract

In this article, we prove the Quantitative Central Limit Theorem (QCLT) for the spatial average of the solution of the nonlinear stochastic heat equation with constant initial condition, driven by space-time Gaussian white noise in dimension 1. The novelty is that the equation contains a drift term. We assume that the drift and diffusion coefficients are twice differentiable with bounded first and second order derivatives. For the proof, we use Malliavin calculus, and the second-order Poincar\'e inequality due to Vidotto (2020). To estimate the moment of the second Malliavin derivative of the solution, we develop a novel estimate for the product of two heat kernels, which is of independent interest. Finally, we provide the functional result corresponding to this CLT.