Gaussian fluctuations for the nonlinear stochastic heat equation with drift

TL;DR

This paper studies Gaussian fluctuations for the spatial average of the one-dimensional nonlinear stochastic heat equation with drift under space-time white noise. Using Malliavin calculus and Vidotto's second-order Gaussian Poincaré inequality, it derives sharp first- and second-Malliavin-derivative bounds, including a novel heat-kernel product estimate, to establish a quantitative central limit theorem for the spatial average $F_R(t)$. It further proves a functional CLT, showing that $R^{-1/2}F_R(\cdot)$ converges in $C([0,T])$ to a zero-mean Gaussian process with covariance given by the limiting spatial covariances, and it demonstrates ergodicity with a finite limiting covariance. The methods introduce a kernel-product tool of independent interest and extend QCLT/FCLT techniques to SPDEs with drift, providing a robust framework for fluctuations in nonlinear SPDEs.

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 Poincaré 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.

Theorems & Definitions (26)

• Definition 1.1
• Theorem 1.4
• Theorem 1.5
• Lemma 3.1
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof