Quantitative normal approximations for the stochastic fractional heat equation

Obayda Assaad,David Nualart,Lauri Viitasaari,Ciprian A Tudor

doi:10.1007/s40072-021-00198-7

Abstract

In this article we present a quantitative central limit theorem for the stochastic fractional heat equation driven by a a general Gaussian multiplicative noise, including the cases of space–time white noise and the white-colored noise with spatial covariance given by the Riesz kernel or a bounded integrable function. We show that the spatial average over a ball of radius R converges, as R tends to infinity, after suitable renormalization, towards a Gaussian limit in the total variation distance. We also provide a functional central limit theorem. As such, we extend recently proved similar results for stochastic heat equation to the case of the fractional Laplacian and to the case of general noise.

Highlights

In this article we consider the stochastic fractional heat equation ∂u (t, x) = −(− ) α 2 u (t, x + σ (u ))W), t ≥ 0, x ∈ Rd (1.1)∂t with initial condition u(0, x) ≡ 1
Our main contribution is in providing quantitative limit theorems in a general context. These results cover three different important situations: when Wis a standard space–time white noise, when Wis a white-colored noise, i.e. a Gaussian field that behaves as a Wiener process in time and it has a non-trivial spatial covariance given by the Riesz kernel of order β < min(α, d), and when Wis a white-colored noise with spatial covariance given by an integrable and bounded function γ
Our results continue the line of research initiated in [12,13] where a similar problem for the stochastic heat equation on R driven by a space–time white noise was considered

Summary

Introduction

∂t with initial condition u(0, x) ≡ 1. In the present article we provide a general existence and uniqueness result to equation (1.1) that covers many different choices of the (Gaussian) random perturbation W. Our main contribution is in providing quantitative limit theorems in a general context These results cover three different important situations: when Wis a standard space–time white noise, when Wis a white-colored noise, i.e. a Gaussian field that behaves as a Wiener process in time and it has a non-trivial spatial covariance given by the Riesz kernel of order β < min(α, d), and when Wis a white-colored noise with spatial covariance given by an integrable and bounded function γ. Our results continue the line of research initiated in [12,13] where a similar problem for the stochastic heat equation on R (or Rd , respectively) driven by a space–time white noise (or spatial covariance given by the Riesz kernel, respectively) was considered.

Main results

Preliminaries

Malliavin calculus and Stein’s method

On fractional Green kernel

A basic inequality

Bound for the Malliavin derivative