# Gaussian fluctuations for the stochastic heat equation with colored noise

## TL;DR

This paper establishes a quantitative central limit theorem for the spatial average of solutions to a d-dimensional stochastic heat equation driven by Gaussian noise with Riesz kernel covariance, demonstrating Gaussian fluctuations as the averaging domain grows.

## Contribution

It introduces a new quantitative CLT for the stochastic heat equation with colored noise, using Malliavin calculus and Stein's method, including a functional version.

## Key findings

- Spatial averages converge to Gaussian distribution as domain size increases.
- Quantitative bounds in total variation distance are provided.
- A functional CLT for the solution process is established.

## Abstract

In this paper, we present a quantitative central limit theorem for the d-dimensional stochastic heat equation driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. We show that the spatial average of the solution over an Euclidean ball is close to a Gaussian distribution, when the radius of the ball tends to infinity. Our central limit theorem is described in the total variation distance, using Malliavin calculus and Stein's method. We also provide a functional central limit theorem.