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.

中文翻译：

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有色噪声随机热方程的高斯涨落

在本文中，我们提出了由高斯乘性噪声驱动的d维随机热方程的定量中心极限定理，该方程时间为白色，并且具有Riesz核给出的空间协方差。我们表明，当球的半径趋于无穷大时，欧氏球上解的空间平均值接近于高斯分布。使用Malliavin微积分和Stein方法，在总变化距离中描述了我们的中心极限定理。我们还提供了一个功能性中心极限定理。