Gaussian fluctuations for the stochastic heat equation with colored noise

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. (2010): 60H15, 60H07, 60G15, 60F05. Mathematics Subject Classifications

Quantitative normal approximations for the stochastic fractional heat equation

“…Our results continue the line of research initiated in [12,13] where a similar problem for the stochastic heat equation on R (or R d , respectively) driven by a space-time white noise (or spatial covariance given by the Riesz kernel, respectively) was considered. As such, we extend the results presented in [12,13] as the main theorems of [12,13] can be recovered from ours by simply plugging in α = 2. Proof-wise our methods are similar to those of these two references.…”

“…For a detailed exposure on the topic in the case of the stochastic heat equation (α = 2), we refer to [4]. Similarly, in the spirit of [13,Corollary 3.3], our approximation results can be generalised to the case of u(0, x) = f (x) with suitable assumptions on the function f , once a comparison principle is established.…”

Averaging Gaussian functionals

“…Remark 1.8. Unlike previous studies, we consider a noise that is colored in time, and our results complement, in particular, those in [14,15]. In [14] where the noise is white in space and time, the authors were able to obtain the chaotic central limit theorem for the linear equation (parabolic Anderson model), proving also a rate of convergence in the total variation distance.…”

Spatial averages for the parabolic Anderson model driven by rough noise

“…Observe that unlike in the papers Huang et al (2020b); Delgado-Vences et al (2020), the variance order of A t (R) is R, which does not depend on the parameters of the covariance, for example, the Hurst index H 1 in the setting (H2). This is due to our assumption ϕ(0) = 0 in both settings of (H1) and (H2), that forces the negligibility of the first chaotic component of A t (R) in the limit, while the other chaoses contribute to the order R. This situation is completely different from the case H 1 > 1/2, where a nonchaotic behavior occurs.…”

“…Other key tools used in Huang et al (2020a) are Clark-Ocone formula, Burkholder's inequality and they essentially rely on the assumption that the underlying Gaussian noise is white in time so as to render us a martingale structure. Soon later, the authors of Huang et al (2020b) consider the same equation with spatial dimension d ≥ 1; while imposing that the Gaussian noise is white in time and it has a spatial covariance given by the Riesz kernel, they establish a functional CLT and a quantitative CLT for spatial averages. We also refer interested readers to several other investigations on stochastic heat equations in Chen et al (2019Chen et al ( , 2020; Gu and Li (2020); Khoshnevisan et al (2020); Pu (2020) and on stochastic wave equations in Nualart and Zheng (2020b); Delgado-Vences et al (2020); Nualart and Zheng (2020c); these papers more or less follow the path paved by the work Huang et al (2020a), although the nature of the problems and the techniques differ.…”