Quantitative central limit theorems for the parabolic Anderson model driven by colored noises

Abstract

In this paper, we study the spatial averages of the solution to the parabolic Anderson model driven by a space-time Gaussian homogeneous noise that is colored in time and space. We establish quantitative central limit theorems (CLT) of this spatial statistics under some mild assumptions, by using the Malliavin-Stein approach. The highlight of this paper is the obtention of rate of convergence in the colored-in-time setting, where one can not use Itô's calculus due to the lack of martingale structure. In particular, modulo highly technical computations, we apply a modified version of second-order Gaussian Poincaré inequality to overcome this lack of martingale structure and our work improves the results by Nualart-Zheng (2020 Electron.

Central limit theorems for heat equation with time-independent noise: the regular and rough cases

“…The method of [26] uses the multivariate chaotic central limit theorem, which yields the normal approximation, but does not give the rate of the convergence. This rate was obtained in the recent preprint [24], where it was shown that d T V ≤ CR −d/2 in case (a) and d T V ≤ CR −β/2 in case (b), using a different method based on an improved version of the second-order Gaussian Poincaré inequality due to [28], which was used for the first time in this context in [5]. In addition, [24] considers the more difficult case (c) of the fractional noise in space with index H < 1/2 (in dimension d = 1), which is also fractional in time with index H 0 > 1/2 satisfying H 0 + H > 3/4.…”

“…This rate was obtained in the recent preprint [24], where it was shown that d T V ≤ CR −d/2 in case (a) and d T V ≤ CR −β/2 in case (b), using a different method based on an improved version of the second-order Gaussian Poincaré inequality due to [28], which was used for the first time in this context in [5]. In addition, [24] considers the more difficult case (c) of the fractional noise in space with index H < 1/2 (in dimension d = 1), which is also fractional in time with index H 0 > 1/2 satisfying H 0 + H > 3/4. In this case, the authors discovered the surprising rates σ 2 R (t) ∼ CR and d T V ≤ CR −1/2 , which show that once we descend below the value 1/2, the spatial index H does not have any impact on the rates in the QCLT.…”

“…A similar inequality holds for g n . To see this, we recall the following estimate which was proved in [24]:…”

“…In this case, the authors discovered the surprising rates σ 2 R (t) ∼ CR and d T V ≤ CR −1/2 , which show that once we descend below the value 1/2, the spatial index H does not have any impact on the rates in the QCLT. In the present article, using similar methods to [24], we consider the QCLT problem for the parabolic Anderson model with time-independent noise (which corresponds formally to the case H 0 = 1), and spatial covariance given by cases (a),(b),(c) above. In addition, we prove the corresponding functional limit result.…”

Spatial integral of the solution to hyperbolic Anderson model with time-independent noise

“…The same problem for the fractional heat equation (in which the Laplacian is replaced by its fractional power) has been considered in [1]. The case of the parabolic Anderson model driven by a Gaussian noise colored in time was treated in [26,25], and the same model with rough noise in space appeared in [24].…”