Quantitative normal approximations for the stochastic fractional heat equation

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.

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

“…In order to do this, we follow the same general strategy as in [6], namely we first identify the order of magnitude of σ 2 R (t), and then use the bound given by Proposition 1.8 of [6] for the distance d T V (F R (t)/σ R (t), Z), which is valid also for the time-independent noise. 1 A key idea, which is common to all references who studied this problem, is to show that the moments of the first and second Malliavin derivatives of u(t, x) are dominated, respectively, by the first two chaos kernels f 1 (•, x; t) and f 2 (•, x; t) which appear in the chaos expansion of the solution. We will achieve this too, in relations ( 24) and (38) below.…”

“…The parabolic Anderson model (corresponding to the case σ(u) = u) with the same noise and delta initial condition was studied in [11]. 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].…”