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

Abstract

In this article, we study the asymptotic behavior of the spatial integral of the solution to the hyperbolic Anderson model in dimension d ≤ 2, as the domain of the integral gets large (for fixed time t). This equation is driven by a spatially homogeneous Gaussian noise, whose covariance function is either integrable, or is given by the Riesz kernel. The novelty is that the noise does not depend on time, which means that Itô's martingale theory for stochastic integration cannot be used. Using a combination of Malliavin calculus with Stein's method, we show that with proper normalization and centering, the spatial integral of the solution converges to a standard normal distribution, by estimating the speed of this convergence in the total variation distance. We also prove the corresponding functional limit theorem for the spatial integral process.

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

“…We first prove the convergence of series ( 21) in L 2 (Ω). By relation ( 26) of [6] (which obviously continues to hold for the heat equation), we have:…”

“…In order to do this, we first note that, as in the case of the wave equation (studied in [6]), we have the decomposition:…”

“…As in the case of Theorem 3.4, the main effort will be dedicated to proving that the series (33) converges in L 2 (Ω). To this end, we will use the following estimate, which is proved exactly as in the case of the wave equation (see [6]):…”

“…(i) (Limiting covariance) We use the same argument as in the proof of Theorem 1.3. (i) of [6] (for the wave equation), with some minor differences which we include below. It is enough to prove that…”

“…The QCLT problem for this model with colored noise in time was considered in [5] in dimension d ≤ 2, in cases (a) and (b) above: in case (a), σ 2 R (t) ∼ CR d and d T V ≤ CR −d/2 , while in case (b), σ 2 R (t) ∼ CR 2d−β and d T V ≤ CR −β/2 . The same problem for the time-independent noise has been considered in the recent preprint [6].…”