The hyperbolic Anderson model: moment estimates of the Malliavin derivatives and applications

Abstract

In this article, we study the hyperbolic Anderson model driven by a space-time colored Gaussian homogeneous noise with spatial dimension $$d=1,2$$ d = 1 , 2 . Under mild assumptions, we provide $$L^p$$ L p -estimates of the iterated Malliavin derivative of the solution in terms of the fundamental solution of the wave solution. To achieve this goal, we rely heavily on the Wiener chaos expansion of the solution. Our first application are quantitative central limit theorems for spatial averages of the solution to the hyperbolic Anderson model, where the rates of convergence are described by the total variation distance. These quantitative results have been elusive so far due to the temporal correlation of the noise blocking us from using the Itô calculus. A novel ingredient to overcome this difficulty is the second-order Gaussian Poincaré inequality coupled with the application of the aforementioned $$L^p$$ L p -estimates of the first two Malliavin derivatives. Besides, we provide the corresponding functional central limit theorems. As a second application, we establish the absolute continuity of the law for the hyperbolic Anderson model. The $$L^p$$ L p -estimates of Malliavin derivatives are crucial ingredients to verify a local version of Bouleau-Hirsch criterion for absolute continuity. Our approach substantially simplifies the arguments for the one-dimensional case, which has been studied in the recent work by [2].

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.…”

“…and we say that δ(u) is the Skorohod integral of u with respect to W . If F has the chaos expansion (5), we define the Ornstein-Uhlenbeck generator…”

“…for any non-negative function h such that h ∈ P 0 ∩ L 1 (R d ) or F h ≥ 0 (see Lemma 2.6 of [5] and Lemma 3.6 of [11]). In particular, this holds for h(t) = e −t|ξ| 2 for any t > 0.…”

“…(ii) (QCLT) We apply a version of Proposition 1.8 of [5] for the time-independent noise, and we use the key estimates for Du and D 2 u given by ( 29) and (35). We omit the details, as they are very similar to the case of the wave equation (see the proof of Theorem 1.3.…”

“…Similarly to the heat equation, for the wave equation with colored noise in time, the existence of solution is an open problem, except in the case σ(u) = u (known as the hyperbolic Anderson model). 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].…”