Spatial averages for the parabolic Anderson model driven by rough noise

Nualart, David; Song, Xiaoming; Zheng, Guangqu

doi:10.30757/alea.v18-33

Cited by 7 publications

(15 citation statements)

References 23 publications

(31 reference statements)

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“…(i) (Limiting Covariance) We use a similar method as in the proof of Proposition 1.2 of [23], based on the Feynman-Kac formula given in Appendix A. For simplicity, we assume that t = s. The general case is similar.…”

Section: Proof Of Theorem 11 Under Assumption Cmentioning

confidence: 99%

“…(iii) Use the same method as Theorem 1.1 of [23]. We first show the tightness, and then establish the finite dimensional convergence.…”

Section: Proof Of Theorem 11 Under Assumption Cmentioning

confidence: 99%

“…Lemma 4.6 (Lemma 3.2 of [23]). For any ε > 0, there exists a constant C(ε) > 0 depending on ε such that…”

Section: Proof Of Theorem 11 Under Assumption Cmentioning

confidence: 99%

“…Together with the sum over ρ, this yields the factor s n . For the first parenthesis, we use relation (5.5) of [23]: E e i n j=1 (Bs−Bt j )ξ j − E e i n j=1 (Bt−Bt j )ξ j ≤ t − s 2 ( n j=1 ξ j ) 2 E e i n j=1 (Bs−Bt j )ξ j .…”

Section: Proof Of Theorem 11 Under Assumption Cmentioning

confidence: 99%

“…Note that E e i n j=1 B t−t j ξ j = E e −Var n j=1 B t−t j ξ j . Using relation (3.9) of [23] (with H 0 = 1/2), we obtain:…”

Section: A Feyman-kac Formulamentioning

confidence: 99%

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

Balan¹,

Yuan²

2022

Preprint

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In this article, we investigate the asymptotic behaviour of the spatial integral of the solution to the parabolic Anderson model with time independent noise in dimension d ≥ 1, as the domain of the integral becomes large. We consider 3 cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the rough noise, i.e. fractional noise with index H ∈ ( 1 4 , 1 2 ) in dimension d = 1. In each case, we identify the order of magnitude of the variance of the spatial integral, we prove a quantitative central limit theorem for the normalized spatial integral by estimating its total variation distance to a standard normal distribution, and we give the corresponding functional limit result.

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Section: Proof Of Theorem 11 Under Assumption Cmentioning

confidence: 99%

“…(iii) Use the same method as Theorem 1.1 of [23]. We first show the tightness, and then establish the finite dimensional convergence.…”

Section: Proof Of Theorem 11 Under Assumption Cmentioning

confidence: 99%

“…Lemma 4.6 (Lemma 3.2 of [23]). For any ε > 0, there exists a constant C(ε) > 0 depending on ε such that…”

Section: Proof Of Theorem 11 Under Assumption Cmentioning

confidence: 99%

Section: Proof Of Theorem 11 Under Assumption Cmentioning

confidence: 99%

“…Note that E e i n j=1 B t−t j ξ j = E e −Var n j=1 B t−t j ξ j . Using relation (3.9) of [23] (with H 0 = 1/2), we obtain:…”

Section: A Feyman-kac Formulamentioning

confidence: 99%

See 3 more Smart Citations

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

Balan¹,

Yuan²

2022

Preprint

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

Balan

Nualart

Quer-Sardanyons

et al. 2022

Stoch PDE: Anal Comp

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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].

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Central limit theorems for nonlinear stochastic wave equations in dimension three

Ebina

2023

Stoch PDE: Anal Comp

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We consider stochastic wave equations in spatial dimensions d ≥ 4. We assume that the driving noise is given by a Gaussian noise that is white in time and has some spatial correlation. We show that the solution process is spatially ergodic under the spatial shift using the tools of Malliavin calculus. When the spatial correlation is given by the Riesz kernel, we also establish that the spatial integral of the solution with proper normalization converges to the standard normal distribution under the Wasserstein distance. The convergence is obtained by first constructing the approximation sequence to the solution and then applying Malliavin-Stein's method to the normalized spatial integral of the sequence. The corresponding functional central limit theorem is presented as well.

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Spatial averages for the parabolic Anderson model driven by rough noise

Cited by 7 publications

References 23 publications

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

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

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

Central limit theorems for nonlinear stochastic wave equations in dimension three

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