Solving the hyperbolic Anderson model 1: Skorohod setting

Chen, Xia; Deya, Aurélien; Song, Jian; Tindel, Samy

doi:10.48550/arxiv.2112.04954

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The Regular Case1

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2022

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“…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.…”

Section: The Regular Casementioning

confidence: 99%

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: The Regular Casementioning

confidence: 99%