Temporal asymptotics for fractional parabolic Anderson model

Chen, Xia; Hu, Yaozhong; Song, Jian; Song, Xiaoming

doi:10.1214/18-ejp139

Cited by 14 publications

(8 citation statements)

References 25 publications

(40 reference statements)

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“…We expect that, by developing approximation methods that are similar to those in [23,18] and by making use of the results in this paper, one can establish the exact uniform and local regularity results for the stochastic heat equation with multiplicative fractional Gaussian noises, particularly those that have been studied in [3,7,8,9,19,20,21,22,34]. We plan to pursue this line of research in a subsequent paper.…”

Section: Introductionmentioning

confidence: 92%

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Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise

Herrell

Song

et al. 2020

Stochastic Analysis and Applications

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In this paper, we study the following stochastic heat equationwhere L is the generator of a Lévy process X taking value in R d , B is a fractionalcolored Gaussian noise with Hurst index H ∈ 1 2 , 1 for the time variable and spatial covariance function f which is the Fourier transform of a tempered measure µ.After establishing the existence of solution for the stochastic heat equation, we study the regularity of the solution {u(t, x), t ≥ 0, x ∈ R d } in both time and space variables. Under mild conditions, we give the exact uniform modulus of continuity and a Chungtype law of iterated logarithm for the sample function (t, x) → u(t, x). Our results generalize and strengthen the corresponding results of Balan and Tudor (2008) and Tudor and Xiao (2017).Running head: Sharp space-time regularity of the solution to a stochastic heat equation 2000 AMS Classification numbers: 60G15, 60J55, 60G18, 60F25.

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Section: Introductionmentioning

confidence: 92%

“…The theoretical studies of SPDEs driven by fBm or other fractional Gaussian noises have been growing rapidly. We refer to, for example, [3,4,7,9,8,17,19,20,21,22,29,31,32,34,35] for recent developments.…”

Section: Introductionmentioning

confidence: 99%

Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise

Herrell

Song

et al. 2020

Stochastic Analysis and Applications

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“…This property is related the moment bounds of the solution. When (1.1) is parabolic Anderson model, namely, when L is a heat operator or fractional heat operator, then the sharp (both lower and upper) moment bounds are known, see [2,6,5,7,8,17,19,25], and we also refer to [15] and references therein. However, when L is wave operators (namely the hyperbolic Anderson model) or when L is (temporal) fractional differential operators, the situation is different and as far as we know here are the progress achieved.…”

Section: Introductionmentioning

confidence: 99%

Intermittency properties for a large class of stochastic PDEs driven by fractional space-time noises

Hu,

Wang

2021

Preprint

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In this paper, we study intermittency properties for various stochastic PDEs with varieties of space-time Gaussian noises via matching upper and lower moment bounds of the solution. Due to the absence of the powerful Feynman-Kac formula, the lower moment bounds have been missing for many interesting equations except for the stochastic heat equation. This work introduces and explores the Feynman diagram formula for the moments of the solution and the small ball nondegeneracy for the Green's function to obtain the lower bounds for all moments which match the upper moment bounds. Our upper and lower moments are valid for various interesting equations, including stochastic heat equations, stochastic wave equations, stochastic heat equations with fractional Laplcians, and stochastic diffusions which are both fractional in time and in space.

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“…We refer the reader to [19] for a study of a general parabolic Anderson models with spacetime colored noise, and to [6] for the precise calculation of the Lyapunov exponents of order p ≥ 2 of the solution to a fractional parabolic Anderson model. (1) and (2) with spacetime white noise in spatial dimension d = 1.…”

Section: Introductionmentioning

confidence: 99%

Second order Lyapunov exponents for parabolic and hyperbolic Anderson models

Balan¹,

Song²

2019

Bernoulli

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In this article, we consider the hyperbolic and parabolic Anderson models in arbitrary space dimension d, with constant initial condition, driven by a Gaussian noise which is white in time. We consider two spatial covariance structures: (i) the Fourier transform of the spectral measure of the noise is a non-negative locallyintegrable function; (ii) d = 1 and the noise is a fractional Brownian motion in space with index 1/4 < H < 1/2. In both cases, we show that there is striking similarity between the Laplace transforms of the second moment of the solutions to these two models. Building on this connection and the recent powerful results of [16] for the parabolic model, we compute the second order (upper) Lyapunov exponent for the hyperbolic model. In case (i), when the spatial covariance of the noise is given by the Riesz kernel, we present a unified method for calculating the second order Lyapunov exponents for the two models.MSC 2010: Primary 60H15; Secondary 37H15

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Temporal asymptotics for fractional parabolic Anderson model

Cited by 14 publications

References 25 publications

Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise

Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise

Intermittency properties for a large class of stochastic PDEs driven by fractional space-time noises

Second order Lyapunov exponents for parabolic and hyperbolic Anderson models

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