Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Moment asymptotics for parabolic Anderson equation with fractional time-space noise: In Skorokhod regime

Xia Chen

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Abstract

In this paper, we consider the parabolic Anderson equation that is driven by a Gaussian noise fractional in time and white or fractional in space, and is solved in a mild sense defined by Skorokhod integral. Our objective is the precise moment Lyapunov exponent and high moment asymptotics. As far as the long term asymptotics are concerned, some feature given in our theorems is different from what have been observed in the Stratonovich-regime and in the setting of the white time noise. While the difference disappears when it comes to the high moment asymptotics. To achieve our goal, we introduce a variational inequality and use some newly developed tools such as time-space LDP of Feynman–Kac type, linearization by tangent approximation, together with some techniques developed along the line of probability in Banach spaces.

Résumé

Nous considérons l’équation d’Anderson parabolique conduite par un bruit gaussien, fractionnaire en temps, et blanc ou fractionnaire en espace, qu’on résout dans un sens faible défini par une intégrale de Skorokhod. Notre objectif est de donner l’exposant de Lyapounov pour les moments, et les asymptotiques des grands moments. Pour les asymptotiques en temps long, nos résultats mettent en évidence des phénomènes différents de ceux observés pour le régime Stratonovich, et dans le cas d’un bruit blanc en temps. Ces différences s’effacent néanmoins lorsque l’on considère les asymptotiques des grands moments. Nos résultats sont obtenus en introduisant une nouvelle inégalité variationnelle, et à l’aide d’outils nouveaux tels qu’un principe de grandes déviations de type Feynman–Kac, la linéarisation par des approximations tangentes, et des techniques inspirées des probabilités dans les espaces de Banach.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 819-841.

Dates
Received: 8 August 2015
Revised: 27 December 2015
Accepted: 28 December 2015
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897747

Digital Object Identifier
doi:10.1214/15-AIHP738

Mathematical Reviews number (MathSciNet)
MR3634276

Zentralblatt MATH identifier
06729834

Subjects
Primary: 60F10: Large deviations 60H15: Stochastic partial differential equations [See also 35R60] 60H40: White noise theory 60J65: Brownian motion [See also 58J65] 81U10: $n$-body potential scattering theory

Keywords
Lyapunov exponent High moment asymptotics White and fractional noise Brownian motion Parabolic Anderson equation Feynman–Kac’s representation

Citation

Chen, Xia. Moment asymptotics for parabolic Anderson equation with fractional time-space noise: In Skorokhod regime. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 819--841. doi:10.1214/15-AIHP738. https://projecteuclid.org/euclid.aihp/1491897747


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