Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Parabolic Anderson model with a fractional Gaussian noise that is rough in time

Xia Chen

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Abstract

This paper concerns the parabolic Anderson equation \begin{equation*}\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+u\frac{\partial^{d+1}W^{\mathbf{H}}}{\partial t\,\partial x_{1}\cdots \,\partial x_{d}} \end{equation*} generated by a $(d+1)$-dimensional fractional noise with the Hurst parameter $\mathbf{H}=(H_{0},H_{1},\ldots ,H_{d})$ with special interest in the setting that some of $H_{0},\ldots ,H_{d}$ are less than half. In the recent work (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 941–976), the case of the spatial roughness has been investigated. To put the last piece of the puzzle in place, this work investigates the case when $H_{0}<1/2$ with the concern on solvability, Feynman–Kac’s moment formula and intermittency of the system.

Résumé

Cet article concerne l’équation d’Anderson parabolique \begin{equation*}\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+u\frac{\partial^{d+1}W^{\mathbf{H}}}{\partial t\,\partial x_{1}\cdots \,\partial x_{d}} \end{equation*} engendrée par un bruit fractionnaire de dimension $d+1$ avec un paramètre de Hurst $\mathbf{H}=(H_{0},H_{1},\ldots ,H_{d})$, en portant une attention particulière au cas où certains des paramètres $H_{0},\ldots ,H_{d}$ sont inférieurs à $1/2$. Le cas rugueux en espace avait fait l’objet du travail récent (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 941–976). Pour mettre en place la dernière pièce du puzzle, cet article examine le cas $H_{0}<1/2$ en se penchant sur les problèmes de résolution, de la formule des moments de Feynman–Kac et de l’intermittence du système.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 792-825.

Dates
Received: 20 August 2018
Revised: 8 March 2019
Accepted: 21 March 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1584345620

Digital Object Identifier
doi:10.1214/19-AIHP983

Mathematical Reviews number (MathSciNet)
MR4076766

Zentralblatt MATH identifier
07199880

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
Parabolic Anderson equation Dalang’s condition Fractional Rough and critical Gaussian noises Feynman–Kac’s representation Brownian motion Moment asymptotics

Citation

Chen, Xia. Parabolic Anderson model with a fractional Gaussian noise that is rough in time. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 792--825. doi:10.1214/19-AIHP983. https://projecteuclid.org/euclid.aihp/1584345620


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