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May 2019 Parabolic Anderson model with rough or critical Gaussian noise
Xia Chen
Ann. Inst. H. Poincaré Probab. Statist. 55(2): 941-976 (May 2019). DOI: 10.1214/18-AIHP904

Abstract

This paper considers the parabolic Anderson equation

\[{\frac{\partial u}{\partial t}}={\frac{1}{2}}\Delta u+u{\frac{\partial^{d+1}W^{\mathbf{H}}}{\partial t\partial x_{1}\cdots\partial x_{d}}}\] generated by a $(d+1)$-dimensional fractional noise with the Hurst parameter $\mathbf{H}=(H_{0},H_{1},\ldots,H_{d})$. The existence/uniqueness, Feynman–Kac’s moment formula and the precise intermittency exponents are formulated in the case when some of $H_{1},\ldots,H_{d}$ are less than one half, and in the case when the Dalang’s condition

\[d-\sum_{k=1}^{n}H_{j}<1\quad\mbox{is replaced by }d-\sum_{k=1}^{n}H_{j}=1.\] Some partial result is also achieved for the case when $H_{0}<1/2$ which brings insight on what to expect as the Gaussian noise is rough in time.

Cet article s’intéresse à l’équation d’Anderson parabolique

\[{\frac{\partial u}{\partial t}}={\frac{1}{2}}\Delta u+u{\frac{\partial^{d+1}W^{\mathbf{H}}}{\partial t\partial x_{1}\cdots\partial x_{d}}}\] engendrée par un bruit fractionnaire de dimension $(d+1)$ et de paramètre de Hurst $\mathbf{H}=(H_{0},H_{1},\ldots,H_{d})$. L’existence et l’unicité, la formule des moments de Feynman–Kac et les exposants précis d’intermittence sont formulés dans le cas où l’un des paramètres $H_{1},\ldots,H_{d}$ est inférieur à un demi, et dans le cas où la condition de Dalang

\[d-\sum_{k=1}^{n}H_{j}<1\quad\mbox{est remplacée par }d-\sum_{k=1}^{n}H_{j}=1.\] Des résultats partiels sont aussi obtenus dans la cas $H_{0}<1/2$, ce qui donne une intuition de ce qui doit être attendu dans le cas où le bruit Gaussien est rugueux en temps.

Citation

Download Citation

Xia Chen. "Parabolic Anderson model with rough or critical Gaussian noise." Ann. Inst. H. Poincaré Probab. Statist. 55 (2) 941 - 976, May 2019. https://doi.org/10.1214/18-AIHP904

Information

Received: 5 January 2018; Revised: 26 March 2018; Accepted: 5 April 2018; Published: May 2019
First available in Project Euclid: 14 May 2019

zbMATH: 07097337
MathSciNet: MR3949959
Digital Object Identifier: 10.1214/18-AIHP904

Subjects:
Primary: 60F10, 60H15, 60H40, 60J65, 81U10

Rights: Copyright © 2019 Institut Henri Poincaré

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Vol.55 • No. 2 • May 2019
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