Electronic Journal of Probability

Temporal asymptotics for fractional parabolic Anderson model

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

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Abstract

In this paper, we consider fractional parabolic equation of the form $ \frac{\partial u} {\partial t}=-(-\Delta )^{\frac{\alpha } {2}}u+u\dot W(t,x)$, where $-(-\Delta )^{\frac{\alpha } {2}}$ with $\alpha \in (0,2]$ is a fractional Laplacian and $\dot W$ is a Gaussian noise colored both in space and time. The precise moment Lyapunov exponents for the Stratonovich solution and the Skorohod solution are obtained by using a variational inequality and a Feynman-Kac type large deviation result for space-time Hamiltonians driven by $\alpha $-stable process. As a byproduct, we obtain the critical values for $\theta $ and $\eta $ such that $\mathbb{E} \exp \left (\theta \left (\int _0^1 \int _0^1 |r-s|^{-\beta _0}\gamma (X_r-X_s)drds\right )^\eta \right )$ is finite, where $X$ is $d$-dimensional symmetric $\alpha $-stable process and $\gamma (x)$ is $|x|^{-\beta }$ or $\prod _{j=1}^d|x_j|^{-\beta _j}$.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 14, 39 pp.

Dates
Received: 18 January 2017
Accepted: 9 January 2018
First available in Project Euclid: 21 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1519182022

Digital Object Identifier
doi:10.1214/18-EJP139

Mathematical Reviews number (MathSciNet)
MR3771751

Zentralblatt MATH identifier
1390.60101

Subjects
Primary: 60F10: Large deviations 60H15: Stochastic partial differential equations [See also 35R60] 60G15: Gaussian processes 60G52: Stable processes

Keywords
Lyapunov exponent Gaussian noise $\alpha $-stable process fractional parabolic Anderson model Feynman-Kac representation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Chen, Xia; Hu, Yaozhong; Song, Jian; Song, Xiaoming. Temporal asymptotics for fractional parabolic Anderson model. Electron. J. Probab. 23 (2018), paper no. 14, 39 pp. doi:10.1214/18-EJP139. https://projecteuclid.org/euclid.ejp/1519182022


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