Translator Disclaimer
2018 Temporal asymptotics for fractional parabolic Anderson model
Xia Chen, Yaozhong Hu, Jian Song, Xiaoming Song
Electron. J. Probab. 23: 1-39 (2018). DOI: 10.1214/18-EJP139

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 $\prod _{j=1}^d|x_j|^{-\beta _j}$.

Citation

Download Citation

Xia Chen. Yaozhong Hu. Jian Song. Xiaoming Song. "Temporal asymptotics for fractional parabolic Anderson model." Electron. J. Probab. 23 1 - 39, 2018. https://doi.org/10.1214/18-EJP139

Information

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

zbMATH: 1390.60101
MathSciNet: MR3771751
Digital Object Identifier: 10.1214/18-EJP139

Subjects:
Primary: 60F10, 60G15, 60G52, 60H15

JOURNAL ARTICLE
39 PAGES


SHARE
Vol.23 • 2018
Back to Top