## Electronic Journal of Probability

### Temporal asymptotics for fractional parabolic Anderson model

#### 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
Accepted: 9 January 2018
First available in Project Euclid: 21 February 2018

https://projecteuclid.org/euclid.ejp/1519182022

Digital Object Identifier
doi:10.1214/18-EJP139

Mathematical Reviews number (MathSciNet)
MR3771751

Zentralblatt MATH identifier
1390.60101

#### 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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