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

Temporal asymptotics for fractional parabolic Anderson model. (English) Zbl 1390.60101

Electron. J. Probab. 23, Paper No. 14, 39 p. (2018).
Summary: 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 \(\prod_{j=1}^d|x_j|^{-\beta_j}\).

Cited in 11 Documents

MSC:

60F10 Large deviations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
60G52 Stable stochastic processes

Keywords:

Lyapunov exponent; Gaussian noise; \(\alpha\)-stable process; fractional parabolic Anderson model; Feynman-Kac representation
PDFBibTeX XMLCite
\textit{X. Chen} et al., Electron. J. Probab. 23, Paper No. 14, 39 p. (2018; Zbl 1390.60101)
Full Text: DOI arXiv Euclid

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