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July 2019 Intermittency for the stochastic heat equation with Lévy noise
Carsten Chong, Péter Kevei
Ann. Probab. 47(4): 1911-1948 (July 2019). DOI: 10.1214/18-AOP1297

Abstract

We investigate the moment asymptotics of the solution to the stochastic heat equation driven by a $(d+1)$-dimensional Lévy space-time white noise. Unlike the case of Gaussian noise, the solution typically has no finite moments of order $1+2/d$ or higher. Intermittency of order $p$, that is, the exponential growth of the $p$th moment as time tends to infinity, is established in dimension $d=1$ for all values $p\in (1,3)$, and in higher dimensions for some $p\in (1,1+2/d)$. The proof relies on a new moment lower bound for stochastic integrals against compensated Poisson measures. The behavior of the intermittency exponents when $p\to 1+2/d$ further indicates that intermittency in the presence of jumps is much stronger than in equations with Gaussian noise. The effect of other parameters like the diffusion constant or the noise intensity on intermittency will also be analyzed in detail.

Citation

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Carsten Chong. Péter Kevei. "Intermittency for the stochastic heat equation with Lévy noise." Ann. Probab. 47 (4) 1911 - 1948, July 2019. https://doi.org/10.1214/18-AOP1297

Information

Received: 1 July 2017; Revised: 1 March 2018; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114707
MathSciNet: MR3980911
Digital Object Identifier: 10.1214/18-AOP1297

Subjects:
Primary: 37H15, 60H15
Secondary: 35B40, 60G51

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • July 2019
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