The Annals of Probability

Intermittency for the stochastic heat equation with Lévy noise

Carsten Chong and Péter Kevei

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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.

Article information

Ann. Probab., Volume 47, Number 4 (2019), 1911-1948.

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

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Mathematical Reviews number (MathSciNet)

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 37H15: Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx]
Secondary: 60G51: Processes with independent increments; Lévy processes 35B40: Asymptotic behavior of solutions

Comparison principle intermittency intermittency fronts Lévy noise moment Lyapunov exponents stochastic heat equation stochastic PDE


Chong, Carsten; Kevei, Péter. Intermittency for the stochastic heat equation with Lévy noise. Ann. Probab. 47 (2019), no. 4, 1911--1948. doi:10.1214/18-AOP1297.

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