The Annals of Probability
- Ann. Probab.
- Volume 47, Number 4 (2019), 1911-1948.
Intermittency for the stochastic heat equation with Lévy noise
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.
Ann. Probab., Volume 47, Number 4 (2019), 1911-1948.
Received: July 2017
Revised: March 2018
First available in Project Euclid: 4 July 2019
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.aop/1562205693