The stochastic heat equation with multiplicative Lévy noise: Existence, moments, and intermittency

Abstract

We study the stochastic heat equation (SHE) ${\partial }_{t}u=\frac{1}{2}\Delta u+\beta u\xi$ driven by a multiplicative Lévy noise $\xi$ with positive jumps and amplitude $\beta >0$, in arbitrary dimension $d\ge 1$. We prove the existence of solutions under an optimal condition if $d=1,2$ and a close-to-optimal condition if $d\ge 3$. Under an assumption that is general enough to include stable noises, we further prove that the solution is unique. By establishing tight moment bounds on the multiple Lévy integrals arising in the chaos decomposition of $u$, we further show that the solution has finite $p$th moments for $p>0$ whenever the noise does. Finally, for any $p>0$, we derive upper and lower bounds on the moment Lyapunov exponents of order $p$ of the solution, which are asymptotically sharp in the limit as $\beta \to 0$. One of our most striking findings is that the solution to the SHE exhibits a property called strong intermittency (which implies moment intermittency of all orders $p>1$ and pathwise mass concentration of the solution), for any non-trivial Lévy measure, at any disorder intensity $\beta >0$, in any dimension $d\ge 1$.

References in corpus (18)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• The one-dimensional KPZ equation and its universality class
• The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1
• The Continuum Directed Random Polymer
• Fractional moment bounds and disorder relevance for pinning models
• A simple construction of the continuum parabolic Anderson model on ${\mathbf{R}}^2$
• Marginal relevance of disorder for pinning models
• Geometric characterization of intermittency in the parabolic Anderson model
• Majorizing multiplicative cascades for directed polymers in random media
• Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift
• Hierarchical pinning models, quadratic maps and quenched disorder
• Differing averaged and quenched large deviations for random walks in random environments in dimensions two and three
• Strong disorder implies strong localization for directed polymers in a random environment
• Stochastic PDEs with heavy-tailed noise
• Intermittency for the stochastic heat equation with Lévy noise
• Spatial ergodicity for SPDEs via a Poincaré-type inequality
• Brownian Polymers in Poissonian Environment: a survey
• The almost-sure asymptotic behavior of the solution to the stochastic heat equation with Lévy noise