The Stochastic Heat Equation with Multiplicative Levy Noise: Existence, Moments, and Intermittency

We study the stochastic heat equation (SHE) partial derivative(t)u = 1/2 Delta u + beta u xi driven by a multiplicative Levy noise xi with positive jumps and coupling constant beta > 0, in arbitrary dimension d >= 1. We prove the existence of solutions under an optimal condition if d = 1, 2 and a close-to-optimal condition if d >= 3. Under an assumption that is general enough to include stable noises, wefurther prove that the solution is unique. By establishing tight moment bounds on the multiple Levy integrals arising in the chaos decomposition of u, we further show that the solution has finite pth 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 -> 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 Levy measure, at any disorder intensity beta > 0, in any dimension d >= 1. This behavior contrasts with that observed for the SHE on Z(d) and for the SHE on R-d with Gaussian noise, for which intermittency does not occur in high dimensions for small beta.