# Preprint: Non-linear noise excitation of intermittent stochastic PDEs and the topology of LCA groups

## Davar Khoshnevisan and Kunwoo Kim

Abstract. Consider the stochastic heat equation $$\partial_t u = \mathscr{L}u + \lambda\sigma(u)\xi$$, where $$\mathscr{L}$$ denotes the generator of a Lévy process on a locally compact Hausdorff abelian group $$G$$, $$\sigma:{\bf R}\to{\bf R}$$ is Lipschitz continuous, $$\lambda\gg1$$ is a large parameter, and $$\xi$$ denotes space-time white noise on $${\bf R}_+\times G$$. The main result of this paper contains a near-dichotomy for the [expected squared] energy $${\rm E}(\|u_t\|_{L^2(G)}^2)$$ of the solution. Roughly speaking, that dichotomy says that, in all known cases where $$u$$ is intermittent, the energy of the solution behaves generically as $$\exp\{\text{const}\cdot\lambda^2\}$$ when $$G$$ is discrete and $$\ge \exp\{\text{const}\cdot\lambda^4\}$$ when $$G$$ is connected.

Keywords. The stochastic heat equation, intermittency, non-linear noise excitation, Lévy processes, locally compact abelian groups.

AMS Classification (2000). Primary 60H15, 60H25; Secondary 35R60, 60K37, 60J30, 60B15.

Support. Research supported in part by a grant from the U.S. National Science Foundation.

Pre/E-Prints. This paper is available in

 Davar Khoshnevisan & Kunwoo Kim Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu & kkim@math.utah.edu

Last Update: February 13, 2013
© 2013 - Davar Khoshnevisan & Kunwoo Kim