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. &

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