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
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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
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2013 - Davar Khoshnevisan & Kunwoo Kim