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

**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

- Pdf Format (posted on February 13, 2013)

Davar Khoshnevisan & Kunwoo KimDepartment 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