The Annals of Probability

Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups

Davar Khoshnevisan and Kunwoo Kim

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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 $G$, $\sigma:\mathbf{R}\to\mathbf{R}$ is Lipschitz continuous, $\lambda\gg1$ is a large parameter, and $\xi$ denotes space–time white noise on $\mathbf{R}_{+}\times G$.

The main result of this paper contains a near-dichotomy for the (expected squared) energy $\mathrm{E}(\|u_{t}\|_{L^{2}(G)}^{2})$ 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 $\exp\{\operatorname{const}\cdot\,\lambda^{2}\}$ when $G$ is discrete and $\ge\exp\{\operatorname{const}\cdot\,\lambda^{4}\}$ when $G$ is connected.

Article information

Source
Ann. Probab., Volume 43, Number 4 (2015), 1944-1991.

Dates
Received: March 2013
Revised: February 2014
First available in Project Euclid: 3 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1433341324

Digital Object Identifier
doi:10.1214/14-AOP925

Mathematical Reviews number (MathSciNet)
MR3353819

Zentralblatt MATH identifier
1322.60116

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60H25: Random operators and equations [See also 47B80]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60K37: Processes in random environments 60J30 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Keywords
Stochastic heat equation intermittency nonlinear noise excitation Lévy processes locally compact Abelian groups

Citation

Khoshnevisan, Davar; Kim, Kunwoo. Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups. Ann. Probab. 43 (2015), no. 4, 1944--1991. doi:10.1214/14-AOP925. https://projecteuclid.org/euclid.aop/1433341324


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