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July 2015 Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups
Davar Khoshnevisan, Kunwoo Kim
Ann. Probab. 43(4): 1944-1991 (July 2015). DOI: 10.1214/14-AOP925
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.
Davar Khoshnevisan. Kunwoo Kim. "Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups." Ann. Probab. 43 (4) 1944 - 1991, July 2015. https://doi.org/10.1214/14-AOP925
Received: 1 March 2013; Revised: 1 February 2014; Published: July 2015
First available in Project Euclid: 3 June 2015
Digital Object Identifier: 10.1214/14-AOP925
Primary: 60H15, 60H25
Secondary: 35R60, 60B15, 60J30, 60K37
Keywords: Intermittency, Lévy processes, locally compact Abelian groups, nonlinear noise excitation, Stochastic heat equation
Ann. Probab.
Vol.43 • No. 4 • July 2015
Davar Khoshnevisan, Kunwoo Kim "Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups," The Annals of Probability, Ann. Probab. 43(4), 1944-1991, (July 2015)