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Intermittence and nonlinear parabolic stochastic partial differential equations

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1.	Title	Title of document	Intermittence and nonlinear parabolic stochastic partial differential equations
2.	Creator	Author's name, affiliation, country	Mohammud Foondun; University of Utah
2.	Creator	Author's name, affiliation, country	Davar Khoshnevisan; University of Utah
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Stochastic partial differential equations, Levy processes
3.	Subject	Subject classification	60H15; 82B44
4.	Description	Abstract	We consider nonlinear parabolic SPDEs of the form $\partial_t u={\cal L} u + \sigma(u)\dot w$, where $\dot w$ denotes space-time white noise, $\sigma:R\to R$ is [globally] Lipschitz continuous, and $\cal L$ is the $L^2$-generator of a L'evy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when $\sigma$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,'' provided that the symmetrization of $\cal L$ is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for $\cal L$ in dimension $(1+1)$. When ${\cal L}=\kappa\partial_{xx}$ for $\kappa>0$, these formulas agree with the earlier results of statistical physics (Kardar (1987), Krug and Spohn (1991), Lieb and Liniger (1963)), and also probability theory (Bertini and Cancrini (1995), Carmona and Molchanov (1994)) in the two exactly-solvable cases. That is when $u_0=\delta_0$ or $u_0\equiv 1$; in those cases the moments of the solution to the SPDE can be computed (Bertini and Cancrini (1995)).
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	Research supported in part by NSF grant DMS-0706728
7.	Date	(YYYY-MM-DD)	2009-02-24
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/614
10.	Identifier	Digital Object Identifier	10.1214/EJP.v14-614
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 14
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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