Electronic Journal of Probability

Intermittence and nonlinear parabolic stochastic partial differential equations

Mohammud Foondun and Davar Khoshnevisan

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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)).

Article information

Electron. J. Probab. Volume 14 (2009), paper no. 21, 548-568.

Accepted: 24 February 2009
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Stochastic partial differential equations Levy processes

This work is licensed under a Creative Commons Attribution 3.0 License.


Foondun, Mohammud; Khoshnevisan, Davar. Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14 (2009), paper no. 21, 548--568. doi:10.1214/EJP.v14-614. http://projecteuclid.org/euclid.ejp/1464819482.

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