Intermittence and nonlinear parabolic stochastic partial differential equations

Mohammud Foondun, Davar Khoshnevisan

Research output: Contribution to journalArticle

61 Citations (Scopus)

Abstract

We consider nonlinear parabolic SPDEs of the form ∂ tu = ℒu+σ(u)w, where ?w denotes spacetime white noise, σ : R → R is [globally] Lipschitz continuous, and ℒ is the L 2-generator of a Lévy 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 σ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is "weakly intermittent," provided that the symmetrization of 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 ℒ in dimension (1+1). When ℒ = Κ∂ xx for κ > 0, these formulas agree with the earlier results of statistical physics [28; 32; 33], and also probability theory [1; 5] in the two exactly-solvable cases. That is when u 0 = δ 0 or u 0 = 1; in those cases the moments of the solution to the SPDE can be computed [1].

LanguageEnglish
Pages548-568
Number of pages21
JournalElectronic Journal of Probability
Volume14
DOIs
Publication statusPublished - 7 Jan 2009
Externally publishedYes

Fingerprint

Stochastic Partial Differential Equations
Parabolic Partial Differential Equations
Lipschitz
Moment
Nonlinear Heat Equation
Anderson Model
Symmetrization
Uniqueness of Solutions
Statistical Physics
Probability Theory
Lévy Process
White noise
Thing
Existence of Solutions
Space-time
Exponent
Generator
Denote
Partial differential equations
Form

Keywords

  • lévy processes
  • liapounov exponents
  • stochastic partial differential equations
  • Burkholder-Davis-Gundy inequality
  • weak intermittence

Cite this

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Intermittence and nonlinear parabolic stochastic partial differential equations. / Foondun, Mohammud; Khoshnevisan, Davar.

In: Electronic Journal of Probability, Vol. 14, 07.01.2009, p. 548-568.

Research output: Contribution to journalArticle

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