### Abstract

We consider nonlinear parabolic SPDEs of the form ∂ _{t}u = 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].

Original language | English |
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Pages (from-to) | 548-568 |

Number of pages | 21 |

Journal | Electronic Journal of Probability |

Volume | 14 |

DOIs | |

Publication status | Published - 7 Jan 2009 |

Externally published | Yes |

### Keywords

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

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## Cite this

*Electronic Journal of Probability*,

*14*, 548-568. https://doi.org/10.1214/EJP.v14-614