### 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].

Language | English |
---|---|

Pages | 548-568 |

Number of pages | 21 |

Journal | Electronic Journal of Probability |

Volume | 14 |

DOIs | |

Publication status | Published - 7 Jan 2009 |

Externally published | Yes |

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### Keywords

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

### Cite this

*Electronic Journal of Probability*,

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

}

*Electronic Journal of Probability*, vol. 14, pp. 548-568. https://doi.org/10.1214/EJP.v14-614

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Intermittence and nonlinear parabolic stochastic partial differential equations

AU - Foondun, Mohammud

AU - Khoshnevisan, Davar

PY - 2009/1/7

Y1 - 2009/1/7

N2 - 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].

AB - 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].

KW - lévy processes

KW - liapounov exponents

KW - stochastic partial differential equations

KW - Burkholder-Davis-Gundy inequality

KW - weak intermittence

UR - http://www.scopus.com/inward/record.url?scp=63549093138&partnerID=8YFLogxK

UR - http://www.emis.ams.org/journals/EJP-ECP/index-2.html

U2 - 10.1214/EJP.v14-614

DO - 10.1214/EJP.v14-614

M3 - Article

VL - 14

SP - 548

EP - 568

JO - Electronic Journal of Probability

T2 - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -