### Abstract

We study discrete nonlinear parabolic stochastic heat equations of the form, u_{n+1}(x)-u_{n}(x) = (Lu_{n})(x)+ σ(u _{n}(x))ζ_{n}(x), for n ∈ Z_{+} and x ∈ Z^{d}, where ζ := {ζ_{n}(x)} _{n≥0,x∈Zd} denotes random forcing and L the generator of a random walk on Z^{d}. Under mild conditions, we prove that the preceding stochastic PDE has a unique solution that grows at most exponentially in time. And that, under natural conditions, it is "weakly intermittent." Along the way, we establish a comparison principle as well as a finite support property.

Original language | English |
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Pages (from-to) | 1042-1060 |

Number of pages | 19 |

Journal | Bernoulli |

Volume | 18 |

Issue number | 3 |

Early online date | 28 Jun 2012 |

DOIs | |

Publication status | Published - 31 Aug 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- intermittency
- stochastic heat equations

### Cite this

*Bernoulli*,

*18*(3), 1042-1060. https://doi.org/10.3150/11-BEJ357

}

*Bernoulli*, vol. 18, no. 3, pp. 1042-1060. https://doi.org/10.3150/11-BEJ357

**An asymptotic theory for randomly forced discrete nonlinear heat equations.** / Foondun, Mohammud; Khoshnevisan, Davar.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An asymptotic theory for randomly forced discrete nonlinear heat equations

AU - Foondun, Mohammud

AU - Khoshnevisan, Davar

PY - 2012/8/31

Y1 - 2012/8/31

N2 - We study discrete nonlinear parabolic stochastic heat equations of the form, un+1(x)-un(x) = (Lun)(x)+ σ(u n(x))ζn(x), for n ∈ Z+ and x ∈ Zd, where ζ := {ζn(x)} n≥0,x∈Zd denotes random forcing and L the generator of a random walk on Zd. Under mild conditions, we prove that the preceding stochastic PDE has a unique solution that grows at most exponentially in time. And that, under natural conditions, it is "weakly intermittent." Along the way, we establish a comparison principle as well as a finite support property.

AB - We study discrete nonlinear parabolic stochastic heat equations of the form, un+1(x)-un(x) = (Lun)(x)+ σ(u n(x))ζn(x), for n ∈ Z+ and x ∈ Zd, where ζ := {ζn(x)} n≥0,x∈Zd denotes random forcing and L the generator of a random walk on Zd. Under mild conditions, we prove that the preceding stochastic PDE has a unique solution that grows at most exponentially in time. And that, under natural conditions, it is "weakly intermittent." Along the way, we establish a comparison principle as well as a finite support property.

KW - intermittency

KW - stochastic heat equations

UR - http://www.scopus.com/inward/record.url?scp=84873403136&partnerID=8YFLogxK

UR - http://projecteuclid.org/current/euclid.bj

U2 - 10.3150/11-BEJ357

DO - 10.3150/11-BEJ357

M3 - Article

VL - 18

SP - 1042

EP - 1060

JO - Bernoulli Society for Mathematical Statistics and Probability

JF - Bernoulli Society for Mathematical Statistics and Probability

SN - 1350-7265

IS - 3

ER -