### 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 |

### Keywords

- intermittency
- stochastic heat equations

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## Cite this

Foondun, M., & Khoshnevisan, D. (2012). An asymptotic theory for randomly forced discrete nonlinear heat equations.

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