### Abstract

Consider a stochastic heat equation ∂_{t}u = κ ∂_{xx}^{2}u+σ(u) ω for a space-time white noise ω and a constant κ >0. Under some suitable conditions on the initial function u0 and σ, we show that the quantities lim sup _{t→∞}*t*^{−1}sup _{x∈R}ln E*l*(|*u*_{t}(*x*)|^{2}) and lim sup _{t→∞}*t*^{−1}ln E(sup _{x∈R}|*u*_{t}(*x*)|^{2}) are equal, as well as bounded away from zero and infinity by explicit multiples of 1/κ. Our proof works by demonstrating quantitatively that the peaks of the stochastic process x → u _{t} (x) are highly concentrated for infinitely-many large values of t. In the special case of the parabolic Anderson model - where σ(u) = λu for some λ > 0 - this "peaking" is a way to make precise the notion of physical intermittency.

Original language | English |
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Pages (from-to) | 895-907 |

Number of pages | 13 |

Journal | Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques |

Volume | 46 |

Issue number | 4 |

DOIs | |

Publication status | Published - 4 Nov 2010 |

Externally published | Yes |

### Keywords

- intermittency
- stochastic heat equation
- stochastic process