On the global maximum of the solution to a stochastic heat equation with compact-support initial data

Mohammud Foondun; Davar Khoshnevisan

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 4, page 895-907
  • ISSN: 0246-0203

Abstract

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Consider a stochastic heat equation ∂tu=κ  ∂xx2u+σ(u)ẇ for a space–time white noise ẇ 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−1sup x∈Rln El(|ut(x)|2) and lim sup t→∞t−1ln E(sup x∈R|ut(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↦ut(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.

How to cite

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Foondun, Mohammud, and Khoshnevisan, Davar. "On the global maximum of the solution to a stochastic heat equation with compact-support initial data." Annales de l'I.H.P. Probabilités et statistiques 46.4 (2010): 895-907. <http://eudml.org/doc/243484>.

@article{Foondun2010,
abstract = {Consider a stochastic heat equation ∂tu=κ  ∂xx2u+σ(u)ẇ for a space–time white noise ẇ and a constant κ&gt;0. Under some suitable conditions on the initial function u0 and σ, we show that the quantities lim sup t→∞t−1sup x∈Rln El(|ut(x)|2) and lim sup t→∞t−1ln E(sup x∈R|ut(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↦ut(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&gt;0 – this “peaking” is a way to make precise the notion of physical intermittency.},
author = {Foondun, Mohammud, Khoshnevisan, Davar},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic heat equation; intermittency},
language = {eng},
number = {4},
pages = {895-907},
publisher = {Gauthier-Villars},
title = {On the global maximum of the solution to a stochastic heat equation with compact-support initial data},
url = {http://eudml.org/doc/243484},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Foondun, Mohammud
AU - Khoshnevisan, Davar
TI - On the global maximum of the solution to a stochastic heat equation with compact-support initial data
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 4
SP - 895
EP - 907
AB - Consider a stochastic heat equation ∂tu=κ  ∂xx2u+σ(u)ẇ for a space–time white noise ẇ and a constant κ&gt;0. Under some suitable conditions on the initial function u0 and σ, we show that the quantities lim sup t→∞t−1sup x∈Rln El(|ut(x)|2) and lim sup t→∞t−1ln E(sup x∈R|ut(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↦ut(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&gt;0 – this “peaking” is a way to make precise the notion of physical intermittency.
LA - eng
KW - stochastic heat equation; intermittency
UR - http://eudml.org/doc/243484
ER -

References

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