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

Mohammud Foondun, Davar Khoshnevisan

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

Consider a stochastic heat equation ∂tu = κ ∂xx2u+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−1sup xRln El(|ut(x)|2) and lim sup t→∞t−1ln E(sup xR|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 → 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.

LanguageEnglish
Pages895-907
Number of pages13
JournalAnnales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
Volume46
Issue number4
DOIs
Publication statusPublished - 4 Nov 2010
Externally publishedYes

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Space-time White Noise
Stochastic Heat Equation
Anderson Model
Intermittency
Compact Support
Stochastic Processes
Infinity
Zero
Stochastic processes

Keywords

  • intermittency
  • stochastic heat equation
  • stochastic process

Cite this

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AU - Khoshnevisan, Davar

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N2 - Consider a stochastic heat equation ∂tu = κ ∂xx2u+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−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 → 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.

AB - Consider a stochastic heat equation ∂tu = κ ∂xx2u+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−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 → 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.

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