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

Foondun, Mohammud; Khoshnevisan, Davar

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 895-907.

Nous considérons l'équation de la chaleur stochastique ∂tu=κ∂xx2u+σ(u)ẇ avec un bruit blanc spatio-temporel ẇ et une constante κ>0. Sous des conditions adéquates sur la condition initiale u0 et sur σ, nous montrons que les quantités lim sup t→∞t-1sup x∈Rln E(|ut(x)|2) et lim sup t→∞t-1ln E(sup x∈R|ut(x)|2) sont égales. Par ailleurs, nous les bornons inférieurement et supérieurement par des constantes strictement positives et finies dépendant explicitement de 1/κ. Nos démonstrations reposent sur la preuve quantitative de la forte concentration des pics du processus x↦ut(x) pour de grandes valeurs de t infiniment nombreuses. Dans le cas particulier du modèle d'Anderson parabolique-où σ(u)=λu pour un λ>0 - ce phénomène de pics est une façon de préciser la notion physique d'intermittence.

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.

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