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, Volume 46 (2010) no. 4, p. 895-907

Consider a stochastic heat equation tu=κxx2u+σ(u) for a space-time white noise 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 xut(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.

Nous considérons l'équation de la chaleur stochastique tu=κ∂xx2u+σ(u) avec un bruit blanc spatio-temporel 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 xRln E(|ut(x)|2) et lim sup t→∞t-1ln E(sup xR|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 xut(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.

DOI : https://doi.org/10.1214/09-AIHP328
Classification:  35R60,  37H10,  60H15,  82B44
Keywords: stochastic heat equation, intermittency
@article{AIHPB_2010__46_4_895_0,
     author = {Foondun, Mohammud and Khoshnevisan, Davar},
     title = {On the global maximum of the solution to a stochastic heat equation with compact-support initial data},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     year = {2010},
     pages = {895-907},
     doi = {10.1214/09-AIHP328},
     zbl = {1210.35305},
     mrnumber = {2744876},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2010__46_4_895_0}
}
Foondun, Mohammud; 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, Volume 46 (2010) no. 4, pp. 895-907. doi : 10.1214/09-AIHP328. http://www.numdam.org/item/AIHPB_2010__46_4_895_0/

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