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
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lim sup t→∞t−1sup x∈Rln El(|ut(x)|2) and lim sup t→∞t−1ln E(sup x∈R|ut(x)|2)
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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.
@article{1288878328,
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 = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {46},
number = {1},
year = {2010},
pages = { 895-907},
language = {en},
url = {http://dml.mathdoc.fr/item/1288878328}
}
Foondun, Mohammud; Khoshnevisan, Davar. On the global maximum of the solution to a stochastic heat equation with compact-support initial data. Ann. Inst. H. Poincaré Probab. Statist., Tome 46 (2010) no. 1, pp. 895-907. http://gdmltest.u-ga.fr/item/1288878328/