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.
|Number of pages||13|
|Journal||Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques|
|Publication status||Published - 4 Nov 2010|
- stochastic heat equation
- stochastic process