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

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(|u_t(x)|²) and lim sup _t→∞t⁻¹ln E(sup _x∈R|u_t(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.