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

Consider a stochastic heat equation partial derivative(t)u = kappa partial derivative(2)(xx)u + sigma(u)(omega) over dot for a space-time white noise (omega) over dot and a constant kappa > 0. Under some suitable conditions on the initial function u(0) and sigma, we show that the quantities lim(t ->infinity)sup t(-1) sup(x is an element of R) InE(vertical bar u(t)(x)vertical bar(2)) and lim(t ->infinity)sup t(-1) InE(sup(x is an element of R)vertical bar u(t)(x)vertical bar(2)) are equal, as well as bounded away from zero and infinity by explicit multiples of 1/kappa. Our proof works by demonstrating quantitatively that the peaks of the stochastic process x bar right arrow u(t)(x) are highly concentrated for infinitely-many large values of t. In the special case of the parabolic Anderson model - where sigma(u) = lambda u for some lambda > 0 - this "peaking" is a way to make precise the notion of physical intermittency.