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

Consider a stochastic heat equation $\partial_t u = \kappa \partial^2_{xx}u+\sigma(u)\dot{w}$ for a space-time white noise $\dot{w}$ and a constant $\kappa>0$. Under some suitable conditions on the the initial function $u_0$ and $\sigma$, we show that the quantity \limsup_{t\to\infty}t^{-1}\ln\E(\sup_{x\in\R} |u_t(x)|^2) is 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 $x\mapsto u_t(x)$ 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.