On the chaotic character of the stochastic heat equation, before the onset of intermitttency

We consider a nonlinear stochastic heat equation ∂ₜu=1/2∂ₓₓu+σ(u)∂ₓₜW, where ∂ₓₜW denotes space–time white noise and σ:R is Lipschitz continuous. We establish that, at every fixed time $t>0$, the global behavior of the solution depends in a critical manner on the structure of the initial function u₀: under suitable conditions on u₀ and σ, ₓuₜ(x) is a.s. finite when u₀ has compact support, whereas with probability one, |x|→∞uₜ(x)/(log|x|)1/6>0 when u₀ is bounded uniformly away from zero. This sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at fixed times, well before the onset of intermittency.