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

We consider a nonlinear stochastic heat equation @tu = 1 @xxu + (u)@xtW , where @xtW denotes space-time white noise and : R ! R is Lipschitz continuous. We establish that, at every xed time t > 0, the global behavior of the solution depends in a critical manner on the structure of the initial function u0: Under suitable technical conditions on u0 and , supx2Rut(x) is a.s. nite when u0 has compact support, whereas with probability one, lim supjxj!1ut(x)=(logjxj) 1=6 > 0 when u0 is bounded uniformly away from zero. The mentioned 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 xed times, well before the onset of intermittency.