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

We consider a nonlinear stochastic heat equation ${\mathrm{\partial }}_{t}u=\frac{1}{2}{\mathrm{\partial }}_{xx}u+\sigma (u){\mathrm{\partial }}_{xt}W$, where ${\mathrm{\partial }}_{xt}W$ denotes space-time white noise and $\sigma:\mathbf {R}\to \mathbf {R}$isLipschitzcontinuous\mathrm{.}Weestablishthat,ateveryfixedtime$t>0$, the global behavior of the solution depends in a critical manner on the structure of the initial function $u_0$: under suitable conditions on $u_0$ and $\sigma$, ${\sup }_{x\in \mathbf{R}}{u}_t(x)$ is a.s. finite when $u_0$ has compact support, whereas with probability one, ${\mathrm{lim\hspace{0.17em}sup}}_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}{u}_t(x)\mathrm{/}(\log \mathrm{\mid }x\mathrm{\mid }{)}^{1\mathrm{/}6}>0$ when $u_0$ 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.