A Boundedness Trichotomy for the Stochastic Heat Equation

We consider the stochastic heat equation with a multiplicative white noise forcing term under standard “intermitency conditions.” The main finding of this paper is that, under mild regularity hypotheses, the a.s.-boundedness of the solution $x\mapsto u(t,x)$ can be characterized generically by the decay rate, at $\pm \mathrm{\infty }$, of the initial function $u_0$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $\mathrm{\Lambda }:={\lim }_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}\mathrm{\mid }\log u_0(x)\mathrm{\mid }\mathrm{/}(\log \mathrm{\mid }x\mathrm{\mid }{)}^{2\mathrm{/}3}$.