A Boundedness Trichotomy for the Stochastic Heat Equation

Abstract: 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 $\pm\infty$, of the initial function $u_0$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $\Lambda:= \lim_{|x|\to\infty} |\log u_0(x)|/(\log|x|)^{2/3}$.