A Boundedness Trichotomy for the Stochastic Heat Equation

TL;DR

This paper classifies the almost sure boundedness of solutions to the stochastic heat equation with multiplicative noise into three regimes based on the decay rate of the initial condition at infinity, under standard intermittency conditions.

Contribution

It establishes a generic trichotomy of boundedness regimes for the stochastic heat equation solutions based on initial decay rates, under mild regularity assumptions.

Findings

Three boundedness regimes depending on the decay rate of initial data.

Characterization of boundedness via the limit involving the initial function.

Solution boundedness determined by the decay rate of initial condition at infinity.

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 $\Lambda:= \lim_{|x|\to\infty} |\log u_0(x)|/(\log|x|)^{2/3}$.