Finite-Time Transition to Intermittency for a Stochastic Heat Equation Driven by the Square of a Gaussian Field

Abstract

In this paper, we study the spatial behavior of the solution $\psi(x,t)$ to the stochastic heat equation $\partial_t\psi(x,t)-\frac{1}{2}\partial^2_{x^2} \psi(x,t)=g\, S(x,t)^2\, \psi(x,t)$, with $0\le t\le T$, $x\in\mathbb{R}$, and $\psi(x,0)=1$. Here, $g>0$ is a coupling constant and $S(x,t)$ is a stationary, homogeneous, and ergodic Gaussian field. Focusing on $\mathcal{E}(x,g)\equiv \psi(x,T)$ at a finite time $T>0$, we identify the critical coupling $g_c(T)$ above which the average of $\mathcal{E}(0,g)$ diverges. We show that in the subcritical regime $g<g_c(T)$, $\mathcal{E}(x,g)$ is spatially ergodic, with no intermittency, while in the supercritical regime $g>g_c(T)$ it becomes spatially intermittent and loses ergodicity. Our results differ from the extensively studied case where $S(x,t)^2$ is replaced by $S(x,t)$, in which intermittency appears only asymptotically as $T\to +\infty$, with no finite-time intermittency.