Finite-Time Transition to Intermittency for a Stochastic Heat Equation Driven by the Square of a Gaussian Field
Philippe Mounaix
TL;DR
This work analyzes the finite-time spatial behavior of the stochastic heat equation driven by the square of a Gaussian field, focusing on the finite-time field $\mathcal{E}(x,g)=\psi(x,T)$. It establishes a critical coupling $g_c(T)$ defined by the divergence of $\langle \mathcal{E}(0,g)\rangle_S$ and shows a sharp phase transition: for $g<g_c(T)$ the field is spatially ergodic and non-intermittent, while for $g>g_c(T)$ it becomes spatially intermittent with a breakdown of ergodicity. The critical coupling is characterized via the maximal first eigenvalue $\mu_{\max}$ of the Gaussian field’s covariance operator, $g_c=\frac{1}{2\mu_{\max}}$, derived through a Feynman–Kac representation and spectral analysis. Unlike the well-studied case with linear noise in $S$, which exhibits asymptotic intermittency as $T\to\infty$, this work reveals a finite-time onset of intermittency and discusses open questions about the behavior at $g=g_c$ and extensions to higher dimensions and the diffractive setting.
Abstract
In this paper, we study the spatial behavior of the solution $ψ(x,t)$ to the stochastic heat equation $\partial_tψ(x,t)-\frac{1}{2}\partial^2_{x^2} ψ(x,t)=g\, S(x,t)^2\, ψ(x,t)$, with $0\le t\le T$, $x\in\mathbb{R}$, and $ψ(x,0)=1$. Here, $g>0$ is a coupling constant and $S(x,t)$ is a stationary, homogeneous, and ergodic Gaussian field. Focusing on $\mathcal{E}(x,g)\equiv ψ(x,T)$ 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.