Noise sensitivity for stochastic heat and Schrödinger equation
Yu Gu, Tomasz Komorowski
TL;DR
The paper addresses noise sensitivity and chaos onset in linear SPDEs driven by a Gaussian field with covariance $E[V(t,x)V(s,y)]=\delta(t-s)R(x-y)$. Using Wiener chaos expansions and the Fourier spectrum $\mathcal{N}_t$, it proves a central limit theorem for $\mathcal{N}_t$, i.e., $(\mathcal{N}_t-\mu t)/(\sigma\sqrt{t}) \Rightarrow N(0,1)$ as $t\to\infty$, with $\mu,\sigma>0$ depending on the model. For the Itô-Schrödinger equation the limiting fluctuation is Gaussian with mean $R(0)t$ and variance $R(0)t$, while for the stochastic heat equation with inverse temperature $\beta$ the limit is Gaussian with mean $\tfrac{1}{2}\beta^4 t$ and variance $\beta^4$, reflecting a Poisson-like chaos intensity of order $t$. The onset of chaos occurs on the perturbation scale $s\sim t^{-1}$, as quantified by decorrelation under the Ornstein–Uhlenbeck perturbation, and the analysis links the second-mMoment behavior to a kinetic/Poisson framework, suggesting broader implications for nonlinear SPDEs and continuum directed polymers.
Abstract
In this note, we consider the stochastic heat and Schrödinger equation, and show that, at time $t$, the onset of the chaos occurs on the scale of $1/t$, and the Fourier spectrum of the solution is asymptotically Gaussian after centering and rescaling.