Noise sensitivity for stochastic heat and Schr\"odinger equation

TL;DR

This paper investigates the noise sensitivity of solutions to the stochastic heat and Schrödinger equations, revealing that chaos emerges on a scale of 1/t and the Fourier spectrum becomes Gaussian after appropriate normalization.

Contribution

It demonstrates the precise scale at which chaos appears and characterizes the asymptotic Gaussianity of the Fourier spectrum for these stochastic equations.

Findings

Chaos onset occurs on the scale of 1/t

Fourier spectrum becomes asymptotically Gaussian

Results apply to both heat and Schrödinger equations

Abstract

In this note, we consider the stochastic heat and Schr\"odinger 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.