Log Log Fluctuations of the Stochastic Heat Flow

TL;DR

This paper investigates the fluctuations of the stochastic heat flow with constant initial data, demonstrating that its spatial average exhibits a Gaussian limit after specific logarithmic centering and scaling.

Contribution

It establishes a pointwise central limit theorem for the logarithm of the spatial average of the stochastic heat flow, revealing its Gaussian fluctuation behavior.

Findings

Logarithm of the spatial average converges to Gaussian distribution.

Centered by -0.5 log log epsilon^{-1} and scaled by sqrt(log log epsilon^{-1}).

Provides a precise asymptotic description of fluctuations.

Abstract

We study the stochastic heat flow with constant initial data and analyze its spatial average on the scale of $\varepsilon\ll1$. We prove that the logarithm of the averaged process satisfies a pointwise central limit theorem: After being centered by $-\tfrac{1}{2}\log\log \varepsilon^{-1}$ and scaled down by $\sqrt{\log\log \varepsilon^{-1}}$, it converges in distribution to a standard Gaussian.