Log Log Fluctuations of the Stochastic Heat Flow

Abstract

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

31 pages; 1 figure