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 $\varepsilon\ll1$. 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.

Figures (1)

• Figure 1: From left to right, the graphs are $G^1$ for $\alpha=12$ and $\alpha=23$, $G^1$ for $\alpha=12$ and $\alpha=34$, $G^2$ for $\alpha=12$ and $\alpha=23$, and $G^2$ for $\alpha=12$ and $\alpha=34$.

Theorems & Definitions (34)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Lemma 2.2
• Lemma 3.1
• proof : Proof of Lemma \ref{['l.DZz.int']}
• Lemma 3.2
• proof