Sharp Heat Kernel Bounds and Entropy in Metric Measure Spaces

TL;DR

This paper derives sharp Gaussian bounds for heat kernels in metric measure spaces with Ricci curvature bounds, and applies these to analyze entropy and Green functions, extending classical Riemannian results.

Contribution

It establishes sharp heat kernel bounds and entropy asymptotics in $ ext{RCD}(0,N)$ spaces, generalizing previous Riemannian and metric measure space results.

Findings

Sharp Gaussian bounds for heat kernels in $ ext{RCD}(0,N)$ spaces.

Monotonicity of Perelman entropy at large times.

Asymptotic behavior of Green functions and entropy.

Abstract

We establish sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying $\RCD(0,N)$ ( equivalently, $\RCD^\ast(0,N)$) condition with $N\in \mathbb{N}\setminus\{1\}$ and having maximum volume growth, and then show its application on the large-time asymptotics of the heat kernel, sharp bounds on the (minimal) Green function, and above all, the large-time asymptotics of the Perelman entropy and the Nash entropy, where for the former the monotonicity of the Perelman entropy is proved. The results generalize the corresponding ones in Riemannian manifolds and also in metric measure spaces obtained recently by the author with R. Jiang and H. Zhang in \cite{JLZ2014}.