Uniform heat kernel asymptotics for the Grushin operator

Abstract

Adopting the powerful methods introduced in \cite{li2021carnotcaratheodory, LZ2025}, we investigate the asymptotic behaviour at infinity for the heat kernel associated with the Grushin operator $\hat{I}{”}_G=\hat{I}{”}_x+|x{|}^2\hat{I}{”}_u$ on ${\mathbb{R}}_x^n\times {\mathbb{R}}_u^{n^{\prime }}$ for all $n,n^{\prime }\ge 1$. We further establish sharp bounds for its spatial derivatives. As a by-product, precise estimates and small-time asymptotic behaviour for the Grushin heat kernel will be provided, as explicit as one can possibly hope for.

15 pages