Uniform heat kernel asymptotics for the Grushin operator

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 ${\mathrm{\Delta }}_G={\mathrm{\Delta }}_x+\mathrm{\mid }x{\mathrm{\mid }}^2{\mathrm{\Delta }}_u$ on ${\mathbb{R}}_x^n\times {\mathbb{R}}_u^{n^{\mathrm{\prime }}}$ for all $n,n^{\mathrm{\prime }}\ge 1$. 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.