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 $\Delta_G = \Delta_x + |x|^2 \Delta_u$ on $\mathbb{R}_x ^n \times \mathbb{R}_u ^{n'}$ for all $n , \, n' \ge 1$. 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.