Heat kernel estimates for the Grusin operator

TL;DR

This paper derives heat kernel estimates for the Grusin operator by analyzing its associated geometry and geodesics, providing Gaussian bounds in terms of the Carnot-Caratheodory distance, and connecting to known results on the Heisenberg group.

Contribution

It introduces explicit heat kernel bounds for the Grusin operator using geometric analysis, extending known results to this specific subelliptic setting.

Findings

Derived explicit Gaussian bounds for the heat kernel.

Connected the geometry of the Grusin operator to the Carnot-Caratheodory distance.

Showed the results align with known estimates on the Heisenberg group for n≥2.

Abstract

We study the geometry associated to the Grusin operator G=\Delta_{x}+|x|^{2}\partial_{u}^{2} on \mathbb{R}_{x}^{n}\times\mathbb{R}_{u}, to obtain heat kernel estimates for this operator. The main work is to find the shortest geodesics connecting two given points in $\mathbb{R}^{n+1}$. This gives the Carnot-Caratheodory distance d_{CC}, associated to this operator. The main result in the second part is to give Gaussian bounds for the heat kernel K_{t} in terms of the Carnot-Caratheodory distance. In particular we obtain the following estimate |k_{t}(\zeta,\eta)|\leq C t^{-\frac{n}{2}-1}\min(1+\frac{d_{CC}(\zeta,\eta)} {|x+\xi|},1+\frac{d_{CC}(\zeta,\eta)^{2}}{4t})^{\alpha}e^{-\frac{1}{4t}d_{CC} (\zeta,\eta)^{2}} for all $\zeta=(x,u_{1}), \eta=(\xi,u)\in\mathbb{R}^{n+1}$, where $\alpha = \max{\frac{n}{2}-1,0}$. Here the homogeneous dimension is q=n+2, so that $\frac{n}{2}-1=\frac{q-4}{2}$. This shows that our result for $n\geq2$ corresponds with the result on the Heisenberg group, which was given by Beals, Gaveau, Greiner in [1].