Heat Kernels for High Step Subelliptic Operators

Subelliptic operators are at the intersection of many areas of analysis and geome- try — including nilpotent Lie group theory, strongly pseudo-convex domains in several complex variables, probability theory of degenerate diffusion process, sub-Riemannian geometry, control theory, Brownian motion, and semiclassical analysis in quantum me- chanics. Subelliptic operators have been studied extensively by many mathematicians in the past several decades, and there have been many related investigations on constructing explicit formulas for fundamental solutions and heat kernels. On the other hand, most if not all of the existing results refer to step two operators — such as the Heisenberg subLaplacian, the step two Grushin operator and subLaplacians on spheres.

Consider the Heisenberg subLaplacian for example, which is a step two subelliptic operator. There are several different known methods to derive its heat kernel, including the probabilistic method, the pseudo-differential operator technique, Fourier transform, stochastic calculus, path integrals, and the geometric method. However, all of these methods fail to derive formulas for the heat kernels of higher step operators (step > 2).

Consequently, an interesting issue is how to derive closed-form formulas for the heat kernels of such higher step (step > 2) subelliptic operators, to be considered in this research project. This will include the step three Grushin operator, the subLaplacian on the Engel group, and the Martinet operator. We will start by extending the geometric method (complex Hamilton-Jacobi theory) for step two operators to higher step cases, and then apply the theory to the step three Grushin operator and the Engel subLaplacian. After the heat kernel of the step three Grushin operator is obtained, the heat kernels for the quartic oscillator and double-well potential will be treated to elucidate some important physical models.