Heat Kernels, Old and New

This article is a resume of ongoing investigations into the nature and form of heat kernels of second order partial differential operators. Our operators are given as a sum of squares of bracket generating vector fields; thus they are (sub)elliptic and induce a (sub)Riemannian geometry. The principal part of a heat kernel of an elliptic operator is an exponential whose exponent is a solution of the associated Hamilton-Jacobi equation. Genuinely subelliptic heat kernels are given by integrals, where the integrands are similar in form to elliptic heat kernels. There are differences. In particular, some of the exponents in the known subelliptic integrands are solutions of a modified Hamilton-Jacobi equation. To clarify this difference we propose a calculation which may lead to an invariant interpretation of the modification.