Heat Kernels, Old and New
Vol. 12 No. 1 (2017) P.1~P.37
DOI: || https://doi.org/10.21915/BIMAS.2017101 |
| ||10.21915/BIMAS.2017101 |
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.
Heat kernels, complex spheres, subLaplacians, Cayley transform.
MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35H20, 35K08; Secondary: 32W30, 53C17.
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Revised : 2015-04-01