Heat kernels for a class of degenerate elliptic operators using stochastic method

Abstract

Formulas for heat kernels are found for degenerate elliptic operators by finding the probability density of the associated Ito diffusion. The formulas involve an integral of a product between a volume function and an exponential term.

Keywords:

• heat kernel
• Brownian motion
• degenerate elliptic operator

AMS Subject Classifications::

• Primary: 53C17
• Secondary: 34K10, 35H20

Acknowledgements

This article was written while Der-Chen Chang and Ovidiu Calin were visiting the Hong Kong University of Science and Technology, during the Winter of 2010–2011. The authors were partially supported by the NSF Grant DMS nos. 0631541, 0802803, by the Hong-Kong RGC grant nos. 600607, 601410, and a competitive research grant at Georgetown University.