Heat kernels for generalized Ornstein–Uhlenbeck operators

Abstract

In this article, we use the Hamiltonian and Lagrangian formalism to study the -dimensional extended Ornstein–Uhlenbeck operator where is an real matrix, is a real parameter and means the gradient. Given the boundary conditions, we find the solutions of the associated Hamiltonian system of . Then, we construct the action function by the Lagrangian function and use the van Vleck’s formula to obtain the volume element of the heat kernel. Finally, we discuss the regular and singular regions of this operator.

Keywords:

• heat kernel
• Ornstein–Uhlenbeck operators
• Hamiltonian system
• action function
• volume element

AMS Subject Classifications:

• 35K05
• 35K08
• 35K15

Acknowledgements

This research project was initiated when the first author visited the National Center for Theoretical Sciences, Hsinchu, Taiwan, during January 2013 and the final version of the paper was completed while the first author visited NCTS during May–August 2014. He would like to express his profound gratitude to the Director of NCTS, professor Winnie Li, for her invitation and for the warm hospitality extended to him during his stay in Taiwan.

Notes

Dedicated to Professor Wei Lin on the occasion of his 80th birthday.

The first author is partially supported by an NSF [grant number DMS-1203845]; Hong Kong RGC competitive earmarked research [grant number 601410].