Variational analysis for the heat kernels of Chandrasekhar-type operators

Abstract

In this paper, we study a class of degenerate elliptic operators with quadratic potentials by Hamiltonian formalism. Geodesics induced by the operators are explicitly characterized. With the help of a probabilistic ansatz, we generalize our kernel formulae for Ornstein–Uhlenbeck operator with quadratic potential and hence derive the heat kernel for our target operators in terms of integral representation.

Keywords:

• Hamiltonian system
• degenerate elliptic operator
• quadratic potential
• geodesics
• heat kernel

AMS Subject Classifications:

• Primary: 35J05
• Secondary: 35F21
• 15A24

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

The author is partially supported by the National Natural Science Foundation of China  [grant number 11501203], [grant number 11426109]; the Fundamental Research Funds for the Central Universities of China  [grant number 22A201514035], as well as the China Postdoctoral Science Foundation  [grant number 2016M600278].