Calin O., Chang D.-C., Furutani K., Iwasaki C. Heat Kernels for Elliptic and Sub-elliptic Operators: Methods and Techniques

Springer, 2010. — 456 p. — (Applied and Numerical Harmonic Analysis). — ISBN: 978-0-8176-4994-4.This monograph is a unified presentation of several theories of finding explicit formulas for heat kernels for both elliptic and sub-elliptic operators. These kernels are important in the theory of parabolic operators because they describe the distribution of heat on a given manifold as well as evolution phenomena and diffusion processes.
Heat Kernels for Elliptic and Sub-elliptic Operators is an ideal reference for graduate students, researchers in pure and applied mathematics, and theoretical physicists interested in understanding different ways of approaching evolution operators.Traditional Methods for Computing Heat KernelsA Brief Introduction to the Calculus of Variations
The GeometricMethod
Commuting Operators
The Fourier Transform Method
The Eigenfunction Expansion Method
The Path Integral Approach
The Stochastic Analysis Method
Heat Kernel on Nilpotent Lie Groups and Nilmanifolds
Laplacians and Sub-Laplacians
Heat Kernels for Laplacians and Step-2 Sub-Laplacians
Heat Kernel for the Sub-Laplacian on the Sphere S3
Laguerre Calculus and the FourierMethod
Finding Heat Kernels Using the Laguerre Calculus
Constructing Heat Kernels for Degenerate Elliptic Operators
Heat Kernel for the Kohn Laplacian on the Heisenberg Group
Pseudo-Differential Operators
The Pseudo-Differential Operator Technique