Heat kernels for elliptic and sub-elliptic operators methods and techniques - Calin, Ovidiu; Chang, Der-Chen; Furutani, Kenro; Iwasaki, Chisato (Applied and numerical harmonic analysis., 2011)
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. The work is divided into four main parts: Part I treats the heat kernel by traditional methods, such as the Fourier transform method, paths integrals, variational calculus, and eigenvalue expansion; Part II deals with the heat kernel on nilpotent Lie groups and nilmanifolds; Part III examines Laguerre calculus applications; Part IV uses the method of pseudo-differential operators to describe heat kernels. Topics and features: ?comprehensive treatment from the point of view of distinct branches of mathematics, such as stochastic processes, differential geometry, special functions, quantum mechanics, and PDEs; ?novelty of the work is in the diverse methods used to compute heat kernels for elliptic and sub-elliptic operators; ?most of the heat kernels computable by means of elementary functions are covered in the work; ?self-contained material on stochastic processes and variational methods is included. 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.