Heat Kernels for Elliptic and Sub-elliptic Operators [electronic resource] : Methods and Techniques / by Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki.
Contributor(s): Chang, Der-Chen [author.] | Furutani, Kenro [author.] | Iwasaki, Chisato [author.] | SpringerLink (Online service)Material type: TextSeries: Applied and Numerical Harmonic Analysis: Publisher: Boston : Birkhäuser Boston, 2011Edition: 1Description: XVIII, 436 p. 25 illus. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9780817649951Subject(s): Mathematics | Harmonic analysis | Operator theory | Partial differential equations | Differential geometry | Probabilities | Physics | Mathematics | Partial Differential Equations | Mathematical Methods in Physics | Operator Theory | Differential Geometry | Probability Theory and Stochastic Processes | Abstract Harmonic AnalysisAdditional physical formats: Printed edition:: No titleDDC classification: 515.353 LOC classification: QA370-380Online resources: Click here to access online
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Part I. Traditional Methods for Computing Heat Kernels -- Introduction -- Stochastic Analysis Method -- A Brief Introduction to Calculus of Variations -- The Path Integral Approach -- The Geometric Method -- Commuting Operators -- Fourier Transform Method -- The Eigenfunctions Expansion Method -- Part II. Heat Kernel on Nilpotent Lie Groups and Nilmanifolds -- Laplacians and Sub-Laplacians -- Heat Kernels for Laplacians and Step 2 Sub-Laplacians -- Heat Kernel for Sub-Laplacian on the Sphere S^3 -- Part III. Laguerre Calculus and Fourier Method -- Finding Heat Kernels by Using Laguerre Calculus -- Constructing Heat Kernel for Degenerate Elliptic Operators -- Heat Kernel for the Kohn Laplacian on the Heisenberg Group -- Part IV. Pseudo-Differential Operators -- The Psuedo-Differential Operators Technique -- Bibliography -- Index.
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.