Overview
- Replete with examples to facilitate understanding
- Explores a diverse arsenal of methods for finding explicit formulas for heat kernels
- Contains most of the heat kernels computable by means of elementary functions
- Approach of the authors unifies a number of different branches of mathematics and physics, including stochastic processes, differential geometry, special functions, quantum mechanics, and PDEs
- Includes supplementary material: sn.pub/extras
Part of the book series: Applied and Numerical Harmonic Analysis (ANHA)
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Table of contents (15 chapters)
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Back Matter
Keywords
- Brownian motion
- Fourier transform
- Laguerre calculus
- Van Vleck formula
- diffusion processes
- elliptic and sub-elliptic operators
- evolution operators
- heat kernel
- nilmanifolds
- nilpotent Lie groups
- parabolic operators
- psuedo-differential operators
- quantum mechanics
- quartic oscillator
- stochastic processes
- sub-Riemannian manifolds
- partial differential equations
About this book
With each methodology treated in its own chapter, this monograph is a thorough exploration of several theories that can be used to find explicit formulas for heat kernels for both elliptic and sub-elliptic operators. The authors show how to find heat kernels for classical operators by employing a number of different methods. Some of these methods come from stochastic processes, others from quantum physics, and yet others are purely mathematical.
What is new about this work is the sheer diversity of methods that are used to compute the heat kernels. It is interesting that such apparently distinct branches of mathematics, including stochastic processes, differential geometry, special functions, quantum mechanics and PDEs, all have a common concept – the heat kernel. This unifying concept, that brings together so many domains of mathematics, deserves dedicated study.
Heat Kernels for Elliptic and Sub-elliptic Operators is an ideal resource for graduate students, researchers, and practitioners in pure and applied mathematics as well as theoretical physicists interested in understanding different ways of approaching evolution operators.
Reviews
From the reviews:
“The present book provides a comprehensive presentation of several theories for finding explicit formulas for heat kernels for both elliptic and sub-elliptic operators. One of the nice features of the book is the diversity of methods used, coming from the theory of stochastic processes, differential geometry, special functions, quantum mechanics and PDEs. … the book is very well organized and essentially self-contained. Hence, it is perfect reference material for graduate students and researchers in harmonic analysis and sub-Riemannian geometry, as well as theoretical physicists.” (Fabio Nicola, Mathematical Reviews, Issue 2011 i)
“Authors of the present monograph devote themselves to finding the explicit formulas of heat kernels for elliptic and sub-elliptic operators. … there are plenty of exact formulas of the heat kernels for elliptic and sub-elliptic operators in this work. Most of them are represented by means of elementary functions. … Those results in this book are important to the experts for further studying the diffusion phenomena or the properties of solutions of parabolic equations with initial data. … this is also a good reference book.” (Jun-Qi Hu, Zentralblatt MATH, Vol. 1207, 2011)
Authors and Affiliations
Bibliographic Information
Book Title: Heat Kernels for Elliptic and Sub-elliptic Operators
Book Subtitle: Methods and Techniques
Authors: Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
Series Title: Applied and Numerical Harmonic Analysis
DOI: https://doi.org/10.1007/978-0-8176-4995-1
Publisher: Birkhäuser Boston, MA
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Birkhäuser Boston 2011
Hardcover ISBN: 978-0-8176-4994-4Published: 21 October 2010
eBook ISBN: 978-0-8176-4995-1Published: 10 October 2010
Series ISSN: 2296-5009
Series E-ISSN: 2296-5017
Edition Number: 1
Number of Pages: XVIII, 436
Number of Illustrations: 25 b/w illustrations
Topics: Partial Differential Equations, Mathematical Methods in Physics, Operator Theory, Differential Geometry, Probability Theory and Stochastic Processes, Abstract Harmonic Analysis