Ebook: Heat Kernels for Elliptic and Sub-elliptic Operators: Methods and Techniques

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.

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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.

---

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.


Content:
Front Matter....Pages i-xviii
Front Matter....Pages 1-1
Introduction....Pages 3-11
A Brief Introduction to the Calculus of Variations....Pages 13-26
The Geometric Method....Pages 27-70
Commuting Operators....Pages 71-74
The Fourier Transform Method....Pages 75-88
The Eigenfunction Expansion Method....Pages 89-104
The Path Integral Approach....Pages 105-144
The Stochastic Analysis Method....Pages 145-197
Front Matter....Pages 199-199
Laplacians and Sub-Laplacians....Pages 201-223
Heat Kernels for Laplacians and Step-2 Sub-Laplacians....Pages 225-271
Heat Kernel for the Sub-Laplacian on the Sphere S 3 ....Pages 273-286
Front Matter....Pages 287-287
Finding Heat Kernels Using the Laguerre Calculus....Pages 289-331
Constructing Heat Kernels for Degenerate Elliptic Operators....Pages 333-348
Heat Kernel for the Kohn Laplacian on the Heisenberg Group....Pages 349-358
Front Matter....Pages 359-359
The Pseudo-Differential Operator Technique....Pages 361-416
Back Matter....Pages 417-436

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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.


Content:
Front Matter....Pages i-xviii
Front Matter....Pages 1-1
Introduction....Pages 3-11
A Brief Introduction to the Calculus of Variations....Pages 13-26
The Geometric Method....Pages 27-70
Commuting Operators....Pages 71-74
The Fourier Transform Method....Pages 75-88
The Eigenfunction Expansion Method....Pages 89-104
The Path Integral Approach....Pages 105-144
The Stochastic Analysis Method....Pages 145-197
Front Matter....Pages 199-199
Laplacians and Sub-Laplacians....Pages 201-223
Heat Kernels for Laplacians and Step-2 Sub-Laplacians....Pages 225-271
Heat Kernel for the Sub-Laplacian on the Sphere S 3 ....Pages 273-286
Front Matter....Pages 287-287
Finding Heat Kernels Using the Laguerre Calculus....Pages 289-331
Constructing Heat Kernels for Degenerate Elliptic Operators....Pages 333-348
Heat Kernel for the Kohn Laplacian on the Heisenberg Group....Pages 349-358
Front Matter....Pages 359-359
The Pseudo-Differential Operator Technique....Pages 361-416
Back Matter....Pages 417-436
....