Heat kernels for elliptic and sub-elliptic operators: methods and techniques
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Birkhäuser (Springer)
2011
|
| Schriftenreihe: | Applied and numerical harmonic analysis
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (p. 425-429) and index |
| Beschreibung: | XVIII, 433 S. graph. Darst. |
| ISBN: | 0817649948 9780817649944 |
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| 245 | 1 | 0 | |a Heat kernels for elliptic and sub-elliptic operators |b methods and techniques |c Ovidiu Calin ... [et al.] |
| 264 | 1 | |a New York, NY |b Birkhäuser (Springer) |c 2011 | |
| 300 | |a XVIII, 433 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Applied and numerical harmonic analysis | |
| 500 | |a Includes bibliographical references (p. 425-429) and index | ||
| 650 | 4 | |a Heat equation | |
| 650 | 4 | |a Operator theory | |
| 650 | 0 | 7 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Calin, Ovidiu L. |d 1971- |e Sonstige |0 (DE-588)128990554 |4 oth | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-0-8176-4995-1 |
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Datensatz im Suchindex
| _version_ | 1804149684851179520 |
|---|---|
| adam_text | Titel: Heat Kernels for Elliptic and Sub-elliptic Operators
Autor: Calin, Ovidiu
Jahr: 2011
Contents
Part I Traditional Methods for Computing Heat Kernels
1 Introduction................................................................... 3
1.1 Physical Significance of the Heat Equation.......................... 3
1.2 The Fundamental Solution ........................................... 5
1.3 The Heat Equation in Its Setting..................................... 6
1.4 Methodology.......................................................... 7
1.5 Theta-Functions....................................................... 7
1.6 Some Useful Integrals................................................ 10
2 A Brief Introduction to the Calculus of Variations....................... 13
2.1 Lagrangian Mechanics................................................ 13
2.1.1 The First Variation.......................................... 14
2.1.2 The Second Variation....................................... 15
2.1.3 Geometrical Interpretation.................................. 16
2.1.4 The Case of RiemannianGeometry........................ 18
2.1.5 Examples of Lagrangian Dynamics........................ 20
2.2 Hamiltonian Mechanics.............................................. 23
2.3 The Hamilton-Jacobi Equation...................................... 24
3 The Geometrie Method...................................................... 27
3.1 Heat Kernel for L = jTü,j~iaiJBi9 i.............................. 27
3.2 Interpretation of the Volume Function............................... 30
3.3 The Operator L= jY11=ihiB% ..................................... 3I
3.4 The Generalized Transport Equation................................. 33
3.5 Solving the Generalized Transport Equation........................ 38
3.6 The Operator L = ±ym(dj + d*),/» ? N.......................... 40
3.7 Heat Kernels on Curved Spaces...................................... 45
3.8 Heat Kernel at theCut-Locus........................................ 46
3.9 Heat Kernel at the Conjugate Locus................................. 47
3.10 Heat Kernel on the Half-Line........................................ 47
3.11 Heat Kernel on S .................................................... 48
3.12 Heat Kernel on the Segment [0, T]................................... 49
xiv Contents
3.13 Heat Kernel on the Cylinder.......................................... 50
3.14 Operators with Potentials............................................. 52
3.15 The Linear Potential.................................................. 54
3.16 The Quadratic Potential............................................... 58
3.17 The Operator j£35,- ± a2 x 2 .................................... 60
3.18 The Operator L = ffi + j? ........................................ 64
4 Commuting Operators....................................................... 71
4.1 Commuting Operators................................................ 71
4.1.1 The Operator L = ±(d2 + d2)............................. 71
4.1.2 The Operator L= i(3j + yd2)........................... 72
4.1.3 The Operator L = L(xa2 + _yfl2).......................... 72
4.1.4 Sum of Squares of Linear Potentials....................... 72
4.1.5 The Operator IL = ±(jc2 + .y2)(d2 + d2)................. 73
5 The Fourier Transform Method............................................ 75
5.1 The Algorithm ........................................................ 75
5.2 Heat Kernel for the Grushin Operator............................... 76
5.3 Heat Kernel for aJ+jcMj, ......................................... 77
5.4 The Formula of Beals, Gaveau and Greiner......................... 78
5.5 Kolmogorov Operators............................................... 80
5.6 The Operator (r + xd2............................................. 85
5.7 The Generalized Grushin Operator ££ =i d2. + 2|jc|2d2.......... 85
5.8 The Heisenberg Laplacian............................................ 87
6 The Eigenfunction Expansion Method..................................... 89
6.1 General Results....................................................... 89
6.2 Mehler s Formula and Applications.................................. 91
6.3 Hille-Hardy s Formula and Applications............................ 94
6.4 Poisson s Summation Formula and Applications................... 98
6.4.1 Heat Kerneion Sl........................................... 99
6.4.2 Heat Kernel on the Segment [0, T].........................100
6.5 Legendre s Polynomials and Applications...........................102
6.6 Legendre Functions and Applications...............................104
7 The Path Integral Approach................................................105
7.1 Introducing Path Integrals............................................105
7.2 Paths Integrals via Trotter s Formula................................107
7.3 Formal Algorithm for Obtaining Heat Kernels......................112
7.4 The Operator |02.....................................................118
7.5 The Operator i(dj + 3*)............................................123
7.6 Heat Kernel for L = ±(d2 - Bx)....................................125
7.7 Heat Kernel for L = ±x2d2x + xöx..................................126
7.8 The Hermite Operator i(3j - aV)................................126
Contents xv
7.9 Evaluating Path Integrals.............................................132
7.9.1 Van Vleck s Formula........................................132
7.9.2 Applications of van Vleck s Formula......................133
7.9.3 The Feynman-Kac Formula................................138
7.10 Non-Commutativity of Sums of Squares............................140
7.11 Path Integrals and Sub-Elliptic Operators...........................143
8 The Stochastic Analysis Method............................................145
8.1 Elements of Stochastic Processes....................................145
8.2 Ito Diffusion...........................................................152
8.3 The Generator of an Ito Diffusion....................................154
8.4 Kolmogorov s Backward Equation and Heat Kernel................156
8.5 Algorithm for Finding the Heat Kernel..............................157
8.6 Finding the Transition Density.......................................159
8.7 Kolmogorov s Forward Equation....................................166
8.8 The Operator a2x - xdx.............................................167
8.9 Generalized Brownian Motion.......................................168
8.10 Linear Noise...........................................................170
8.11 Geometrie Brownian Motion.........................................171
8.12 Mean Reverting Ornstein-Uhlenbeck Process......................174
8.13 Bessel Operator and Bessel Process.................................176
8.14 Brownian Motion on a Circle........................................181
8.15 An Example of a Heat Kernel........................................183
8.16 A Two-Dimensional Case............................................185
8.17 Kolmogorov s Operator...............................................188
8.18 The Operator ijc2322.................................................189
8.19 Grushin s Operator....................................................191
8.20 Squared Bessel Process...............................................192
8.21 CIR Processes.........................................................193
8.22 Limitation of the Method............................................193
8.23 Operators with Potential and the Feynman-Kac Formula..........195
Part II Heat Kernel on Nilpotent Lie Groups and Nilmanifolds
9 Laplacians and Sub-Laplacians.............................................201
9.1 Sub-Riemannian Structure and Heat Kernels........................201
9.1.1 Sub-Riemannian Structure..................................201
9.1.2 Heat Kernel of the Sub-Laplacian and Laplacian..........203
9.1.3 The Volume Form...........................................206
9.2 Laplacian and Sub-Laplacian on Nilpotent Lie Groups.............207
9.2.1 Nilpotent Lie Groups........................................207
9.2.2 The Heisenberg Group......................................209
9.2.3 Higher-Dimensional Heisenberg Groups..................211
9.2.4 Quaternionic Heisenberg Group............................213
9.2.5 Heisenberg-Type Lie Algebra..............................216
xvi Contents
9.2.6 Free Two-Step Nilpotent Lie Algebra......................219
9.2.7 The Engel Group............................................220
10 Heat Kernels for Laplacians and Step-2 Sub-Laplacians................225
10.1 Spectral Decomposition and Heat Kernel............................225
10.2 Complex Hamilton-Jacobi Theory...................................229
10.2.1 Path Integrals and Integral Expression
ofa Heat Kernel.............................................230
10.2.2 A Solution of a Hamilton-Jacobi Equation................232
10.2.3 The Generalized Transport Equation.......................237
10.2.4 Heat Kernel for the Sub-Laplacian.........................244
10.2.5 Heat Kernel for Laplacian 1.................................251
10.2.6 Heat Kernel for Laplacian II................................252
10.2.7 Examples....................................................255
10.3 Grushin-Type Operators and the Heat Kernel.......................263
10.3.1 Grushin-Type Operators....................................263
10.3.2 Heisenberg Group Case.....................................265
10.3.3 Heat Kernel of the Grushin Operator.......................267
11 Heat Kernel for the Sub-Laplacian on the Sphere S3....................273
11.1 Sub-Riemannian Structure and Sub-Laplacian on the Sphere S3 . .273
11.1.1 S3 in the Quaternion Number Field........................273
11.1.2 Heat Kernel of the Sub-Laplacian on S3...................277
11.1.3 Spherical Grushin Operator and the Heat Kernel..........282
Part III Laguerre Calculus and the Fourier Method
12 Finding Heat Kernels Using the Laguerre Calculus........ .............289
12.1 Introduction...........................................................289
12.2 Laguerre Calculus.....................................................291
12.2.1 Twisted Convolution........................................292
12.2.2 P.V. Convolution Operators.................................293
12.2.3 Laguerre Functions..........................................294
12.2.4 Laguerre Calculus on Hi....................................295
12.2.5 Laguerre Calculus on H« ...................................296
12.2.6 Left Invariant Differential Operators.......................297
12.3 The Heisenberg Sub-Laplacian.......................................300
12.4 Powers of the Sub-Laplacian.........................................303
12.5 Heat Kernel for the Operator C%.....................................306
12.6 Fundamental Solution of the Paneitz Operator......................309
12.6.1 Laguerre Tensor of the Paneitz Operator...................311
12.6.2 Fundamental Solution: The Case n 2....................313
12.6.3 Fundamental Solution: The Case n = 1 ...................316
12.7 Heat Kernel of the Paneitz Operator.................................318
12.8 Projection and Relative Fundamental Solution......................321
Contents xvii
12.9 The Kernel 4 m(z, t) for m n.......................................326
12.10 The Isotropie Heisenberg Group.....................................328
12.11 Conclusions...........................................................330
13 Constructing Heat Kernels for Degenerate Elliptic Operators..........333
13.1 Introduction...........................................................333
13.2 Finding Heat Kernels for Operators L}¦¦, j = 1.....5..............335
13.3 Some Explicit Calculations...........................................340
13.3.1 Laplace Operator............................................341
13.3.2 Kolmogorov Operator.......................................342
13.3.3 The Operator L3 ............................................344
13.3.4 The Operator L4 ............................................346
13.3.5 The Operator L5 ............................................347
14 Heat Kernel for the Kohn Laplacian on the Heisenberg Group.........349
14.1 The Kohn Laplacian on the Heisenberg Group......................349
14.2 Füll Fourier Transform on the Group................................351
14.3 Solving the Transformed Heat Equation
Using Hermite Functions.............................................353
14.4 Conclusions...........................................................358
Part IV Pseudo-Differential Operators
15 The Pseudo-Differential Operator Technique.............................361
15.1 Basic Results of Pseudo-Differential Operators.....................361
15.1.1 Definition of Pseudo-Differential Operators ..............362
15.1.2 Calculus with Pseudo-Differential Operators..............363
15.1.3 Proofof Lemma 15.1.3.....................................366
15.2 Fundamental Solution by Symbolic Calculus:
The Nondegenerate Case.............................................372
15.3 Basic Results for Pseudo-Differential Operators of Weyl Symbols 376
15.3.1 Calculus with Pseudo-Differential Operators..............376
15.4 Fundamental Solution by Symbolic Calculus:
The Degenerate Case.................................................382
15.4.1 Construction of the Symbol.................................382
15.4.2 The Symbol pm = jE5=i 7;..............................396
15.4.3 The Special Case of Quadratic Symbols...................397
15.4.4 A Key Theorem for Eigenfunction Expansion.............400
15.5 The Hermite Operator................................................400
15.5.1 Exact Form of the Symbol
of the Fundamental Solution................................401
15.5.2 Eigenfunction Expansion...................................403
15.6 The Grushin Operator.................................................407
15.7 Exact Form of the Symbol of the Fundamental
Solution for the Sub-Laplacian.......................................408
xviii Contents
15.8 The Sub-Laplacian on Step-2 Nilpotent Lie Groups................410
15.9 The Kolmogorov Operator...........................................415
Appendix............................................................................417
Conclusions.........................................................................419
List of Frequently Used Notations and Symbols................................423
References...........................................................................425
Index.................................................................................431
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| id | DE-604.BV040596828 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:26:57Z |
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| isbn | 0817649948 9780817649944 |
| language | English |
| lccn | 2010937587 |
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| spelling | Heat kernels for elliptic and sub-elliptic operators methods and techniques Ovidiu Calin ... [et al.] New York, NY Birkhäuser (Springer) 2011 XVIII, 433 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied and numerical harmonic analysis Includes bibliographical references (p. 425-429) and index Heat equation Operator theory Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Wärmeleitungskern (DE-588)4781182-1 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s Wärmeleitungskern (DE-588)4781182-1 s DE-604 Calin, Ovidiu L. 1971- Sonstige (DE-588)128990554 oth Erscheint auch als Online-Ausgabe 978-0-8176-4995-1 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025424654&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Heat kernels for elliptic and sub-elliptic operators methods and techniques Heat equation Operator theory Elliptische Differentialgleichung (DE-588)4014485-9 gnd Wärmeleitungskern (DE-588)4781182-1 gnd |
| subject_GND | (DE-588)4014485-9 (DE-588)4781182-1 |
| title | Heat kernels for elliptic and sub-elliptic operators methods and techniques |
| title_auth | Heat kernels for elliptic and sub-elliptic operators methods and techniques |
| title_exact_search | Heat kernels for elliptic and sub-elliptic operators methods and techniques |
| title_full | Heat kernels for elliptic and sub-elliptic operators methods and techniques Ovidiu Calin ... [et al.] |
| title_fullStr | Heat kernels for elliptic and sub-elliptic operators methods and techniques Ovidiu Calin ... [et al.] |
| title_full_unstemmed | Heat kernels for elliptic and sub-elliptic operators methods and techniques Ovidiu Calin ... [et al.] |
| title_short | Heat kernels for elliptic and sub-elliptic operators |
| title_sort | heat kernels for elliptic and sub elliptic operators methods and techniques |
| title_sub | methods and techniques |
| topic | Heat equation Operator theory Elliptische Differentialgleichung (DE-588)4014485-9 gnd Wärmeleitungskern (DE-588)4781182-1 gnd |
| topic_facet | Heat equation Operator theory Elliptische Differentialgleichung Wärmeleitungskern |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025424654&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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