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From Brownian Motion to Schroedinger's Equation : Grundlehren Der Mathematischen Wissenschaften - Kai Lai Chung

From Brownian Motion to Schroedinger's Equation

Grundlehren Der Mathematischen Wissenschaften

Hardcover

Published: 1st July 2001
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Die Theorie der Brown'schen Bewegung hat sich in letzter Zeit zu einem sehr wichtigen Werkzeug bei der L”sung von Problemen in der mathematischen Physik entwickelt. In der vorliegenden, in sich abgeschlossenen und sehr gut lesbaren Abhandlung haben die zwei auf diesem Gebiet f�hrenden Autoren besonderen Wert auf Klarheit der Ausf�hrung gelegt. Das Werk richtet sich an all diejenigen Leser, deren Interessengebiet die Wahrscheinlichkeitstheorie mit Anwendungen in der Analysis und Math. Physik umfasst. Groáe Teile dieses Werkes stammen aus der Forschung der Autoren selbst (einiges davon zum 1. Mal in ver”ffentlichter Form), teilweise wurden wichtige Ergebnisse anderer Autoren zu diesem Gebiet in ausgefeilter, verbesserter Form aufgenommen. Interessenten: Studenten h”herer Semester und Forscher in der Mathematik und mathematischen Physik.

"This book is an excellent contribution to potential theory and stochastic processes, and recommended to researchers and graduate students of mathematics and mathematical physics." - M. Nagasawa, Universitat Zurich

Preparatory Materialp. 1
Basic Notationp. 1
Markov Process and Strong Markov Propertyp. 3
Construction of Brownian Motionp. 9
Harmonic Function and Poisson Kernelp. 13
Exit Time and Placep. 19
Dirichlet Boundary Value Problemp. 23
Appendix to Chapter 1p. 27
Notes on Chapter 1p. 28
Killed Brownian Motionp. 31
Feller Propertiesp. 31
Transition Densityp. 33
Green Potential and Functionp. 37
Compactness and Spectrump. 46
Laplacian as Generatorp. 48
Notes on Chapter 2p. 59
Schrodinger Operatorp. 61
The Schrodinger Equation and Class Jp. 61
Semigroup with Multiplicative Functionalp. 71
Potential Operator and its Inversep. 81
Schrodinger Infinitesimal Generatorp. 88
Notes on Chapter 3p. 101
Stopped Feynman-Kac Functionalp. 103
Harnack Inequality and Gauge Theoremp. 103
Properties of GDqp. 107
Fundamental Properties of the Gaugep. 112
Dirichlet Boundary Value Problem for the Schrodinger Equationp. 114
Representation Theoremp. 117
Equivalence Theorem for Gaugeabilityp. 124
Notes on Chapter 4p. 128
Conditional Brownian Motion and Conditional Gaugep. 131
Conditional Brownian Motionp. 131
Life in a Lipschitz Domainp. 135
Conditional Gauge for a Small Ballp. 144
General Gauge Theoremp. 149
Continuity of Weak Solutionsp. 152
New Approach to the Gauge Theoremp. 155
Notes on Chapter 5p. 157
Green Functionsp. 159
Basic Properties of the q-Green Functionp. 159
Inequalites for Green Functions in a Lipschitz Domain in Rd (d = 3)p. 163
Inequalities for Green Functions in a Jordan Domain in R2p. 175
Notes on Chapter 6p. 190
Conditional Gauge and q-Green Functionp. 191
Conditional Gauge Theoremp. 191
Approximation and Continuity of the Conditional Gaugep. 197
Extended Conditional Gauge Theoremp. 200
Representation of the Conditional Gaugep. 207
Notes on Chapter 7p. 213
Various Related Developmentsp. 215
Variation of Gaugep. 215
Variation of the Principal Eigenvaluep. 220
Principal Eigenfunction and Sharp Variationp. 224
Boundary Harnack Principle and Applicationp. 230
Schrodinger Equation in the Classical Settingp. 234
Dirichlet Problem and Truncated Gaugep. 236
Notes on Chapter 8p. 239
The Case of One Dimensionp. 241
Fundamental Expectationsp. 241
Gauge for a Finite or Infinite Intervalp. 248
Special Cases and Examplesp. 262
Local Time and Densityp. 270
Derivatives and Neumann's Problemp. 274
Notes on Chapter 9p. 278
Referencesp. 281
Indexp. 285
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540570301
ISBN-10: 3540570306
Series: Grundlehren Der Mathematischen Wissenschaften
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 292
Published: 1st July 2001
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 0.61
Edition Number: 2
Edition Type: Revised