Functional Integrals: Approximate Evaluation and Applications

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Springer Science & Business Media, Mar 31, 1993 - Mathematics - 419 pages
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Integration in infinitely dimensional spaces (continual integration) is a powerful mathematical tool which is widely used in a number of fields of modern mathematics, such as analysis, the theory of differential and integral equations, probability theory and the theory of random processes. This monograph is devoted to numerical approximation methods of continual integration. A systematic description is given of the approximate computation methods of functional integrals on a wide class of measures, including measures generated by homogeneous random processes with independent increments and Gaussian processes. Many applications to problems which originate from analysis, probability and quantum physics are presented. This book will be of interest to mathematicians and physicists, including specialists in computational mathematics, functional and statistical physics, nuclear physics and quantum optics.
 

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Contents

Backgrounds from Analysis on Linear Topological Spaces
1
12 Definition of Functional Integrals with Respect to Measure Quasimeasure and Pseudomeasure Relations with Random Process Theory
5
13 Characteristic Functionals of Measures
7
14 Moments Semiinvariants Integrals of Cylindric Functions
11
Integrals with Respect to Gaussian Measures and Some Quasimeasures Exact Formulae Wick Polynomials Diagrams
15
22 Exact Formulae for Integrals of Special Functionals Infinitesimal Change of Measure
20
23 Integrals of Variations and of Derivatives of Functionals Wick Ordering Diagrams
26
24 Integration with Respect to Gaussian Measure in Particular Spaces
34
85 Cubature Formulae for Multiple Probabilistic Integrals
200
Approximations which Agree with Diagram Approaches
211
92 Approximate Integration of Functionals of Wick Exponents
215
93 Formulae which are Exact for Diagrams of a Given Type
219
94 Approximate Formulae for Integrals with Respect to Quasimeasures
226
95 Some Extensions Composite Formulae
229
Approximations of Integrals Based on Interpolation of Measure
235
102 Integrals with Respect to Wiener Measure Conditional Wiener Measure and Modular Measure
241

Integration in Linear Topological Spaces of Some Special Classes
47
32 Projective Limits of Linear Topological Spaces
48
33 Generalized Function Spaces
52
34 Integrals in Product Spaces
55
Approximate InterpolationType Formulae
65
42 Repeated Interpolation Taylors Formula
67
43 Construction Rules for Divided Difference Operators
68
44 Approximate Interpolation Formulae
77
Formulae Based on Characteristic Functional Approximations which Preserve a Given Number of Moments
81
52 Reducing the Number of Terms in Approximations
89
53 Approximate Formulae
101
Integrals with Respect to Gaussian Measures
109
62 Formulae Based on Approximations of the Correlation Functional
119
63 Stationary Gaussian Measures
128
64 Error Estimates for Approximate Formulae Based on Approximations of the Argument
130
65 Formulae which are Exact for Special Kinds of Functionals
134
66 Convergence of Functional Quadrature Processes
139
Integrals with Respect to Conditional Wiener Measure
147
72 Formulae of First Accuracy Degree
155
73 Third Accuracy Degree
158
74 Arbitrary Accuracy Degree
161
Integrals With Respect to Measures which Correspond to Uniform Processes with Independent Increments
167
81 Formulae of First Third and Fifth Accuracy Degrees
168
82 Arbitrary Accuracy Degree
176
83 Integrals with Respect to Measures Generated by Multidimensional Processes
189
84 Convergence of composite Formulae
193
103 Formulae Based on Measure Interpolation for Integrals of NonDifferentiable Functional
245
Integrals with Respect to Measures Generated by Solutions of Stochastic Equations Integrals Over Manifolds
249
112 Approximations of Integrals with Respect to Measures Generated by Stochastic Differential Equations over Martingales
253
113 Formula of Infinitesimal Change of Measure in Integrals with Respect to Measures Generated by Solutions of Ito Equations
260
114 Approximate Formulae for Integrals over Manifolds
266
Quadrature Formulae for Integrals of Special Form
277
122 Formulae Based on Trigonometric Interpolation
282
123 Quadrature Formulae with Equal Coefficients
292
124 Tables of Nodes and Coefficients of Quadrature Formula of Highest Accuracy Degree for Some Integrals
300
125 Formulae with the Minimal Residual Estimate
319
Evaluation of Integrals by MonteCarlo Method
327
132 Estimates for Integrals with Respect to Wiener Measure
331
133 Estimation of Integrals with Respect to Arbitrary Gaussian Measure in Space of Continuous Functions
334
134 A Sharper MonteCarlo Estimate of Functional Integrals
338
Approximate Formulae for Multiple Integrals with Respect to Gaussian Measure
343
141 Formulae of Third Accuracy Degree
344
142 Formulae of Fifth Accuracy Degree
350
143 Formulae of Seventh Accuracy Degree
357
144 Cubature Formulae for Multiple Integrals of a Certain Kind
359
Some Special Problems of Functional Integration
367
152 Application of Approximations Based on Measure Interpolation to Evaluation of GroundState Energy for Certain Quantum Systems
375
153 MeanSquare Approximation of Some Classes of Linear Functionals
378
154 Exact Formulae for Integrals with Respect to Gaussian and Conditional Gaussian Measures of Special Types of Functionals
391
Bibliography
401
Index
417
Copyright

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Common terms and phrases

Popular passages

Page 401 - Slavnov, AA and Faddeev, LD Introduction to Quantum Theory of Gauge Fields, Nauka, Moscow, 1978 (in Russian).
Page 401 - Techniques and Applications of Path Integration, J. Wiley & Sons, New York, 1981.
Page iv - Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, USA In all other countries, sold and distributed by Kluwer Academic Publishers Group, PO Box 322, 3300 AH Dordrecht, The Netherlands.
Page 409 - Formulae of arbitrary given degree of accuracy for the approximate calculation of continual Integrals with respect to Measures generated by homogeneous processes with Independent Increments.
Page 411 - On approximate evaluation of functional integral with respect to measures generated by solutions of stochastic equations, Izvestiya Akademii Nauk BSSR.
Page 409 - Billingsley, P. Convergence of Probability Measures, J. Wiley & Sons, New York, 1968.
Page 406 - Simon, B. The P(4>)2 Euclidean (quantum) field theory, Princeton University Press, Princeton, 1974.

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