## Functional Integrals: Approximate Evaluation and ApplicationsIntegration 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 |

401 | |

417 | |

### Other editions - View all

Functional Integrals: Approximate Evaluation and Applications A.D. Egorov,P.I. Sobolevsky,L.A. Yanovich No preview available - 2012 |

### Common terms and phrases

approximate evaluation arbitrary assumes the form Chapter characteristic functional coefficients composite formulae conditional Wiener measure construction continuous functions convergence correlation function B(t,s cubature cylindric function denote derivative diagrams differential divided difference operator eigenfunctions eigenvalues estimate evaluation of integrals exact for functional exact value example following approximate formula formulae for integrals formulae of form Fredholm determinant function f(t functional integrals functional integrals w.r.t. functional polynomials Hilbert space implies inequality integrals w.r.t. Gaussian integrals w.r.t. measure Integrals with Respect kernel Let us consider linear topological space Monte Carlo method norm obtain orthonormal basis polynomial of degree Proof pt(u quadrature process quasimeasure random process real nodes right side Rn(f satisfy the condition scalar product segment space of functions specified theorem third accuracy degree topological space trace class trigonometric polynomial valid w.r.t. Gaussian measure Wiener integral Wiener process zero mean

### 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.