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.
Inhaltsverzeichnis
1. Backgrounds from Analysis on Linear Topological Space. 2. Integrals with Respect to Gaussian Measures and Some Quasimeasures: Exact Formulae, Wick Polynomials, Diagrams. 3. Integration in Linear Topological Spaces of Some Special Classes. 4. Approximate Interpolation-Type Formulae. 5. Formulae Based on Characteristic Functional Approximations which Preserve a Given Number of Moments. 6. Integrals with Respect to Gaussian Measures. 7. Integrals with Respect to Conditional Wiener Measure. 8. Integrals with Respect to Measures which Correspond to Uniform Processes with Independent Increments. 9. Approximations which Agree with Diagram Approaches. 10. Approximations of Integrals Based on Interpolation of Measure. 11. Integrals with Respect to Measures Generated by Solutions of Stochastic Equations. Integrals over Manifolds. 12. Quadrature Formulae for Integrals of Special Form. 13. Evaluation of Integrals by Monte-Carlo Method. 14. Approximate Formulae for Multiple Integrals with Respect to Gaussian Measures. 15. Some Special Problems of Functional Integration. Bibliography. Index.
