Integration in Hilbert Space

ISBN-13: 9783642656347 / Angielski / Miękka / 2011 / 180 str.

A. V. Skorohod;K. Wickwire

Opis

Integration in function spaces arose in probability theory when a gen eral theory of random processes was constructed. Here credit is cer tainly due to N. Wiener, who constructed a measure in function space, integrals-with respect to which express the mean value of functionals of Brownian motion trajectories. Brownian trajectories had previously been considered as merely physical (rather than mathematical) phe nomena. A. N. Kolmogorov generalized Wiener's construction to allow one to establish the existence of a measure corresponding to an arbitrary random process. These investigations were the beginning of the development of the theory of stochastic processes. A considerable part of this theory involves the solution of problems in the theory of measures on function spaces in the specific language of stochastic pro cesses. For example, finding the properties of sample functions is connected with the problem of the existence of a measure on some space; certain problems in statistics reduce to the calculation of the density of one measure w. r. t. another one, and the study of transformations of random processes leads to the study of transformations of function spaces with measure. One must note that the language of probability theory tends to obscure the results obtained in these areas for mathematicians working in other fields. Another dir, ection leading to the study of integrals in function space is the theory and application of differential equations. A. N."

Spis treści

1. Definition of a Measure in Hubert Space.- § 1. Measurable Hubert Spaces.- § 2. Weak Distributions.- § 3. The Characteristic Functional. Moment Functional.- § 4. The Minlos-Sazonov Theorem.- § 5- Gaussian Measures.- § 6. Generalized Measures in Hubert Space.- 2. Measurable Functions on Hubert Space.- § 7. Measurable Linear Functional.- § 8. Measurable Linear Operators.- § 9. Measurable Polynomial Functions.- § 10. Square-integrable Polynomials.- § 11. Orthogonal Systems of Polynomials.- § 12. Polynomials Orthogonal with Respect to a Weight Function.- 3. Absolute Continuity of Measures.- § 13. The Radon-Nikodym Theorem. Conditional Measures.- § 14. Martingales and Semi-Martingales..- § 15. General Conditions for Absolute Continuity.- § 16. Absolute Continuity of Product Measures.- § 17. Absolute Continuity of Gaussian Measures.- § 18. Absolute Continuity of Mixed Measures.- 4. Admissible Shifts and Quasi-invariant Measures.- § 19. Admissible Shifts of Measures.- § 20. Admissible Directions.- § 24. Differentiation of Measures w.r.t. a Direction.- § 22. An Admissibility Condition for Shifts.- § 23. Quasi-in variant Measures.- 5. Some Questions of Analysis in Hubert Space.- § 24. The Substitution Formula and Absolute Continuity.- § 25. Linear Transformations.- § 26. Absolute Continuity of Measures under Nonlinear Transformation.- § 27. Surface Integrals..- § 28. Gauss'Formula.- Bibliographic Notes.