Introduction to Infinite Dimensional Stochastic Analysis

Introduction to Infinite Dimensional Stochastic Analysis by Zhi-yuan Huang, Jia-an Yan.

The infinite dimensional analysis as a branch of mathematical sciences was formed in the late 19th and early 20th centuries. Motivated by problems in mathematical physics, the first steps in this field were taken by V. Volterra, R. GateallX, P. Levy and M. Frechet, among others (see the preface to L...

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Bibliographic Details
Main Authors: Zhi-yuan Huang (Author), Jia-an Yan (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Dordrecht : Springer Netherlands : Imprint: Springer, 2000.
Edition:1st ed. 2000.
Series:Mathematics and Its Applications ; 502
Springer eBook Collection.
Subjects:
Probabilities.
Functional analysis.
Operator theory.
Applied mathematics.
Engineering mathematics.
Harmonic analysis.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • I Foundations of Infinite Dimensional Analysis
  • §1. Linear operators on Hilbert spaces
  • §2. Fock spaces and second quantization
  • §3. Countably normed spaces and nuclear spaces
  • §4. Borel measures on topological linear spaces
  • II Malliavin Calculus
  • §1. Gaussian probability spaces and Wiener chaos decomposition
  • §2. Differential calculus of functionals, gradient and divergence operators
  • §3. Meyer’s inequalities and some consequences
  • §4. Densities of non-degenerate functionals
  • III Stochastic Calculus of Variation for Wiener Functionals
  • §1. Differential calculus of Itô functionals and regularity of heat kernels
  • §2. Potential theory over Wiener spaces and quasi-sure analysis
  • §3. Anticipating stochastic calculus
  • IV General Theory of White Noise Analysis
  • §1. General framework for white noise analysis
  • §2. Characterization of functional spaces
  • §3. Products and Wick products of functionals
  • §4. Moment characterization of distributions and positive distributions
  • V Linear Operators on Distribution Spaces
  • §1. Analytic calculus for distributions
  • §2. Continuous linear operators on distribution spaces
  • §3. Integral kernel operators and integral kernel representation for operators
  • §4. Applications to quantum physics
  • Appendix A Hermite polynomials and Hermite functions
  • Appendix B Locally convex spaces amd their dual spaces
  • 1. Semi-norms, norms and H-norms
  • 2. Locally convex topological linear spaces, bounded sets
  • 3. Projective topologies and projective limits
  • 4. Inductive topologies and inductive limits
  • 5. Dual spaces and weak topologies
  • 6. Compatibility and Mackey topology
  • 7. Strong topologies and reflexivity
  • 8. Dual maps
  • 9. Uniformly convex spaces and Banach-Saks’ theorem
  • Comments
  • References
  • Index of Symbols.

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