Introduction to Infinite Dimensional Stochastic Analysis

  • Zhi-yuan Huang
  • Jia-an Yan

Part of the Mathematics and Its Applications book series (MAIA, volume 502)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Zhi-yuan Huang, Jia-an Yan
    Pages 1-58
  3. Zhi-yuan Huang, Jia-an Yan
    Pages 59-112
  4. Zhi-yuan Huang, Jia-an Yan
    Pages 113-160
  5. Zhi-yuan Huang, Jia-an Yan
    Pages 161-209
  6. Zhi-yuan Huang, Jia-an Yan
    Pages 210-251
  7. Back Matter
    Pages 252-296

About this book

Introduction

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 Levy[2]). Nevertheless, the most fruitful direction in this field is the infinite dimensional integration theory initiated by N. Wiener and A. N. Kolmogorov which is closely related to the developments of the theory of stochastic processes. It was Wiener who constructed for the first time in 1923 a probability measure on the space of all continuous functions (i. e. the Wiener measure) which provided an ideal math­ ematical model for Brownian motion. Then some important properties of Wiener integrals, especially the quasi-invariance of Gaussian measures, were discovered by R. Cameron and W. Martin[l, 2, 3]. In 1931, Kolmogorov[l] deduced a second partial differential equation for transition probabilities of Markov processes order with continuous trajectories (i. e. diffusion processes) and thus revealed the deep connection between theories of differential equations and stochastic processes. The stochastic analysis created by K. Ito (also independently by Gihman [1]) in the forties is essentially an infinitesimal analysis for trajectories of stochastic processes. By virtue of Ito's stochastic differential equations one can construct diffusion processes via direct probabilistic methods and treat them as function­ als of Brownian paths (i. e. the Wiener functionals).

Keywords

Operator theory Probability theory Stochastic calculus Variation abstract harmonic analysis distribution functional analysis harmonic analysis probability space stochastic processes

Authors and affiliations

  • Zhi-yuan Huang
    • 1
  • Jia-an Yan
    • 2
  1. 1.Department of MathematicsHuazhong University of Science and TechnologyWuhanP. R. China
  2. 2.Institute of Applied MathematicsChinese Academy of SciencesBeijingP. R. China

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-011-4108-6
  • Copyright Information Kluwer Academic Publishers and Science Press 2000
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-94-010-5798-1
  • Online ISBN 978-94-011-4108-6
  • About this book
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