Brownian Motion

Following the publication of the Japanese edition of this book, several inter­ esting developments took place in the area. The author wanted to describe some of these, as well as to offer suggestions concerning future problems which he hoped would stimulate readers working in this field. For these reasons, Chapter 8 was added. Apart from the additional chapter and a few minor changes made by the author, this translation closely follows the text of the original Japanese edition. We would like to thank Professor J. L. Doob for his helpful comments on the English edition. T. Hida T. P. Speed v Preface The physical phenomenon described by Robert Brown was the complex and erratic motion of grains of pollen suspended in a liquid. In the many years which have passed since this description, Brownian motion has become an object of study in pure as well as applied mathematics. Even now many of its important properties are being discovered, and doubtless new and useful aspects remain to be discovered. We are getting a more and more intimate understanding of Brownian motion. Following the publication of the Japanese edition of this book, several interƯ esting developments took place in the area. The author wanted to describe some of these, as well as to offer suggestions concerning future problems which he hoped would stimulate readers working in this field. For these reasons, Chapter 8 was added. Apart from the additional chapter and a few minor changes made by the author, this translation closely follows the text of the original Japanese edition. We would like to thank Professor J.L. Doob for his helpful comments on the English edition. T. Hida T.P. Speed v Preface The physical phenomenon described by Robert Brown was the complex and erratic motion of grains of pollen suspended in a liquid. In the many years which have passed since this description, Brownian motion has become an object of study in pure as well as applied mathematics. Even now many of its important properties are being discovered, and doubtless new and useful aspects remain to be discovered. We are getting a more and more intimate understanding of Brownian motion Front Matter....Pages i-xvi Background....Pages 1-43 Brownian Motion....Pages 44-113 Generalised Stochastic Processes and Their Distributions....Pages 114-131 Functionals of Brownian Motion....Pages 132-184 The Rotation Group....Pages 185-231 Complex White Noise....Pages 232-251 The Unitary Group and Its Applications....Pages 252-279 Causal Calculus in Terms of Brownian Motion....Pages 280-292 Back Matter....Pages 293-327