Brownian Motion 1 Background 1.1 Probability Spaces Random Variables and Expectations 1.2 Examples 1.3 Probability Distributions 1.4 Conditional Expectations 1.5 Limit Theorems 1.6 Gaussian Systems 1.7 Characterisations of Gaussian Distributions 2 Brownian Motion 2.1 Brownian Motion. Wiener Measure 2.2 Sample Path Properties 2.3 Cbnstructions of Brownian Motion 2.4 Markov Properties of Brownian Motion 2.5 Applications of the Hille Yosida Theorem 2.6 Processes Related to Brownian Motio 3 Generalised Stochastic Processes and Their Distributions 3.1 Characteristic Functionals 3.2 The Bochner Minlos Theorem 3.3 Examples of Generalised Stochastic Processes and Their Distributions 3.4 White Noise 4 Functionals of Brownian Motion 4.1 Basic Functionals 4.2 The Wiener Itô Decomposition of (L2) 4.3 Representations of Multiple Wiener Integrals 4.4 Stochastic Processes 4.5 Stochastic Integrals 4.6 Examples of Applications 4.7 The Fourier Wiener Transform 5 The Rotation Group 5.1 Transformations of White Noise (I): Rotations 5.2 Subgroups of the Rotation Group 5.3 The Projective Transformation Group 5.4 Projective Invariance of Brownian Motion 5.5 Spectral Type of One Parameter Subgroups 5.6 Derivation of Properties of White Noise Using the Rotation Group 5.7 Transformations of White Noise (II): Translations 5.8 The Canonical Commutation Relations of Quantum Mechanics 6 Complex White Noise 6.1 Complex Gaussian System 6.2 Complexification of White Noise 6.3 The Complex Multiple Wiener Integral 6.4 Special Functionals in $$(\\text{L}_c^2) $$ 7 The Unitary Group and Its Applications 7.1 The Infinite Dimensional Unitary Group 7.2 The Unitary Group U(c) 7.3 Subgroups of U(c) 7.4 Generators of the Subgroups 7.5 The Symmetry Group of the Heat Equation 7.6 Applications to the Schrödinger Equation 8 Causal Calculus in Terms of Brownian Motion 8.1 Summary of Known Results 8.2 Coordinate Systems in (* µ) 8.3 Generalised Brownian Functionals 8.4 Generalised Random Measures 8.5 Causal Calculus A.l Martingales A.2 Brownian Motion with a Multidimensional Parameter A.3 Examples of Nuclear Spaces A.4 Wiener’s Non Linear Circuit Theory A.5 Formulae for Hermite Polynomials. 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.