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Chapter 0. Preliminaries 1.Basic Notation 2.Monotone Class Theorem 3.Completion 4.Functions of Finite Variation and Stieltjes Integrals 5.Weak Convergence in Metric Spaces 6.Gaussian and Other Random Variables Chapter ¢ñ.Introduction 1.Examples of Stochastic Processes. Brownian Motion 2.Local Properties of Brownian Paths 3.Canonical Processes and Gaussian Processes 4.Filtrations and Stopping Times Notes and Comments Chapter ¢ò.Martingales 1.Definitions, Maximal Inequalities and Applications 2.Convergence and Regularization Theorems 3.Optional Stopping Theorem Notes and Comments Chapter ¢ó.Markov Processes 1.Basic Definitions 2.Feller Processes 3.Strong Markov Property 4.Summary of Results on Levy Processes Notes and Comments Chapter ¢ô.Stochastic Integration 1.Quadratic Variations 2.Stochastic Integrals 3.Ito's Formula and First Applications 4.Burkholder-Davis-Gundy Inequalities 5.Predictable Processes Notes and Comments Chapter ¢õ.Representation of Martingales 1.Continuous Martingales as Time-changed Brownian Motions 2.Conformal Martingales and Planar Brownian Motion 3.Brownian Martingales 4.Integral Representations Notes and Comments Chapter ¢ö.Local Times 1.Definition and First Properties 2.The Local Time of Brownian Motion 3.The Three-Dimensional Bessel Process 4.First Order Calculus 5.The Skorokhod Stopping Problem Notes and Comments Chapter ¢÷.Generators and Time Reversal 1.Infinitesimal Generators 2.Diffusions and It6 Processes 3.Linear Continuous Markov Processes 4.Time Reversal and Applications Notes and Comments Chapter ¢ø.Girsanov's Theorem and First Applications 1.Girsanov's Theorem 2.Application of Girsanov's Theorem to the Study of Wiener's Space 3.Functionals and Transformations of Diffusion Processes Notes and Comments Chapter ¢ù.Stochastic Differential Equations 1.Formal Definitions and Uniqueness 2.Existence and Uniqueness in the Case of Lipschitz Coefficients 3.The Case of Holder Coefficients in Dimension One Notes and Comments Chapter ¢ú.Additive Functionals of Brownian Motion 1.General Definitions 2.Representation Theorem for Additive Functionals of Linear Brownian Motion 3.Ergodic Theorems for Additive Functionals 4.Asymptotic Results for the Planar Brownian Motion Notes and Comments Chapter ¢û.Bessel Processes and Ray-Knight Theorems 1.Bessel Processes 2.Ray-Knight Theorems 3.Bessel Bridges Notes and Comments Chapter ¢ü. Excursions 1.Prerequisites on Poisson Point Processes 2.The Excursion Process of Brownian Motion 3.Excursions Straddling a Given Time 4.Descriptions of It6's Measure and Applications Notes and Comments Chapter ¢ú¢ó.Limit Theorems in Distribution 1.Convergence in Distribution 2.Asymptotic Behavior of Additive Functionals of Brownian Motion 3.Asymptotic Properties of Planar Brownian Motion Notes and Comments Appendix 1.Gronwall's Lemma 2.Distributions 3.Convex Functions 4.Hausdorff Measures and Dimension 5.Ergodic Theory 6.Probabilities on Function Spaces 7.Bessel Functions 8.Sturm-Liouville Equation Bibliography Index of Notation Index of Terms Catalogue