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Luke Bell
Luke Bell

Learn Everything About Brownian Motion with This Handbook of Facts and Formulae

Handbook of Brownian Motion: A Comprehensive Guide for Probabilists and Physicists

Brownian motion is one of the most fundamental and ubiquitous phenomena in nature. It describes the random movement of particles in fluids, such as dust in air or pollen in water. It also plays a key role in many areas of mathematics, physics, engineering, finance, biology, and more. It is a rich source of inspiration for modeling complex systems, studying stochastic processes, exploring fractal geometry, and developing probabilistic methods.

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But how much do you know about Brownian motion? Do you know how to construct it mathematically? Do you know its basic properties and how to prove them? Do you know how to compute the distributions of various functionals of Brownian motion, such as hitting times, maximum values, local times, or occupation measures? Do you know how to use Brownian motion to solve partial differential equations, value financial derivatives, simulate random media, or understand phase transitions?

If you want to learn more about Brownian motion and its applications, then you should read the Handbook of Brownian Motion, by Andrei N. Borodin and Paavo Salminen. This book is a comprehensive and authoritative reference that covers all aspects of Brownian motion, from its historical origins to its latest developments. It consists of two parts: the first part is devoted to the basic properties of Brownian motion, such as its construction, sample path properties, relation with random walk and harmonic functions, and stochastic calculus; the second part consists of tables of distributions of functionals of Brownian motion and related processes, such as Brownian motion with drift, reflecting Brownian motion, Bessel processes, Ornstein-Uhlenbeck processes, and geometric Brownian motion. The book also includes appendices on special functions, inverse Laplace transforms, SLE processes, and intersections of Brownian paths.

The main goals and benefits of reading this book are:

  • To gain a deep understanding of Brownian motion and its mathematical foundations.

  • To learn powerful and elegant methods for analyzing Brownian motion and its functionals.

  • To have an easy access to a large number of facts and formulas associated to Brownian motion.

  • To discover various applications of Brownian motion in different fields.

  • To appreciate the beauty and richness of Brownian motion as a subject of study.

In this article, we will give an overview of the book and highlight some of its main features. We will also provide some examples and exercises to illustrate the concepts and techniques presented in the book. We hope that this article will motivate you to read the book and explore the fascinating world of Brownian motion.

Part I: Basic properties of Brownian motion

The first part of the book consists of six chapters that introduce the reader to the theory of linear diffusions in general and Brownian motion in particular. The chapters are organized as follows:

Chapter 1: Stochastic processes in general

This chapter gives a brief introduction to the general theory of stochastic processes, such as definitions, classifications, filtrations, martingales, Markov processes, and stopping times. It also introduces some basic concepts and tools that will be used throughout the book, such as probability measures, random variables, expectations, conditional expectations, characteristic functions, and Laplace transforms.

Chapter 2: Linear diffusions

This chapter focuses on the class of linear diffusions, which are stochastic processes that satisfy a stochastic differential equation of the form

dXt = b(Xt)dt + σ(Xt)dWt

where Wt is a standard Brownian motion and b and σ are deterministic functions. The chapter discusses the existence and uniqueness of solutions, the infinitesimal generator, the scale function, the speed measure, the invariant measure, and the Feller property. It also introduces some important examples of linear diffusions, such as Ornstein-Uhlenbeck process, Bessel process, and geometric Brownian motion.

Chapter 3: Stochastic calculus

This chapter develops the theory of stochastic calculus for Brownian motion and linear diffusions. It covers the topics of stochastic integration, Itô's formula, Itô's lemma, Itô's chain rule, Girsanov's theorem, Feynman-Kac formula, and stochastic differential equations. It also gives some applications of stochastic calculus to finance, physics, and biology.

Chapter 4: Brownian motion

This chapter is devoted to the study of Brownian motion itself. It starts with the construction and definition of Brownian motion as a Gaussian process with independent increments. It then proceeds to investigate its sample path properties, such as continuity, nowhere differentiability, fractal dimension, self-similarity, and local time. It also explores its relation with random walk and harmonic functions.

Chapter 5: Local time as a Markov process

This chapter introduces the concept of local time of Brownian motion, which is a measure of how much time Brownian motion spends at a given point. It shows that local time is itself a Markov process and derives its infinitesimal generator and transition probabilities. It also discusses some applications of local time to potential theory and excursion theory.

Chapter 6: Differential systems associated to Brownian motion

This chapter studies some differential systems that are related to Brownian motion and its functionals. It covers the topics of Kolmogorov's backward and forward equations, Fokker-Planck equation, heat equation, Schrödinger equation, Dirichlet problem, Poisson equation, Laplace equation, Green's function, harmonic measure, and Harnack inequality.

Part II: Tables of distributions of functionals of Brownian motion

The second part of the book consists of nine chapters that present tables of distributions of functionals of Brownian motion and related processes. The chapters are organized as follows:

Introduction and notation

This chapter explains the general format and notation used in the tables. It also gives some general results and formulas that apply to all processes considered in the book.

Brownian motion without drift

This chapter gives tables of distributions for functionals of standard Brownian motion without drift. The functionals include hitting times, maximum values, minimum values, last exit times, first passage times, overshoots, undershoots, ladder heights, ladder epochs, 71b2f0854b


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