**Mathematical Surveys and Monographs**

Volume: 18;
1981;
201 pp;
Softcover

MSC: Primary 60;

**Print ISBN: 978-0-8218-1518-2
Product Code: SURV/18**

List Price: $76.00

AMS Member Price: $60.80

MAA Member Price: $68.40

**Electronic ISBN: 978-1-4704-1245-6
Product Code: SURV/18.E**

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# Essentials of Brownian Motion and Diffusion

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*Frank B. Knight*

This work was first drafted five years ago at the invitation of the
editors of the Encyclopedia of Mathematics and its Applications. However, it was
found to contain insufficient physical applications for that series; hence,
it has finally come to rest at the doorstep of the American Mathematical
Society. The first half of the work is little changed from the original, a fact
which may partly explain both the allusions to applications and the elementary
approach. It was written to be understood by a reader having minimal
familiarity with continuous time stochastic processes. The most advanced
prerequisite is an understanding of discrete parameter martingale convergence theorem.

General summary and outline:

0. Introduction. Some gratuitous generalities on scientific method as it
relates to diffusion theory.

1. Brownian motion is defined by the characterization of P. Lévy.
Then it is constructed in three basic ways and these are proved to be
equivalent in the appropriate sense. Uniqueness theorem.

2. Projective invariance and the Brownian bridge presented. Probabilistic
and absolute properties are distinguished. Among the former: the distribution
of the maximum, first passage time distributions, and fitting probabilities.
Among the latter: law of created logarithm, quadratic variation, Hölder
continuity, non-recurrence for \(r\geq 2\).

3. General methods of Markov processes adapted to diffusion. Analytic and
probabilistic methods are distinguished. Among the former: transition
functions, semigroups, generators, resolvents. Among the latter: Markov
properties, stopping times, zero-or-one laws, Dynkin's formula, additive
functionals.

4. Classical modifications of Brownian motion. Absorption and the Dirichlet
problem. Space-time process and the heat equation. Killed processes, Green
functions, and the distributions of additive sectionals. Time-change theorem
(classical case), parabolic equations and their solution semigroups, some basic
examples, distribution of passage times.

5. Local time: construction by random walk embedding. Local time processes.
Trotter's theorem. The Brownian flow. Brownian excursions. The zero set and
Lévy's equivalence theorem. Local times of classical diffusions. Sample
path properties.

6. Boundary conditions for Brownian motion. The general boundary conditions.
Construction of the processes using local time. Green functions and
eigenfunction expansions (compact case).

7. The chapter is a “finale” on nonsingular diffusion. The
generators \((d/dm)(d^+/dx^+)\) are characterized. The diffusions on
open intervals are constructed. The conservative boundary conditions are
obtained and their diffusions are constructed. The general additive functionals
and nonconservative diffusions are developed and expressed in terms of Brownian
motions.

The audience for this survey includes anyone who desires an introduction to Markov
processes with continuous paths that is both coherent and elementary. The
approach is from the particular to the general. Each method is first explained
in the simplest case and supported by examples. Therefore, the book should be
readily understandable to anyone with a first course in measure-theoretic
probability.

#### Table of Contents

# Table of Contents

## Essentials of Brownian Motion and Diffusion

- Table of Contents vii8 free
- Preface xi12 free
- Introduction 116 free
- Chapter 1. Definition, Existence, and Uniqueness of the Brownian Motion 520 free
- Chapter 2. Initial Features of the Process 1934
- Chapter 3. General Markovian Methods 3146
- Chapter 4. Absorbing, Killing, and Time Changing: The Classical Cases 6176
- 4.1. Absorption 6176
- 4.2. Killing 7186
- 4.3. Time Changing 89104
- 1. Sectionally Continuous Coefficients 90105
- 2. The Corresponding Diffusions on (a, b) 91106
- 3. The Ornstein-Uhlenbeck Velocity Process 96111
- 4. Stochastic Differential Equations (Heuristic) 98113
- 5. Continuous State Branching Processes 100115
- 6. The Bessel Processes 102117
- 7. Transience, Neighborhood Recurrence, and Passage Times 103118

- Chapter 5. Local Times, Excursions, and Absolute Sample Path Properties 107122
- 5.1. Local Time: Extrinsic Construction 107122
- 5.2. Brownian Excursions 120135
- 5.3. The Zero Set and Intrinsic Local Time 127142
- 1. Distribution of the Zeros 127142
- 2. Construction of Process from Zeros and Excursions 128143
- 3. P. Lévy's Equivalence (Y[sub(1)](t), M(t)) ≡ (B(t), 2s(t, 0)) 130145
- 4. Passage Time Process as Subordinator 132147
- 5. The "Mesure du Voisinage" and Local Time 135150
- 6. The General Sojourn Density Diffusions 137152
- 7. Local Times of Diffusions 139154

- 5.4. Some Absolute Sample Path Properties 142157

- Chapter 6. Boundary Conditions for Brownian Motion (r = 1) 153168
- Chapter 7. Nonsingular Diffusion in R[sup(1)] 169184
- Bibliography 195210
- Index 199214
- Errata 203218