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Books > Science & Mathematics > Physics > Classical mechanics > Analytical mechanics
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Essentials of Brownian Motion and Diffusion (Paperback)
Loot Price: R3,179
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Essentials of Brownian Motion and Diffusion (Paperback)
Series: Mathematical Surveys and Monographs
Expected to ship within 12 - 19 working days
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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.This book
contains a general summary and outline, and an introduction. It
presents some gratuitous generalities on scientific method as it
relates to diffusion theory. Brownian motion is defined by the
characterization of P. Levy. Then it is constructed in three basic
ways and these are proved to be equivalent in the appropriate
sense. Uniqueness theorem. Projective invariance and the Brownian
bridge is presented. Probabilistic and absolute properties are
distinguished. Among the former includes the distribution of the
maximum, first passage time distributions, and fitting
probabilities, and among the latter includes law of created
logarithm, quadratic variation, Holder continuity, non-recurrence
for $r\geq 2$. 3.General methods of Markov processes are adapted to
diffusion. Analytic and probabilistic methods are distinguished.
Among the former include transition functions, semigroups,
generators, resolvents. Among the latter include Markov properties,
stopping times, zero-or-one laws, Dynkin's formula, additive
functionals. The book features 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; and, time-change theorem
(classical case), parabolic equations and their solution
semigroups, some basic examples, distribution of passage times.The
book covers Local time: construction by random walk embedding;
Local time processes; Trotter's theorem; The Brownian flow;
Brownian excursions; The zero set and Levy's equivalence theorem;
Local times of classical diffusions; and, Sample path properties.
It also includes boundary conditions for Brownian motion; the
general boundary conditions; construction of the processes using
local time; and, green functions and eigenfunction expansions
(compact case).Another 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.
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