zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Large deviations and applications. (English) Zbl 0549.60023
CBMS-NSF Reg. Conf. Ser. Appl. Math. 46, 75 p. (1984).

This is an extremely useful and nice introduction into the theory of large deviations in infinite dimensional situations, as it has been developed by Ventsel and Freidlin, Donsker and the author and others. The general situation is of the following type: Let ξ t , t (or t + ) be random elements with values in a topological space X which converge, as t, in probability to a point xX, i.e. the probability that ξ t stays away from x decreases to O. Often, it decreases exponentially fast. The theory of large deviations gives precise information about this behavior.

Why is this important? In applications one often encounters expectations of the form E(F t (ξ t )), F t :X, where the main contribution comes from a part of X where F t is large, but the probability that ξ t is in this part is small. A typical example for the large deviation behavior is the classical one, where one considers a sequence of i.i.d. real valued random variables X 1 ,X 2 ,··. with mean 0 and exponential moments. Then let ξ n = i=1 n X i /n. Then ξ n converge to 0 and for x>0

lim n n -1 logP(ξ n x)=-h(x)

where h is the so-called entropy function, in this case the Legendre transformation of the function λlogE(e λX 1 ). This is (part of) a result of CramĂ©r.

The present book starts with developing this classical theory. Then it discusses the Ventsel-Freidlin theory of large deviation for paths of diffusion processes first in the special case of Brownian motion. The main emphasis is on large deviation probabilities for empirical measures of Markov processes. The theory is developed immediately at the so-called level 3, i.e. where one considers empirical measures of the whole path of the process. The more usual empirical measures of the values of the process in the state space can then be discussed via a contraction principle. Two applications are given. First an asymptotic evaluation of certain expectations of the Wiener sausage, i.e. the ϵ- neighbourhood of the Wiener path. Secondly the so-called polaron problem is discussed, a complicated path integral which appears in solid state physics. The book contains detailed proofs of most of the results. The polaron problem is only sketched.

The book is certainly far the best short introduction into these topics which starting with the most elementary case leads to very recent highlights of the theory. A bit disturbing are the many misprints.

Reviewer: F.Bolthausen

60F10Large deviations
60-02Research monographs (probability theory)
60J65Brownian motion
60J60Diffusion processes