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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 ${\xi }_{t}$, $t\in ℕ$ (or $t\in {ℝ}^{+}\right)$ be random elements with values in a topological space X which converge, as $t\to \infty$, in probability to a point $x\in X$, i.e. the probability that ${\xi }_{t}$ 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 $E\left({F}_{t}\left({\xi }_{t}\right)\right)$, ${F}_{t}:X\to ℝ$, where the main contribution comes from a part of X where ${F}_{t}$ is large, but the probability that ${\xi }_{t}$ 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 ${\xi }_{n}={\sum }_{i=1}^{n}{X}_{i}/n$. Then ${\xi }_{n}$ converge to 0 and for $x>0$

$\underset{n\to \infty }{lim}{n}^{-1}logP\left({\xi }_{n}\ge x\right)=-h\left(x\right)$

where h is the so-called entropy function, in this case the Legendre transformation of the function $\lambda \to logE\left({e}^{\lambda {X}_{1}}\right)$. This is (part of) a result of Cramér.

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

##### MSC:
 60F10 Large deviations 60-02 Research monographs (probability theory) 60J65 Brownian motion 60J60 Diffusion processes