Large deviations and the Malliavin calculus. (English) Zbl 0537.35003

Bismut, Jean-Michel

Progress in Mathematics, Vol. 45. Boston - Basel - Stuttgart: Birkhäuser. VIII, 216 p. DM 59.00 (1984).

This book uses the so-called Malliavin calculus and large deviation techniques to study the asymptotics as \(t\downarrow \downarrow 0\) of the conditional probabilities of bridges associated with certain hypoelliptic diffusions. In particular, stochatic differential equations of the form \(dx=\sum^{m}_{i=1}X_ i(x)\quad dw^ i;\quad x(0)=x_ 0\) are studied, where \(X_ 1,...,X_ m\) are smooth vector fields as on a manifold M. Let be \(x_ s=\psi_ s(w)\). Associated to the latter differential equation is the second order operator \(L=1/2\sum^{m}_{i=1}X^ 2_ i\). The law of \(\psi_ 1(w)\) is given by the kernel \(e^ L(x_ 0,.)\). Under certain standard assumptions of Hörmander, one considers the asymptotics of the law \(p_ t(x)\) of \(\psi_ 1(\sqrt{t}dw)\) as \(t\downarrow \downarrow 0.\)
In the elliptic case, rather complete results are obtained. In case ”hypoelliptic” is substituted for ”elliptic”, some basic machinery is developed and some conjectures are formulated. The special case of the Hessenberg group is considered in some detail.
The book assumes a fair amount of sophistication on the part of the reader - some background in probability, Riemannian geometry, and the basic language of differential equations is advised. However the author is fairly kind to his readers. He works out special cases, writes out most details, and gives copious references. The author is able to keep his book fairly brief and yet provide an introduction to an active area of research.

Reviewer: S.G.Krantz

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35B40	Asymptotic behavior of solutions to PDEs
65H10	Numerical computation of solutions to systems of equations
58J65	Diffusion processes and stochastic analysis on manifolds
60G15	Gaussian processes
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60	Diffusion processes

Keywords:

diffusion processes; stochastic analysis on manifolds; Gaussian processes; Malliavin calculus; large deviation techniques; hypoelliptic diffusions