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Random perturbations of dynamical systems. (English) Zbl 0659.58003
Progress in Probability and Statistics, 16. Boston, MA etc.: Birkhäuser Verlag. 294 p. DM 88.00 (1988).
The major theme of the present book is the characterization of some of those parameters of (deterministic) dynamical systems which are persistent under small random perturbations. The dynamical systems under consideration are mainly \(C^ 2\) maps F: \(M\to M\), M a smooth manifold. A small random perturbation of F is a family of Markov chains whose transition probabilities \(P^{\epsilon}(x,\cdot)\) are close to the Dirac measure \(\delta_{F(x)}(\cdot)\) for \(\epsilon\) small.
Let \(\mu^{\epsilon}\) be an invariant (probability) measure for \(P^{\epsilon}\). Weak limits of \(\mu^{\epsilon}\) with \(\epsilon\) \(\to 0\) are invariant measures for F. A major part of the book is concerned with the asymptotic behaviour of \(\mu^{\epsilon}\) for \(\epsilon\) \(\to 0\). In particular, the nature of limiting measures is investigated. Essentially, two kinds of results are obtained in this direction.
Firstly, all weak limits of \(\mu^{\epsilon}\), \(\epsilon\) \(\to 0\), are shown to have support on attractors of F, provided either the perturbations are “local” (in Section I.4) or a certain large deviations condition is satisfied (excluding “locality”; Section 1.5). Locality of a random perturbation essentially means that \(P^{\epsilon}\) is supported by the \(\epsilon\)-neighbourhoods of F(x) for \(\epsilon\) small. Section I.5 is a generalization of results of [M. I. Frejdlin and A. D. Venttsel’, Random perturbations of dynamical systems (1984; Zbl 0522.60055); see Chap. 6]. Apart from this, there are no specific connections with Frejdlin and Venttsel’s equally titled monograph.
Secondly, in Chapter II F is assumed to be \(C^ 2\) and either to be a diffeomorphism with a hyperbolic attractor \(\Lambda\) \(\subset M\), or to be expanding. In addition, the transition probabilities \(P^{\epsilon}(x,\cdot)\) are assumed to satisfy certain regularity conditions too complicated to be stated here. Then any weak limit \(\mu\) on \(\Lambda\) of invariant measures \(\mu^{\epsilon}\) for \(P^{\epsilon}\) is shown to have absolutely continuous conditional measures on unstable manifolds (hence to be a Sinaï-Bowen-Ruelle (SBR) measure).
Further topics dealt with in Chapters I and II are: \(\bullet\) Criteria for the existence of nontrivial invariant sets for F in a compact \(K\subset M\) in terms of asymptotics of exit times of the perturbations from K for \(\epsilon\) \(\to 0\). - \(\bullet\) An estimate from below for the metric entropy \(h_{\mu}(F)\) in terms of entropies of the dynamical system induced by the Markov chains associated with \(P^{\epsilon}\) (and the invariant measure \(\mu^{\epsilon})\) on the canonical probability space. These must be taken with respect to certain specific partitions only. The entropies themselves, defined to be the supremum over all finite partitions, are usually not finite. This estimate becomes an equality under the assumptions of Chapter II described above. A diffeomorphism F has to be assumed minimal in addition. - \(\bullet\) The topological pressure of a (not necessarily attracting) hyperbolic set \(\Lambda\) of a \(C^ 2\) diffeomorphism can be obtained asymptotically from the escape rates of the random perturbations from a neighbourhood of \(\Lambda\).
Suitably modified versions of most of the results are proved for flows.
Chapter III is devoted to questions of a different nature. It compiles results of the author - partly joint with A. Eizenberg - on asymptotics of the principal eigenvalue and of the spectrum of a family of elliptic second order differential operators for \(\epsilon\) \(\to 0\). The operators are otained from the generators of a diffusion perturbation of the flow induced by some vector field on a manifold M. They are specified by imposing zero boundary conditions on some open set \(G\subset M\) with a sufficiently smooth boundary. In particular, if G contains a hyperbolic set \(\Lambda\) then the principal eigenvalue converges to the topological pressure of \(\Lambda\) for \(\epsilon\) \(\to 0.\)
In Chapter IV, \(\mu^{\epsilon}\to \mu\) is investigated using additional “ad hoc arguments” for random perturbations of piecewise expanding interval maps and for Misiurewicz maps (\(\mu\) absolutely continuous with respect to the Lebesgue measure) as well as for Lorenz type maps (\(\mu\) essentially BSR). These three types of systems lack the shadowing property for all pseudo orbits, which was an essential ingredient of the proofs in Chapter II.
During the last years there has been considerable activity at the border between dynamical systems and stochastic processes. The present book presents results from this area, some of which were not published previously. It concentrates on topics the author has been active on himself. Several other topics are left out. To name but a few: Lyapunov exponents and dimensions, stochastic flows, the general theory of random dynamical systems. The “Pesin formula for products of iid diffeomorphisms” due to F. Ledrappier and L.-S. Young [Probab. Theory Relat. Fields 80, 217-240 (1988; Zbl 0638.60054)], which is very closely connected with Sections II.4-5 of the present book, is not even mentioned.
The book is addressed to mathematicians and mathematical physicists working in probability and/or dynamical systems. Non-specialists in the particular field dealt with in the book may find it hard reading, though. Not only that the exposition is rather technical. Unfortunately, the text has been prepared in camera ready form with the help of a not very sophisticated word processing system together with a not very good printing device.
Altogether, the book is closer to a collection of research papers than to a textbook.
Reviewer: H.Crauel

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
37A99 Ergodic theory
37D99 Dynamical systems with hyperbolic behavior
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
34C40 Ordinary differential equations and systems on manifolds
58J65 Diffusion processes and stochastic analysis on manifolds