This is the third extended edition of the English edition of one of the most influential books on the subject of randomly perturbed systems. The original edition appeared already in 1979 in Russian, see [Zbl 0499.60053]. The English translation by J. Szücs followed in 1984, see [Zbl 0522.60055]. In 1998, an updated and extended second edition appeared, see [Zbl 0922.60006].
This third edition has been revised and minor changes have been made mostly concerning terminology. In Chapter 6, on the long-time behaviour of Markov perturbations, the connection between sublimiting distributions, already discussed in the second edition, and the notions of metastability and stochastic resonance are investigated and embedded in the theory of large deviations.
A new section is added to Chapter 8, where deterministic perturbations to one-dimensional Hamiltonian systems are extensively studied. The investigation reveals that, although the perturbations under consideration are purely deterministic, the system may exhibit a stochastic nature on long time scales if the corresponding Hamiltonian has saddle points.
The third edition also contains the completely new Chapter 9 dedicated to a further investigation of perturbations of systems with many degrees of freedom. Here, the geometry of the slow component is described by means of an open book space.
These valuable extensions incorporate recent developments in the field into this classic monograph, in particular references to results that could only be conjectured in earlier versions. The changes underline the actuality of the book and its significance in many areas ranging from mathematics and physics to theoretical engineering.