Large Deviations for Stochastic Processes
This book is focused on recent results on large deviation for a general class of Markov processes obtained mostly by the authors. It is not intended for beginners and inexperienced readers and a very good mathematical background and a preliminary knowledge of Markov processes and stochastic analysis is an advantage when reading the book. The volume itself is divided into five parts. The book starts with an introduction and overview with many motivating examples and it ends with an extensive appendix containing useful results on operators, semigroups and mass transport theory. The core of the book is in the remaining three sections. Large deviations in general are treated in section 1. The exponential tightness and necessary and sufficient conditions for it are given in an analogous way as for classical tightness of measures. The rate function is introduced and the problem of identifying the rate function is studied. Section 2 gives an overview of the classical semigroup approach to large deviations of Markov processes and an alternative approach using viscosity solutions. The proofs of large deviation results need verification of a comparison principle. Since it is typically the most difficult step, a chapter in section 3 is devoted to this problem. In the rest of section 3, the comparison principle is discussed for different stochastic processes, in particular for nearly deterministic processes with almost negligible perturbation, for stochastic processes in a random environment, for occupation measures of Markov processes and, finally, for solutions of stochastic equations in infinite dimensions. Large deviations have many applications in probability and statistics, hence the book may be recommended to researchers in the field of stochastic processes and their applications or to specialists in statistics of stochastic processes.