"This book is an excellent presentation of the application of martingale theory to the theory of Markov processes, especially multidimensional diffusions. This approach was initiated by Stroock and Varadhan in their famous papers. (...) The proofs and techniques are presented in such a way that an adaptation in other contexts can be easily done. (...) The reader must be familiar with standard probability theory and measure theory which are summarized at the beginning of the book. This monograph can be recommended to graduate students and research workers but also to all interested in Markov...

The theory of large deviations deals with rates at which probabilities of certain events decay as a natural parameter in the problem varies. This book, which is based on a graduate course on large deviations at the Courant Institute, focuses on three concrete sets of examples: (i) diffusions with small noise and the exit problem, (ii) large time behavior of Markov processes and their connection to the Feynman-Kac formula and the related large deviation behavior of the number of distinct sites visited by a random walk, and (iii) interacting particle systems, their scaling limits, and large...