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Mathematical Surveys and Monographs
2006; 410 pp; hardcover
List Price: US$102
Member Price: US$81.60
Order Code: SURV/131
Random Walk Intersections: Large Deviations and Related Topics - Xia Chen
The book is devoted to the results on large deviations for a class of stochastic processes. Following an introduction and overview, the material is presented in three parts. Part 1 gives necessary and sufficient conditions for exponential tightness that are analogous to conditions for tightness in the theory of weak convergence. Part 2 focuses on Markov processes in metric spaces. For a sequence of such processes, convergence of Fleming's logarithmically transformed nonlinear semigroups is shown to imply the large deviation principle in a manner analogous to the use of convergence of linear semigroups in weak convergence. Viscosity solution methods provide applicable conditions for the necessary convergence. Part 3 discusses methods for verifying the comparison principle for viscosity solutions and applies the general theory to obtain a variety of new and known results on large deviations for Markov processes. In examples concerning infinite dimensional state spaces, new comparison principles are derived for a class of Hamilton-Jacobi equations in Hilbert spaces and in spaces of probability measures.
Graduate students and research mathematicians interested in stochastic processes.
"This book is an excellent introduction to the art of large deviations for Markov processes."
-- Zentralblatt MATH
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