Large Deviations for Stochastic Processes (Mathematical Surveys and Monographs No. 131)
By: Jin Feng (author), Thomas G. Kurtz (author)Hardback
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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.
Introduction: Introduction An overview The general theory of large deviations: Large deviations and exponential tightness Large deviations for stochastic processes Large deviations for Markov processes and semigroup convergence: Large deviations for Markov processes and nonlinear semigroup convergence Large deviations and nonlinear semigroup convergence using viscosity solutions Extensions of viscosity solution methods The Nisio semigroup and a control representation of the rate function Examples of large deviations and the comparison principle: The comparison principle Nearly deterministic processes in $R^d$ Random evolutions Occupation measures Stochastic equations in infinite dimensions Appendix: Operators and convergence in function spaces Variational constants, rate of growth and spectral theory for the semigroup of positive linear operators Spectral properties for discrete and continuous Laplacians Results from mass transport theory Bibliography Index.
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- ID: 9780821841457
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