Softcover ISBN:  9781470418700 
Product Code:  SURV/131.S 
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AMS Member Price:  $81.60 
Electronic ISBN:  9781470413583 
Product Code:  SURV/131.E 
List Price:  $102.00 
MAA Member Price:  $91.80 
AMS Member Price:  $81.60 

Book DetailsMathematical Surveys and MonographsVolume: 131; 2006; 410 ppMSC: Primary 60; 47; Secondary 49;
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 HamiltonJacobi equations in Hilbert spaces and in spaces of probability measures.
ReadershipGraduate students and research mathematicians interested in stochastic processes.

Table of Contents

Chapters

1. Introduction

2. An overview

3. Large deviations and exponential tightness

4. Large deviations for stochastic processes

5. Large deviations for Markov processes and nonlinear semigroup convergence

6. Large deviations and nonlinear semigroup convergence using viscosity solutions

7. Extensions of viscosity solution methods

8. The Nisio semigroup and a control representation of the rate function

9. The comparison principle

10. Nearly deterministic processes in $R^d$

11. Random evolutions

12. Occupation measures

13. Stochastic equations in infinite dimensions

Appendix A. Operators and convergence in function spaces

Appendix B. Variational constants, rate of growth and spectral theory for the semigroup of positive linear operators

Appendix C. Spectral properties for discrete and continuous Laplacians

Appendix D. Results from mass transport theory


Additional Material

Reviews

This book is an excellent introduction to the art of large deviations for Markov processes.
Zentralblatt MATH


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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 HamiltonJacobi equations in Hilbert spaces and in spaces of probability measures.
Graduate students and research mathematicians interested in stochastic processes.

Chapters

1. Introduction

2. An overview

3. Large deviations and exponential tightness

4. Large deviations for stochastic processes

5. Large deviations for Markov processes and nonlinear semigroup convergence

6. Large deviations and nonlinear semigroup convergence using viscosity solutions

7. Extensions of viscosity solution methods

8. The Nisio semigroup and a control representation of the rate function

9. The comparison principle

10. Nearly deterministic processes in $R^d$

11. Random evolutions

12. Occupation measures

13. Stochastic equations in infinite dimensions

Appendix A. Operators and convergence in function spaces

Appendix B. Variational constants, rate of growth and spectral theory for the semigroup of positive linear operators

Appendix C. Spectral properties for discrete and continuous Laplacians

Appendix D. Results from mass transport theory

This book is an excellent introduction to the art of large deviations for Markov processes.
Zentralblatt MATH