**Mathematical Surveys and Monographs**

Volume: 131;
2006;
410 pp;
Hardcover

MSC: Primary 60; 47;
Secondary 49

Print ISBN: 978-0-8218-4145-7

Product Code: SURV/131

List Price: $102.00

Individual Member Price: $81.60

**Electronic ISBN: 978-1-4704-1358-3
Product Code: SURV/131.E**

List Price: $102.00

Individual Member Price: $81.60

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#### Supplemental Materials

# Large Deviations for Stochastic Processes

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*Jin Feng; Thomas G. Kurtz*

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.

#### Table of Contents

# Table of Contents

## Large Deviations for Stochastic Processes

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface ix10 free
- Notation xi12 free
- Introduction 114 free
- Chapter 1. Introduction 316 free
- Chapter 2. An overview 2942 free
- Part 1. The general theory of large deviations 3952
- Chapter 3. Large deviations and exponential tightness 4154
- Chapter 4. Large deviations for stochastic processes 5770
- Part 2. Large deviations for Markov processes and semigroup convergence 7790
- Chapter 5. Large deviations for Markov processes and nonlinear semigroup convergence 7992
- Chapter 6. Large deviations and nonlinear semigroup convergence using viscosity solutions 97110
- Chapter 7. Extensions of viscosity solution methods 109122
- Chapter 8. The Nisio semigroup and a control representation of the rate function 135148
- Part 3. Examples of large deviations and the comparison principle 163176
- Chapter 9. The comparison principle 165178
- Chapter 10. Nearly deterministic processes in 𝑅^{𝑑} 199212
- Chapter 11. Random evolutions 229242
- Chapter 12. Occupation measures 283296
- Chapter 13. Stochastic equations in infinite dimensions 293306
- Appendix 343356
- Appendix A. Operators and convergence in function spaces 345358
- Appendix B. Variational constants, rate of growth and spectral theory for the semigroup of positive linear operators 353366
- Appendix C. Spectral properties for discrete and continuous Laplacians 367380
- Appendix D. Results from mass transport theory 371384
- Bibliography 403416
- Index 409422 free
- Back Cover Back Cover1426

#### Readership

Graduate students and research mathematicians interested in stochastic processes.

#### Reviews

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

-- Zentralblatt MATH