x PREFACE picture of this field. These developments make it desirable to write a monograph on this subject which has not been adequately exposed in a systematic way. This book was developed from the lecture notes of a year-long graduate course at the University of Tennessee. Making it accessible to non-experts with only basic knowledge of stochastic processes and functional analysis has been one of my guidelines in writing it. To make it reasonably self-contained, I added Chapter 1 for the general theory of large deviations. Most of the theorems listed in this chapter are not always easy to find in literature. In addition, a few exercises are included in the “Notes and comments” section in each chapter, an effort to promote active reading. Some of them appear as extensions of, or alternative solutions to the main theorems addressed in the chapter. Others are not very closely related to the main results on the topic, such as the exercises concerning small ball probabilities, but are linked to our context by sharing similar ideas and treatments. The challenging exercises are marked with the word “hard”. The mainspring of the book does not logically depend on the results claimed in the exercises. Consequently, skipping any exercise does not compromise understanding the book. The topics and results included in the book do reflect my taste and my involve- ment on the subject. The “Notes and comments” section at the end of each chapter is part of the effort to counterbalance the resulted partiality. Some relevant works not included in the other sections may appear here. In spite of that, I would like to apologize in advance for any possible inaccuracy of historic perspective appearing in the book. In the process of investigating the subject and writing the book, I benefitted from the help of several people. It is my great pleasure to acknowledge the contributions, which appear throughout the whole book, made by my collaborators R. Bass, W. Li, P. orters and J. Rosen in the course of several year’s collaboration. I would like to express my special thanks to D. Khoshnevisan, from whom I learned for the first time the story about intersection local times. I thank A. Dembo, J. Denzler, A. Dorogovtsev, B. Duplantier, X. B. Feng, S. Kwapien, J. Rosinski, A. Freire, J-F. Le Gall, D. S. Wu, M. Yor for discussion, information, and encouragement. I appreciate the comments from the students who preliminary version of this book, Z. Li, J. Grieves and F. Xing in particular, whose comments and suggestions resulted in a considerable reduction of errors. I am grateful to M. Saum for his support in resolving the difficulties I encountered in using latex. I would like to thank the National Science Foundation for the support I received over the years and also the Department of Mathematics and Department of Sta- tistics of Standford University for their hospitality during my sabbatical leave in Fall, 2007. A substantial part of the manuscript was written during my visit at Stanford. Last and most importantly, I wish to express my gratitude to my family, Lin, Amy and Roger, for their unconditional support. a attended a course based on
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