Brownian Motion and Diffusion, Softcover reprint of the original 1st ed. 1983

A long time ago I started writing a book about Markov chains, Brownian motion, and diffusion. I soon had two hundred pages of manuscript and my publisher was enthusiastic. Some years and several drafts later, I had a thot:sand pages of manuscript, and my publisher was less enthusiastic. So we made it a trilogy: Markov Chains Brownian Motion and Diffusion Approximating Countable Markov Chains familiarly - Me, B & D, and ACM. I wrote the first two books for beginning graduate students with some knowledge of probability; if you can follow Sections 3.4 to 3.9 of Brownian Motion and Diffusion you're in. The first two books are quite independent of one another, and completely independent of the third. This last book is a monograph, which explains one way to think about chains with instantaneous states. The results in it are supposed to be new, except where there are spe­ cific disclaimers; it's written in the framework of Markov Chains. Most of the proofs in the trilogy are new, and I tried hard to make them explicit. The old ones were often elegant, but I seldom saw what made them go. With my own, I can sometimes show you why things work. And, as I will argue in a minute, my demonstrations are easier technically. If I wrote them down well enough, you may come to agree.

1. Brownian Motion.- 1. Introduction.- 2. Foundations.- 3. Strong Markov and Reflection.- 4. Sample Function Properties.- 5. Strassen’s Law of the Iterated Logarithm.- 6. The Skorokhod Representation.- 7. Donsker’s Invariance Principle.- 8. Strassen’s Invariance Principle.- 9. Lindeberg’s Theorem for Martingales.- 10. Central Limit Theorem for Random Sums.- 11. Wald’s Identity.- 2. Diffusion.- 1. Introduction.- 2. Regularity.- 3. Scale.- 4. Semigroups.- 5. Green’s Function.- 6. Speed Measure.- 7. Infinitesimal Generator.- 8. Brownian Local Time.- 9. Transformation of Time.- 10. Diffusion Local Time.- 11. First Examples.- 12. An Example of Feller and McKean.- 13. An Example of Breiman.- 14. A Generalization.- 15. Weak Markov Processes.- 3. Appendix.- 1. Notation.- 2. Numbering.- 3. Bibliography.- 4. The Abstract Lebesgue Integral.- 5. Atoms.- 6. Independence.- 7. Conditioning.- 8. Martingales.- 9. Metric Spaces.- 10. Regular Conditional Distributions.- 11. The Kolmogorov Consistency Theorem.- 12. The Diagonal Argument.- 13. Classical Lebesgue Measure.- 14. Real Variables.- 15. Absolute Continuity.- 16. Convex Functions.- 17. Complex Variables.- Symbol Finder.