The interplay of probability theory and partial differential equations forms a fascinating part of mathematics. Among the subjects it has in- spired are the martingale problems of Stroock and Varadhan, the Harnack inequality of Krylov and Safonov, the theory of symmetric diffusion pro- cesses, and the Malliavin calculus. When I first made an outline for my previous book Probabilistic Techniques in Analysis, I planned to devote a chapter to these topics. I soon realized that a single chapter would not do the subject justice, and the current book is the result.
The first chapter provides the probabilistic machine needed to drive the subject, namely, stochastic differential equations. We consider existence, uniqueness, and smoothness of solutions and stochastic differential equa- tions with reflection.