Diffusions and Elliptic Operators
The interplay of probability theory and partial di?erential equations forms a fascinating part of mathematics. Among the subjects it has - spired are the martingale problems of Stroock and Varadhan, the Harnack inequality of Krylov and Safonov, the theory of symmetric di?usion p- cesses, and the Malliavin calculus. When I ?rst 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 ?rst chapter provides the probabilistic machine needed to drive thesubject,namely,stochasticdi?erentialequations.Weconsiderexistence, uniqueness, and smoothness of solutions and stochastic di?erential eq- tions with re?ection. The second chapter is the heart of the subject. We show how many partial di?erential equations can be solved by simple probabilistic expr- sions. The Dirichlet problem, the Cauchy problem, the Neumann problem, the oblique derivative problem, Poisson’s equation, and Schr¨ odinger’s eq- tion all have solutions that are given by appropriate probabilistic expr- sions. Green functions and fundamental solutions also have simple pro- bilistic representations. If an operator has smooth coe?cients, then equations with these - erators will have smooth solutions. This theory is discussed in Chapter III. The chapter is largely analytic, but probability allows some simpli?cation in the arguments. ChapterIVconsidersone-dimensionaldi?usionsandthecorrespo- ingsecond-orderordinarydi?erentialequations.Everyone-dimensionaldif- viii PREFACE fusion can be derived from Brownian motion by changes of time and scale.