Preface

Summary

There are few benefits to growing old, especially if you are a mathematician. However, one of them is that, over the course of time, you accumulate a certain amount of baggage containing information in which, if you are lucky and they are polite, your younger colleagues may express some interest.

Having spent most of my career at the interface between probability and partial differential equations, it is hardly surprising that this is the item in my baggage about which I am asked most often. When I was a student, probabilists were still smitten by the abstract theory of Markov processes which grew out of the beautiful work of G. Hunt, E.B. Dynkin, R.M. Blumenthal, R.K. Getoor, P.A. Meyer, and a host of others. However, as time passed, it became increasingly apparent that the abstract theory would languish if it were not fed a steady diet of hard, analytic facts. As A.N. Kolmogorov showed a long time ago, ultimately partial differential equations are the engine which drives the machinery of Markov processes. Until you solve those equations, the abstract theory remains a collection of “if, then” statements waiting for someone to verify that they are not vacuous.

Unfortunately for probabilists, the verification usually involves ideas and techniques which they find unpalatable. The strength of probability theory is that it deals with probability measures, but this is also its weakness.