Introduction

Summary

These notes grow out of lectures which I gave during the fall semester of 1985 at M.I.T. My purpose has been to provide a reasonably self-contained introduction to some stochastic analytic techniques which can be used in the study of certain analytic problems, and my method has been to concentrate on a particularly rich example rather than to attempt a general overview. The example which I have chosen is the study of second order partial differential operators of parabolic type. This example has the advantage that it leads very naturally to the analysis of measures on function space and the introduction of powerful probabilistic tools like martingales. At the same time, it highlights the basic virtue of probabilistic analysis: the direct role of intuition in the formulation and solution of problems.

The material which is covered has all been derived from my book [S.&V.] (Multidimensional Diffusion Processes. Grundlehren #233, Springer-Verlag, 1979) with S.R.S. Varadhan. However, the presentation here is quite different. In the first place, the emphasis there was on generality and detail; here it is on conceptual clarity. Secondly, at the time when we wrote [S.&V.], we were not aware of the ease with which the modern theory of martingales and stochastic integration can be presented. As a result, our development of that material was a kind of hybrid between the classical ideas of K. Itô and J.L. Doob and the modern theory based on the ideas of P.A. Meyer, H. Kunita, and S. Watanabe. In these notes the modern theory is presented; and the result is, I believe, not only more general but also more unders tandable.