PREFACE

Summary

You have not played as yet? Do not do so: above all avoid a martingale if you do.

W.M. Thaakeray

ABOUT THESE NOTES. ACKNOWLEDGEMENTS

These are a much expanded and revised version of notes of seminars given at the University of Warwick during the year 1973/74. These were given and written up jointly with J. Eells. The audience consisted of non-probabilists with a reasonably good background in manifold theory. The aim was to go through, from the beginning, the basic properties of stochastic differential equations, extend the theory to manifolds and in particular use this in the proof that by polygonal approximation the Cartan development or ‘rolling’ has a ‘stochastic extension’ which gives a geometric construction of Brownian motion on Riemannian manifolds (see §§11 A, B Chapter VII). Another aim was to describe, from this point of view, the stochastic parallel transport discussed in Itô's Stockholm article (1963).

The previous year we had been on leave separately: myself at the University of California at Santa Cruz, and Aarhus University, beginning to learn some Stratonovich calculus after earlier suggestions from R. Curtain; and Eells at I.A.S. Princeton, and I.H.E.S. Bures-sur-Yvette, where he collaborated with P. Malliavin (1972/3) in an examination of diffusions on vector bundles, horizontal lifts, etc. using a different approach. We met up at I.H.E.S., and all these institutions deserve thanks for their hospitality.