Elliptic differential operators and diffusion processes

Abstract

This article develops survey lectures for general mathematical audiences which the author delivered at the 27th Annual Meeting of the Australian Mathematical Society at the University of Queensland, 1983, and at the 10th Austrian Congress of Mathematicians in Innsbruck, 1981. The central theme of these lectures was the use of probabilistic methods in the study of linear elliptic-parabolic differential equations of second order.The starting point will be an orientative discussion of the role of Brownian motion in classical potential theory. It will then be discussed that, given an elliptic-parabolic differential operator L of a certain type, there exists a uniquely determined diffusion process which is linked with L formally in the same way in which Brownian motion is linked with the Laplace operator. The fundamental results of K. Itô, D.W.Stroock and S.R.S. Varadhan will be in the centre of this part of the paper. We will then proceed to the discussion of more refined problems of the same type for differentiable manifolds. A glimpse at stochastic Riemannian geometry will then close our tour d'horizon.