Introduction

In Kendall (1990) it is explained how three nonlinear Dirichlet problems are closely connected to a problem about the existence of a certain convex surrogate distance function. Here we consider an aspect of these relationships in a setting more general than the Riemannian case of Kendall (1990). The problems are as follows. Suppose M is a smooth manifold equipped with a connection Λ and separately with a reference Riemannian structure (the connection need not be compatible with the metric!). Consider B a closed region in M. (In the sequel B is generally compact, but we prefer to state the following properties as applicable to a general region.)

(A): Does Β have Λ-convex geometry? That is to say, does there exist a (product-connection) convex function Q : Β × Β → [0, 1] vanishing only on the diagonal Δ = {(x, x) : x ∈ Β}? Here the “Λ” in “Λ-convex” refers to the use of the connection Λ to build the product-connection, instead of the Levi–Civita connection supplied by the reference Riemannian metric. (In the rest of the paper the prefix “Λ” is omitted; by “convex” we mean “Λ-convex” unless indicated otherwise.)

(B): Dirichlet problem for Λ-martingales lying in Β. This problem requires one to find Λ-martingales X (under a given filtration) attaining a given terminal value X(∞). In the following the heading (B) refers specifically to whether the Dirichlet problem is well-posed and has unique solutions.

This volume contains the proceedings of the Durham Symposium on Stochastic Analysis, held at the University of Durham 11–21 July 1990, under the auspices of the London Mathematical Society.

The core of the Symposium consisted of courses of lectures by six keynote speakers (Aldous, Dawson, Kesten, Meyer, Sznitman and Varadhan), three of which appear here in written form. In addition, there were twenty-six talks by invited speakers; the written versions of eleven of these make up the remainder of the volume.

All the papers in the volume have been refereed.

It is a pleasure to thank here all those individuals and institutions who contributed to the success of the Symposium, and to these Proceedings. We thank the London Mathematical Society for the invitation to organize the meeting, and the Science and Engineering Research Council for financial support under grant GR/F18459. We are grateful for the University of Durham for use of its facilities in Gray College and the Department of Mathematics, to our local organizers, John Bolton and Lyndon Woodward, and to Mrs Susan Nesbitt for her devoted secretarial help. Our main debt is to the speakers and participants at the Symposium, and to the contributors and referees for the Proceedings, and we thank them all. Last but by no means least, we thank David Williams, the organizer of the 1980 Durham Symposium on Stochastic Integrals, for much valuable advice during the planning of this meeting.

INTRODUCTION

Many different models of random trees have arisen in a variety of applied setting, and there is a large but scattered literature on exact and asymptotic results for particular models. For several years I have been interested in what kinds of “general theory” (as opposed to ad hoc analysis of particular models) might be useful in studying asymptotics of random trees. In this paper, aimed at theoretical probabilists, I discuss aspects of this incipient general theory which are most closely related to topics of current interest in theoretical stochastic processes. No prior knowledge of this subject is assumed: the paper is intended as an introduction and survey.

To give the really big picture in a paragraph, consider a tree on n vertices. View the vertices as points in abstract (rather than d-dimensional) space, but let the edges have length (= 1, as a default) so that there is metric structure: the distance between two vertices is the length of the path between them. Consider the average distance between pairs of vertices. As n → ∞ this average distance could stay bounded or could grow as order n, but almost all natural random trees fall into one of two categories. In the first (and larger) category, the average distance grows as order logn. This category includes supercritical branching processes, and most “Markovian growth” models such as those occurring in the analysis of algorithms. This paper is concerned with the second category, in which the average distance grows as order n½.

Introduction.

In 1984 [P1] we introduced the theme of inverse problems for Brownian motion on Riemannian manifolds, in terms of the mean exit time from small geodesic balls. Since that time a number of works have appeared on related stochastic problems as well as on some classical, non-stochastic quantities which may be treated by the same methods. Most recently H.R. Hughes [Hu] has shown that in six dimensions one cannot recover the Riemannian metric from the exit time distribution, thereby answering in a strong sense the main question posed in [P1].

The general area of “inverse spectral theory” was initiated by Mark Kac in his now famous paper [Ka] on the two-dimensional drumhead. In the intervening years a large literature has developed on inverse spectral problems in higher dimensional Euclidean space and differentiable manifolds; for recent surveys see ([Be], [Br], [Go]). In these approaches one is given the entire spectrum of eigenvalues, from which one asks various geometric questions. Our approach, by contrast, is able to obtain strong geometric information from the sole knowledge of the principal eigenvalue of a parametric family of geodesic balls (see section 6, below).

Introduction

The primary aim of this paper is to establish evolution equations for the intersection local time (ILT) of the super Brownian motion and certain super stable processes. We shall proceed by carefully defining the requisite concepts and giving all of our main results in the Introduction, while leaving the proofs for later sections. The Introduction itself is divided into four sections, which treat, in turn, the definition of the superprocesses that will interest us, the definition of ILT and some previous results, our main result – a Tanaka-like evolution equation for ILT – and an Itô formula for measure-valued processes along with a description of how to use it to derive the evolution equation. Some technical lemmas make up Section 2 of the paper, while Section 3 is devoted to proofs.

In order to conserve space, we shall motivate neither the study of superprocesses per se – other than to note that they arise as infinite density limits of infinitely rapidly branching stochastic processes – nor the study of ILT – other than to note that this seems to be important for the introduction of an intrinsic dependence structure for the spatial part of a superprocess. Good motivational and background material on superprocesses can be found in Dawson (1978, 1986), Dawson, Iscoe and Perkins (1989), Ethier and Kurtz (1986), Roelly-Coppoletta (1986), Walsh (1986) and Watanabe (1968), as well as other papers in this volume. Material on ILT can be found in Adler, Feldman and Lewin (1991), Adler and Lewin (1991), Adler and Rosen (1991), Dynkin (1988) and Perkins (1988).

The most important chapter in this book is Chapter E: Exercises. I have left the interesting things for you to do. You can start now on the ‘EG’ exercises, but see ‘More about exercises’ later in this Preface.

The book, which is essentially the set of lecture notes for a third-year undergraduate course at Cambridge, is as lively an introduction as I can manage to the rigorous theory of probability. Since much of the book is devoted to martingales, it is bound to become very lively: look at those Exercises on Chapter 10! But, of course, there is that initial plod through the measure-theoretic foundations. It must be said however that measure theory, that most arid of subjects when done for its own sake, becomes amazingly more alive when used in probability, not only because it is then applied, but also because it is immensely enriched.

You cannot avoid measure theory: an event in probability is a measurable set, a random variable is a measurable function on the sample space, the expectation of a random variable is its integral with respect to the probability measure; and so on. To be sure, one can take some central results from measure theory as axiomatic in the main text, giving careful proofs in appendices; and indeed that is exactly what I have done.