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.

Introduction

This chapter examines local properties of s-sets in ℝn for nonintegral s. The fundamental result is that any such set is irregular, that is, has lower circular or spherical density strictly less than 1 at almost all of its points. Indeed, the stronger result that its density fails to exist at almost all of its points has also been established. As before, we also examine the existence of suitably defined tangents, and show that the set of points at which such tangents exist must have measure zero.

For the case of subsets of the plane, the work is entirely due to Marstrand (1954a, 1955), the former paper providing a very complete account. As with sets of integral dimension, higher-dimensional analogues present formidable difficulties; the natural generalizations were eventually proved by Marstrand (1964).

s-sets with 0 < s < 1

First we consider s-sets in ℝn for s strictly less than 1. In this case the basic properties, including non-existence of the density almost everywhere, are relatively easy to obtain.

The following topological observation about such sets is sometimes useful.

Lemma 4.1

An s-set E in ℝnwith 0 < 5 < 1 is totally disconnected.

Proof. Let x and y be distinct points in the same connected component of E. Define a mapping f:ℝn →[0, ∞) by f(z) = |z–x|. Since f does not increase distances it follows from Lemma 1.8 that ℋs(f(E)) ≤ ℋs(E) < ∞. As 5 < 1 it follows that f(E) is a subset of ℝ of Lebesgue measure zero and, in particular, has dense complement.

This tract provides a rigorous self-contained account of the mathematics of sets of fractional and integral Hausdorff dimension. It is primarily concerned with geometric theory rather than with applications. Much of the contents could hitherto be found only in original mathematical papers, many of which are highly technical and confusing and use archaic notation. In writing this book I hope to make this material more readily accessible and also to provide a useful and precise account for those using fractal sets.

Whilst the book is written primarily for the pure mathematician, I hope that it will be of use to several kinds of more or less sophisticated and demanding reader. At the most basic level, the book may be used as a reference by those meeting fractals in other mathematical or scientific disciplines. The main theorems and corollaries, read in conjunction with the basic definitions, give precise statements of properties that have been rigorously established.

To get a broad overview of the subject, or perhaps for a first reading, it would be possible to follow the basic commentary together with the statements of the results but to omit the detailed proofs. The non-specialist mathematician might also omit the details of Section 1.1 which establishes the properties of general measures from a technical viewpoint.

A full appreciation of the details requires a working knowledge of elementary mathematical analysis and general topology. There is no doubt that some of the proofs central to the development are hard and quite lengthy, but it is well worth mastering them in order to obtain a full insight into the beauty and ingenuity of the mathematics involved.

Introduction

This chapter surveys examples of sets of fractional dimension which result from particular constructions or occur in other branches of mathematics or physics and relates them to earlier parts of the book. The topics have been chosen very much at the author's whim rather than because they represent the most important occurrences of fractal sets. In each section selected results of interest are proved and others are cited. It is hoped that this approach will encourage the reader to follow up some of these topics in greater depth elsewhere. Most of the examples come from areas of mathematics which have a vast literature; therefore in this chapter references are given only to the principal sources and to recent papers and books which contain further surveys and references.

Curves of fractional dimension

In this section we work in the (x,y)-coordinate plane and investigate the Hausdorff dimension of Γ, the set of points (x,f(x)) forming the graph of a function f defined, say, on the unit interval.

If f is a function of bounded variation, that is, if is bounded for all dissections 0 = x0 < x1 < … < xm = l, then we are effectively back in the situation of Section 3.2; Γ is a rectifiable curve and so a regular 1-set. However, if f is a sufficiently irregular, though continuous, function it is possible for Γ to have dimension greater than 1. In such cases it can be hard to calculate the Hausdorff dimension and measure of Γ from a knowledge of f. However, if f satisfies a Lipschitz condition it is easy to obtain an upper bound.

The geometric measure theory of sets of integral and fractional dimension has been developed by pure mathematicians from early in this century. Recently there has been a meteoric increase in the importance of fractal sets in the sciences. Mandelbrot (1975,1977,1982) pioneered their use to model a wide variety of scientific phenomena from the molecular to the astronomical, for example: the Brownian motion of particles, turbulence in fluids, the growth of plants, geographical coastlines and surfaces, the distribution of galaxies in the universe, and even fluctuations of price on the stock exchange. Sets of fractional dimension also occur in diverse branches of pure mathematics such as the theory of numbers and non-linear differential equations. Many further examples are described in the scientific, philosophical and pictorial essays of Mandelbrot. Thus what originated as a concept in pure mathematics has found many applications in the sciences. These in turn are a fruitful source of further problems for the mathematician. This tract is concerned primarily with the geometric theory of such sets rather than with applications.

The word ‘fractal’ was derived from the latin fractus, meaning broken, by Mandelbrot (1975), who gave a ‘tentative definition’ of a fractal as a set with its Hausdorff dimension strictly greater than its topological dimension, but he pointed out that the definition is unsatisfactory as it excludes certain highly irregular sets which clearly ought to be thought of in the spirit of fractals. Hitherto mathematicians had referred to such sets in a variety of ways – ‘sets of fractional dimension’, ‘sets of Hausdorff measure’, ‘sets with a fine structure’ or ‘irregular sets’.