S. James Taylor

The measure theory of random fractals

In 1951 A. S. Besicovitch, who was my research supervisor, suggested that I look at the problem of determining the dimension of the range of a Brownian motion path. This problem had been communicated to him by C. Loewner, but it was a natural question which had already attracted the attention of Paul Levy. It was a good problem to give to an ignorant Ph.D. student because it forced him to learn the potential theory of Frostman [33] and Riesz[75] as well as the Wiener [98] definition of mathematical Brownian motion. In fact the solution of that first problem in [81] used only ideas which were already twenty-five years old, though at the time they seemed both new and original to me. My purpose in this paper is to try to trace the development of these techniques as they have been exploited by many authors and used in diverse situations since 1953. As we do this in the limited space available it will be impossible to even outline all aspects of the development, so I make no apology for giving a biased account concentrating on those areas of most interest to me. At the same time I will make conjectures and suggest some problems which are natural and accessible in the hope of stimulating further research.

Fractal properties of products and projections of measures in d

Borel measures in ℝ d are called fractal if locally at a.e. point their Hausdorff and packing dimensions are identical. It is shown that the product of two fractal measures is fractal and almost all projections of a fractal measure into a lower dimensional subspace are fractal. The results rely on corresponding properties of Borel subsets of ℝ d which we summarize and develop.

Uniform measure results for the image of subsets under Brownian motion

SummaryThe paper obtains bounds on the Hausdorff and packing measures of the imageX(E) of a Borel setE by a transient strictly stable processXt which a.s. hold for allE and for every measure function

The multifractal structure of stable occupation measure

h_{\beta ,\gamma } (s) = s^\beta \left| {\log s} \right|^{\gamma ^ \star }

Multifractal structure of a general subordinator

. In some cases examples are constructed to show that the bounds are sharp.

The local structure of the sample paths of asymmetric cauchy processes

We calculate the multifractal spectrum and mass exponents for super-Brownian motion in three or more dimensions. The former is trivial for points of unusually high density but not for points in the support of unusually low density. This difference is due to the presence of sets of points in the support (of positive dimension) about which there are asymptotically large empty annuli. This behaviour is quite different from that of ordinary Brownian motion and invalidates the multifractal formalism in the physics literature. The mass exponents for packing and Hausdorff measure are distinct, and both are piecewise linear.

THE BEHAVIOUR AND CONSTRUCTION OF LOCAL TIMES FOR LEVY PROCESSES

Let X be a stable subordinator of index [alpha] and [mu] be the occupation measure of X. Denote d([mu],x) and as the lower and upper local dimensions of [mu]. We obtain that the Hausdorff dimension of the set of the points where is (2[alpha]2/[beta]) - [alpha] a.s. and the lower bound of packing dimension is 2[alpha] - [beta] a.s. if [alpha][less-than-or-equals, slant][beta][less-than-or-equals, slant]2[alpha]. When [beta] > 2[alpha], the corresponding set is empty a.s.. And for a.s. [Omega], the set of the points where is the closure of X[0,1].

Packing measure, and its evaluation for a Brownian path

SummaryPrecise conditions are obtained for the packing measure of an arbitrary subordinator to be zero, positive and finite, or infinite. It develops that the packing measure problem for a subordinatorX(t) is equivalent to the upper local growth problem forY(t)=min (Y1(t), Y2(t)), whereY1 andY2 are independent copies ofX. A finite and positive packing measure is possible for subordinators “close to Cauchy”; for such a subordinator there is non-random concave upwards function that exactly describes the upper local growth ofY (although, as is well known, there is no such function for the subordinatorX itself).