Fractals in probability and analysis (Q2833169)

This book focuses on some of the fractal sets that appear in analysis and probability theory and presents the mathematical methods to study them. The prerequisites are a standard course on measure and probability theory. Exercises with varying degrees of difficulty are also included at the end of each chapter.NEWLINENEWLINENEWLINEIn Chapter 1, basic concepts such as the Minkowski and Hausdorff dimension are defined, Billingley's lemma and the law of large numbers are proved, and the concept of dimension of a measure introduced. Chapter 2 deals with self-similar sets and the open set condition, alternate definitions to Minkowski dimension and packing measures and dimension. The next chapter considers Frostman's lemma and capacity and dimension. Self-affine sets and their dimensions are investigated in Chapter 4. The concept of nowhere differentiable functions is introduced in Chapter 5 and several examples are given. The following two chapters deal with Brownian motion and its properties. Dimension doubling, the law of the iterated logarithm, and the relation between harmonic functions and Brownian motion in \(\mathbb{R}^d\) are exhibited. Random walks, Markov chains, and capacity are the topics of Chapter 8. In Chapter 9, deterministic and random constructions of Besicovitch(-Kakeya) sets are presented and an application of such sets to Fourier analysis discussed. The last chapter introduces \(\beta\)-numbers and uses them to estimate the length of the shortest curve containing a given set in \(\mathbb{R}^2\) to within a bounded factor. This is also known as a traveling salesman problem. The three appendices summarize the Banach fixed point theorem, Frostman's lemma for analytic sets, and provide hints and solutions to selected exercises.NEWLINENEWLINENEWLINEThe book is well-suited for researcher who want to acquaint themselves with some fractal sets and the mathematical methodologies surrounding them. It provides the means to go deeper into the subjects the authors introduced and to investigate further topics in the areas of fractal geometry and fractal analysis.