Introduction

The principal objective of this chapter is to present some of the essential basic mathematical background that is employed in later sections. A good deal of this material should be straightforward for a well trained undergraduate mathematics or physics major; however, there are a few slightly more advanced topics that are treated concisely. For these topics, collateral reading in a standard text is highly recommended. Some suggestions of where to start additional reading are provided in the end–notes. The mathematically talented reader could bypass most of this chapter and skip to the derivations in Chapter 3.

Almost all the mathematical notation employed can be found in the List of symbols; please consult that list for the definition or for the first use of a particular symbol. Some common notational devices are reviewed first, and this is followed by a concise description of some of the more important mathematical tools, such as Fourier analysis, complex variable theory, and the basics of integration theory, i.e. topics that are central to later developments. Further extensions of some of these tools are given later as needed.

Introduction

The developments of the previous chapter are continued in this chapter. The principal focus is on the reflectance and the energy loss function. Knowledge of the reflectance and phase allows the real and imaginary parts of the complex refractive index to be evaluated, and hence the real and imaginary parts of the complex dielectric constant can be determined. Measurement of the reflectance and calculation of the phase provides a convenient route to a number of optical properties.

The procedure for dealing with the formulation of dispersion relations for the reflectance and phase has some features that are different from the cases of the dielectric constant and the refractive index. This is tied directly to the fact that the real and imaginary parts of the complex reflectivity depend upon both the reflectance and the phase. To uncouple these quantities requires consideration of the logarithm of the complex reflectivity, which introduces some additional issues into the discussion. This also has a direct bearing on the types of sum rules that can be derived.

Dispersion relations for the normal-incident reflectance and phase

The topic of obtaining dispersion relations for the normal-incident reflectivity is examined in this section. Consideration is restricted to the normal-incidence case, and it will be assumed that light is impinging on the material surface from a vacuum. The reflectivity, R(ω), is the ratio of the reflected to the incident light intensities.

Systems

This chapter is concerned with setting up the foundations that allow the connection between causality and analyticity to be established. The interplay between these two topics and the Hilbert transform is also treated. The material of this chapter lays the basis for many of the applications discussed in the following chapters.

Consider the arrangement in Figure 17.1, where the input to the system is denoted by i, and r designates the output response. The input and output could be of the same nature, for example a voltage, or very different variables, for example a voltage input, with the output being a physical displacement of a mass. As an example of Figure 17.1, consider the input to be the driving force acting on a simple oscillator arrangement with a mass hanging from a fixed point by a spring. The output is the displacement of the mass. The input has some dependence on time, and likewise the output response. Henceforth, the temporal dependence is made explicit. Often the input and output response are continuous functions of the time variable, but this is not a requirement. In later sections, the focus will include consideration of step function and impulse inputs.

The term “system” refers to a device capable of converting an input into some output response. The output response should be totally characterized by the system input and the characteristics of the system.

Some background from classical electrodynamics

The principal intent of this chapter is to arrive at the classical Hilbert transform connections that apply between the real and imaginary components of the generalized (complex) refractive index, and for the complex dielectric constant. Connections of this type are frequently termed dispersion relations in the physics literature. But for the two functions just mentioned, and for many associated results, they are most often referred to as the Kramers–Kronig relations. Historically, these were the first applications of the Hilbert transform concept in the physical sciences, and were discovered by Kronig (1926) and independently by Kramers (1927). These authors were interested in issues connected with the dispersion of light, and from this emerged the term dispersion relation to describe the Hilbert transform relations found by Kramers and Kronig. The reader will recall that dispersion refers to the frequency variation of the refractive index (or some other optical property), and dispersion formulas provide a connection between the refractive index and the frequency. Functions such as the dielectric constant, refractive index, and permeability, which will be defined shortly, are referred to as optical constants. These functions characterize the interaction of electromagnetic radiation with matter. Though in widespread use, this terminology is somewhat of a misnomer, since the optical constants actually depend on the frequency of the incident electromagnetic radiation interacting with the material, and are hence not true constants.

Abstract

Introduction

Muller's ratchet is a mechanism that has been suggested as an explanation for the evolution of sex [13]. The idea is simple; in an asexually reproducing population chromosomes are passed down as indivisible blocks and so the number of deleterious mutations accumulated along any ancestral line in the population can only increase.

Abstract

Introduction

Upper and lower bounds for the (linear) growth rates of the diameter of the image of a bounded set in Rd under the action of a stochastic flow under various conditions have been shown in [4, 5, 6, 16, 17, 20]. In this survey, we will discuss upper bounds only. A well-established class of methods to obtain probability bounds for the supremum of a process are chaining techniques. Typically they transform bounds for the one-and two-dimensional distributions of the process into upper bounds of the supremum (for a real-valued process) or the diameter of the range of the process (for a process taking values in a metric space). In the next section, we will present some of these techniques, the best-known being Kolmogorov's continuity theorem, which not only states the existence of a continuous modification, but also provides explicit probabilistic upper bounds for the modulus of continuity and the diameter of the range of the process. We will also state a result which we call basic chaining.

Abstract

Introduction

We discuss mathematical models for an effect which in population genetics jargon, somewhat orthogonal to diffusion process nomenclature, is called “genetic drift”, namely the phenomenon that the distribution of genetic types in a population changes in the course of time simply due to stochasticity in the individuals' reproductive success and the finiteness of all real populations. We will only consider “neutral” genetic types.

This contrasts and complements the notion of selection, which refers to scenarios in which one or some of the types confer a direct or indirect reproductive advantage to their bearers. Thus, in the absence of demographic stochasticity, the proportion of a selectively advantageous type would increase in the population, whereas that of neutral types would remain constant. The interplay between small fitness differences among types and the stochasticity due to finiteness of populations leads to many interesting and challenging problems, see e.g. the paper by A. Etheridge, P. Pfaffelhuber and A. Wakolbinger in this volume.

This collection of papers on stochastic analysis is dedicated to Professor Heinrich von Weizsäcker on the occasion of his 60th birthday. The papers, written by a group of his students, coauthors, friends and colleagues, capture various important trends in the field, providing overviews of recent developments and often new results. They also give a hint of many of Heinrich's interests, and the profound influence he has, both within the field and on his collaborators. All papers have been peerreviewed.

Heinrich von Weizsäcker began his research in mathematics as a graduate student in the early seventies. At the time, his focus was on real analysis and measure theory. He obtained his Doctorate at the Ludwig-Maximilian-Universtät München in 1973 under the supervision of Professor Hans Richter, for a thesis entitled ‘Vektorverbände und meßbare Funktionen’ (Vector lattices and measurable functions). In 1977 he defended his habilitation with a thesis entitled ‘Einige maßtheoretische Formen der Sätze von Krein-Milman und Choquet’ (Some measure theoretic variants of the theorems of Krein-Milman and Choquet) and after brief spells at the universities of Regensburg and Marburg, he moved to a chair at Universität Kaiserslautern.

In Kaiserslautern he built a strong research group, focusing more and more on stochastic analysis. He supervised a total of 11 PhDs and two habilitations; six of his former students remain in academia today. His current PhD students are Richard Kiefer, Martin Kolb and Yang Zou.

Abstract

Introduction

I am often asked why I am interested in Hausdorff dimension. Are there any important problems that can be solved using Hausdorff dimension? Can Hausdorff dimension really add to our understanding of stochastic processes? I believe that the answer is yes to both questions, and in this paper I attempt to give some evidence in the case of the second question, by means of three examples. I will focus on the notion of a multifractal spectrum or dimension spectrum, which in its broadest form refers to the Hausdorff dimension of a parametrized family of sets, seen as a function of the parameter.

The examples are chosen on the one hand for their relative simplicity, on the other hand to illustrate the diversity of shapes which a multifractal spectrum can take. A common thread in all the examples is the notion of a tree, which either features prominently in the initial description or presents a very valuable reformulation of the model.