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.

Abstract

Introduction

The first of Feynman's papers [7], on what one now calls Feynman path integrals over trajectories in the configuration space, contains three main observations. Firstly, it is shown that the solution of the Cauchy problem for a Schrödinger equation can be represented by a limit of a sequence of integrals over Cartesian products of the classical configuration space when the multiplicity of integrals tends to infinity. Secondly, the limit is interpreted as an integral over trajectories in the configuration space. And finally, it is noticed that the integrand contains the exponent of the classical action. Feynman's definition of the Feynman path integrals over trajectories in the phase space, which is formulated in his second paper on the subject, has a similar structure but, in contrast to the preceding definition, the Lagrangian in the classical action is expressed through the Hamiltonian function.

Abstract

Introduction

In this paper we define the concept of a stochastic ensemble. It is our intention thereby to give an intuitive axiomatic approach to the concept of a random variable. The primary ingredient is a sufficiently rich collection of random variables (with ‘good’ target spaces). The set of observable events will be derived from it.

Among the notions of probability it is the random variable which in our view constitutes the fundamental object of modern probability theory. Albeit in the history of mathematical probability events came first, random variables are closer to the roots of understanding nondeterministic phenomena. Nowadays events typically refer to random variables and are no longer studied for their own sake, and for distributions the situation is not much different. Moreover, random variables turn out to be flexible mathematical objects. They can be handled in other ways than events or distributions (think of couplings), and these ways often conform to intuition.

Abstract

Introduction

In many situations, understanding the behaviour of a stochastic system is greatly aided by understanding its behaviour conditioned on certain events. This allows us, for example, to study rare events by conditioning on the event happening or to analyse the behaviour of a composite system when only some of its components can be observed. Since properties of conditional distributions are often difficult to obtain analytically, it is desirable to be able to study these distributions numerically. This allows us to develop meaningful conjectures about the distribution in question or, in a more applied context, to derive quantitative information about it. In this text we present a general technique to generate samples from conditional distributions on infinite-dimensional spaces. We give several examples to illustrate how this technique can be applied.

Sampling, i.e. finding a mechanism which produces random values distributed according to a prescribed target distribution, is generally a difficult problem. There exist many ‘tricks’ to sample from specific distributions, ranging from very specialized methods, like the Box–Müller method for generating one-dimensional standard Gaussian distributed values, to generic methods, like rejection sampling, which can be applied to whole classes of distributions.

Abstract

Introduction

Stochastic processes have been used as a powerful modelling tool for decades in situations where the evolution of a system has some random component, be it intrinsic or to model the interaction with a complex environment. In its most general form, a stochastic process describes the evolution X(t, ω) of a system, where t denotes the time parameter and ω takes values in some probability space and abstracts the ‘element of chance’ describing the randomness of the process.