Markov Operators on Banach Lattices

Abstract

A brief search on www.ams.org with the keyword “Markov operator” produces some 684 papers, the earliest of which dates back to 1959. This suggests that the term “Markov operator” emerged around the 1950’s, clearly in the wake of Andrey Markov’s seminal work in the area of stochastic processes and Markov chains. Indeed, [17] and [6], the two earliest papers produced by the ams.org search, study Markov processes in a statistical setting and “Markov operators” are only referred to obliquely, with no explicit definition being provided. By 1965, in [7], the situation has progressed to the point where Markov operators are given a concrete definition and studied more directly. However, the way in which Markov operators originally entered mathematical discourse, emerging from Statistics as various attempts to generalize Markov processes and Markov chains, seems to have left its mark on the theory, with a notable lack of cohesion amongst its propagators. The study of Markov operators in the Lp setting has assumed a place of importance in a variety of fields. Markov operators figure prominently in the study of densities, and thus in the study of dynamical and deterministic systems, noise and other probabilistic notions of uncertainty. They are thus of keen interest to physicists, biologists and economists alike. They are also a worthy topic to a statistician, not least of all since Markov chains are nothing more than discrete examples of Markov operators (indeed, Markov operators earned their name by virtue of this connection) and, more recently, in consideration of the connection between copulas and Markov operators. In the realm of pure mathematics, in particular functional analysis, Markov operators have proven a critical tool in ergodic theory and a useful generalization of the notion of a conditional expectation. Considering the origin of Markov operators, and the diverse contexts in which they are introduced, it is perhaps unsurprising that, to the uninitiated observer at least, the theory of Markov operators appears to lack an overall unity. In the literature there are many different definitions of Markov operators defined on L1(μ) and/or L1(μ) spaces. See, for example, [13, 14, 26, 2], all of which manage to provide different definitions. Even at a casual glance, although they do retain the same overall flavour, it is apparent that there are substantial differences in these definitions. The situation is not much better when it comes to the various discussions surrounding ergodic Markov operators: we again see a variety of definitions for an ergodic operator (for example, see [14, 26, 32]), and again the connections between these definitions are not immediately apparent. In truth, the situation is not as haphazard as it may at first appear. All the definitions provided for Markov operator may be seen as describing one or other subclass of a larger class of operators known as the positive contractions. Indeed, the theory of Markov operators is concerned with either establishing results for the positive contractions in general, or specifically for one of the aforementioned subclasses. The confusion concerning the definition of an ergodic operator can also be rectified in a fairly natural way, by simply viewing the various definitions as different possible generalizations of the central notion of a ergodic point-set transformation (such a transformation representing one of the most fundamental concepts in ergodic theory). The first, and indeed chief, aim of this dissertation is to provide a coherent and reasonably comprehensive literature study of the theory of Markov operators. This theory appears to be uniquely in need of such an effort. To this end, we shall present a wealth of material, ranging from the classical theory of positive contractions; to a variety of interesting results arising from the study of Markov operators in relation to densities and point-set transformations; to more recent material concerning the connection between copulas, a breed of bivariate function from statistics, and Markov operators. Our goals here are two-fold: to weave various sources into a integrated whole and, where necessary, render opaque material readable to the non-specialist. Indeed, all that is required to access this dissertation is a rudimentary knowledge of the fundamentals of measure theory, functional analysis and Riesz space theory. A command of measure and integration theory will be assumed. For those unfamiliar with the basic tenets of Riesz space theory and functional analysis, we have included an introductory overview in the appendix. The second of our overall aims is to give a suitable definition of a Markov operator on Banach lattices and provide a survey of some results achieved in the Banach lattice setting, in particular those due to [5, 44]. The advantage of this approach is that the theory is order theoretic rather than measure theoretic. As we proceed through the dissertation, definitions will be provided for a Markov operator, a conservative operator and an ergodic operator on a Banach lattice. Our guide in this matter will chiefly be [44], where a number of interesting results concerning the spectral theory of conservative, ergodic, so-called “stochastic” operators is studied in the Banach lattice setting. We will also, and to a lesser extent, tentatively suggest a possible definition for a Markov operator on a Riesz space. In fact, we shall suggest, as a topic for further research, two possible approaches to the study of such objects in the Riesz space setting. We now offer a more detailed breakdown of each chapter. In Chapter 2 we will settle on a definition for a Markov operator on an L1 space, prove some elementary properties and introduce several other important concepts. We will also put forward a definition for a Markov operator on a Banach lattice. In Chapter 3 we will examine the notion of a conservative positive contraction. Conservative operators will be shown to demonstrate a number of interesting properties, not least of all the fact that a conservative positive contraction is automatically a Markov operator. The notion of conservative operator will follow from the Hopf decomposition, a fundmental result in the classical theory of positive contractions and one we will prove via [13]. We will conclude the chapter with a Banach lattice/Riesz space definition for a conservative operator, and a generalization of an important property of such operators in the L1 case. In Chapter 4 we will discuss another well-known result from the classical theory of positive contractions: the Chacon-Ornstein Theorem. Not only is this a powerful convergence result, but it also provides a connection between Markov operators and conditional expectations (the latter, in fact, being a subclass of theMarkov operators). To be precise, we will prove the result for conservative operators, following [32]. In Chapter 5 we will tie the study of Markov operators into classical ergodic theory, with the introduction of the Frobenius-Perron operator, a specific type of Markov operator which is generated from a given nonsingular point-set transformation. The Frobenius-Perron operator will provide a bridge to the general notion of an ergodic operator, as the definition of an ergodic Frobenius-Perron operator follows naturally from that of an ergodic transformation. In Chapter 6 will discuss two approaches to defining an ergodic operator, and establish some connections between the various definitions of ergodicity. The second definition, a generalization of the ergodic Frobenius-Perron operator, will prove particularly useful, and we will be able to tie it, following [26], to several interesting results concerning the asymptotic properties of Markov operators, including the asymptotic periodicity result of [26, 27]. We will then suggest a definition of ergodicity in the Banach lattice setting and conclude the chapter with a version, due to [5], of the aforementioned asymptotic periodicity result, in this case for positive contractions on a Banach lattice. In Chapter 7 we will move into more modern territory with the introduction of the copulas of [39, 40, 41, 42, 16]. After surveying the basic theory of copulas, including introducing a multiplication on the set of copulas, we will establish a one-to-one correspondence between the set of copulas and a subclass of Markov operators. In Chapter 8 we will carry our study of copulas further by identifying them as a Markov algebra under their aforementioned multiplication. We will establish several interesting properties of this Markov algebra, in parallel to a second Markov algebra, the set of doubly stochastic matrices. This chapter is chiefly for the sake of interest and, as such, diverges slightly from our main investigation of Markov operators. In Chapter 9, we will present the results of [44], in slightly more detail than the original source. As has been mentioned previously, these concern the spectral properties of ergodic, conservative, stochastic operators on a Banach lattice, a subclass of the Markov operators on a Banach lattice. Finally, as a conclusion to the dissertation, we present in Chapter 10 two possible routes to the study of Markov operators in a Riesz space setting. The first definition will be directly analogous to the Banach lattice case; the second will act as an analogue to the submarkovian operators to be introduced in Chapter 2. We will not attempt to develop any results from these definitions: we consider them a possible starting point for further research on this topic. In the interests of both completeness, and in order to aid those in need of more background theory, the reader may find at the back of this dissertation an appendix which catalogues all relevant results from Riesz space theory and operator theory.