Student Number : 0108851W - MSc Dissertation - School of Mathematics - Faculty of Science / 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.

The concept of LR-nets was introduced and investigated firstly by H.P. Lotz in [27] and by F. Raebiger in [30]. Therefore we call such nets Lotz-Raebiger nets, shortly LR-nets. In this thesis we treat two problems on asymptotic behavior of these operator nets. First problem is to generalize well known theorems for Ces`aro averages of a single operator to LR-nets, namely to generalize the Eberlein and Sine theorems. The second problem is related to the strong convergence of Markov LR-nets on L1-spaces. We prove that the existence of a lower-bound functions is necessary and sufficient for asymptotic stability of LR-nets of Markov operators.

A stochastic operator is a positive linear contraction, P : L1 --+ L1, such that llPfII2 = llfll1 for f > 0. It is called asymptotically stable if the iterates pn f of each density converge in the norm to a fixed density. Pf(x) = f K(x,y)f(y)dy, where K( ·, y) is a density, defines a stochastic kernel operator. A general probabilistic/ deterministic model for biological systems is considered. This leads to the LMT operator P f(x) = Jo - Bx H(Q(>.(x)) - Q(y)) dy, where -H'(x) = h(x) is a density. Several particular examples of cell cycle models are examined. An operator overlaps supports iffor all densities f,g, pn f APng of 0 for some n. If the operator is partially kernel, has a positive invariant density and overlaps supports, it is asymptotically stable. It is found that if h( x) > 0 for x ~ xo ~ 0 and ["'" x"h(x) dx < liminf(Q(A(x))" - Q(x)") for a E (0, 1] lo x-oo then P is asymptotically stable, and an opposite condition implies P is sweeping. Many known results for cell cycle models follow from this. / Mathematical Science / M. Sc. (Mathematics)

A stochastic operator is a positive linear contraction, P : L1 --+ L1, such that llPfII2 = llfll1 for f > 0. It is called asymptotically stable if the iterates pn f of each density converge in the norm to a fixed density. Pf(x) = f K(x,y)f(y)dy, where K( ·, y) is a density, defines a stochastic kernel operator. A general probabilistic/ deterministic model for biological systems is considered. This leads to the LMT operator P f(x) = Jo - Bx H(Q(>.(x)) - Q(y)) dy, where -H'(x) = h(x) is a density. Several particular examples of cell cycle models are examined. An operator overlaps supports iffor all densities f,g, pn f APng of 0 for some n. If the operator is partially kernel, has a positive invariant density and overlaps supports, it is asymptotically stable. It is found that if h( x) > 0 for x ~ xo ~ 0 and ["'" x"h(x) dx < liminf(Q(A(x))" - Q(x)") for a E (0, 1] lo x-oo then P is asymptotically stable, and an opposite condition implies P is sweeping. Many known results for cell cycle models follow from this. / Mathematical Science / M. Sc. (Mathematics)

L'objet de cette thèse est l'étude du comportement asymptotique des systèmes de fonctions itérées (IFS). Dans un premier chapitre, nous présenterons les notions liées à l'étude de tels systèmes et nous rappellerons différentes applications possibles des IFS telles que les marches aléatoires sur des graphes ou des pavages apériodiques, les systèmes dynamiques aléatoires, la classification de protéines ou encore les mesures quantiques répétées. Nous nous attarderons sur deux autres applications : les chaînes de Markov d'ordre infini et d'ordre variable. Nous donnerons aussi les principaux résultats de la littérature concernant l'étude des mesures invariantes pour des IFS ainsi que ceux pour le calcul de la dimension de Hausdorff. Le deuxième chapitre sera consacré à l'étude d'une classe d'IFS composés de contractions sur des intervalles réels fermés dont les images se chevauchent au plus en un point et telles que les probabilités de transition sont constantes par morceaux. Nous donnerons un critère pour l'existence et pour l'unicité d'une mesure invariante pour l'IFS ainsi que pour la stabilité asymptotique en termes de bornes sur les probabilités de transition. De plus, quand il existe une unique mesure invariante et sous quelques hypothèses techniques supplémentaires, on peut montrer que la mesure invariante admet une dimension de Hausdorff exacte qui est égale au rapport de l'entropie sur l'exposant de Lyapunov. Ce résultat étend la formule, établie dans la littérature pour des probabilités de transition continues, au cas considéré ici des probabilités de transition constantes par morceaux. Le dernier chapitre de cette thèse est, quant à lui, consacré à un cas particulier d'IFS : les chaînes de Markov de longueur variable (VLMC). On démontrera que sous une condition de non-nullité faible et de continuité pour la distance ultramétrique des probabilités de transitions, elles admettent une unique mesure invariante qui est attractive pour la convergence faible. / The purpose of this thesis is the study of the asymptotic behaviour of iterated function systems (IFS). In a first part, we will introduce the notions related to the study of such systems and we will remind different applications of IFS such as random walks on graphs or aperiodic tilings, random dynamical systems, proteins classification or else $q$-repeated measures. We will focus on two other applications : the chains of infinite order and the variable length Markov chains. We will give the main results in the literature concerning the study of invariant measures for IFS and those for the calculus of the Hausdorff dimension. The second part will be dedicated to the study of a class of iterated function systems (IFSs) with non-overlapping or just-touching contractions on closed real intervals and adapted piecewise constant transition probabilities. We give criteria for the existence and the uniqueness of an invariant probability measure for the IFSs and for the asymptotic stability of the system in terms of bounds of transition probabilities. Additionally, in case there exists a unique invariant measure and under some technical assumptions, we obtain its exact Hausdorff dimension as the ratio of the entropy over the Lyapunov exponent. This result extends the formula, established in the literature for continuous transition probabilities, to the case considered here of piecewise constant probabilities. The last part is dedicated to a special case of IFS : Variable Length Markov Chains (VLMC). We will show that under a weak non-nullness condition and continuity for the ultrametric distance of the transition probabilities, they admit a unique invariant measure which is attractive for the weak convergence.