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.