In sections 1.1, 1.2 and 1.3 of this chapter, we briefly introduce basic notions and some results borrowed from the theory of discrete time homogeneous countable Markov chains (MC).

In section 1.4, some well known examples of MCs are given, for which a complete classification can be obtained by elementary methods: simple probabilistic arguments in 1.4.1, explicit solution of recurrent equations in 1.4.2, generating functions in 1.4.3.

It is not our intention to devote a detailed section to the fundamentals of probability theory, which are presented in a plethora of excellent text-books. Thus, we only introduce in fact the minimal basic notions and notation useful for our purpose.

• The events are the subsets of some abstract set Ω, which belong to Σ, the σ-algebra defined on Ω.

• The couple (Ω, Σ) is a measurable space and the sets belonging to Σ are

Σ-measurable sets.

• The triple (Ω, Σ, µ), where µ, is a positive measure defined on Σ, is a measure space. A probability space is a measure space of total measure 1, i.e. µ(Σ) = 1, and in this case most of the time we shall write (Ω, Σ, P).

• A Σ-measurable real-valued function f with domain Ω is called a random variable. More generally a random element ϕ with values in a measurable space (X, B) is a measurable mapping of (Ω, Σ, P) into (X, B). For X = RN or ZN, B being the σ-algebra of Borel sets, we shall speak of random vectors.

The year 1885 has a special significance in the history of approximation theory. It was then that Weierstrass published his famous result which says that a continuous function on a closed bounded interval of the real line can be uniformly approximated by polynomials. The same year saw the birth of holomorphic approximation in the celebrated paper of Runge [Run]. Given an open set Ω in the complex plane C, which compact subsets K have the property that any holomorphic function defined on a neighbourhood of K can be uniformly approximated on K by functions holomorphic on Ω? Runge's Theorem supplies the answer: precisely the sets K such that Ω\K has no components which are relatively compact in Ω. Since Runge's original work holomorphic approximation has developed into a significant research area. We mention particularly the contributions of Carleman [CarT], Alice Roth [Rot1], [Rot3], Mergelyan [Mer], Arakelyan [Ara1] and Nersesyan [Ner]. A helpful account of these and other results can be found in the book by Gaier [Gai]. The purpose of these notes is to give a corresponding account of the theory of harmonic approximation in Euclidean space Rn (n ≥ 2).

The starting point in the history of harmonic approximation is not as easy to identify. In the case of approximation in higher dimensions, the paper of Walsh [Wal] in 1929 seems a reasonable choice, but for approximation in the plane mention must also be made of work of Lebesgue [Leb] in 1907.

Introduction

The following is Arakelyan's generalization of Mergelyan's Theorem (see §1.1) to non-compact sets. It can be found in [Ara1] or [Ara2].

Arakelyan's Theorem (1968).Let Ω be an open set in C and E be a relatively closed subset of Ω. The following are equivalent:

1. (a) for each f in C(E) ∩ Hol(E°) and each positive number ∈, there exists g in Hol(Ω) such that |g – f| < ∈ on E;

2. (b) Ω*\E is connected and locally connected.

The above local connectedness condition will be discussed in §3.2. Its first appearance (at least, in an equivalent form) in the context of holomorphic approximation occurs in early work of Alice Roth which is not as well known as it should be. It is remarkable that, as early as 1938, Roth [Rot1] had shown that (when Ω = C) condition (b) above is sufficient for uniform approximation of functions in Hol(E) by entire holomorphic functions. (See [Rot2] for the generalization to other choices of Ω.) Of course, Arakelyan's Theorem is an improvement of Roth's result.

This chapter presents corresponding results for uniform approximation by harmonic functions on relatively closed sets. In fact, we will obtain generalizations of Theorems 1.3, 1.7, 1.10, 1.15, and Corollary 1.16. Further, it will be shown that, whenever uniform approximation is possible, something rather better is also true (at least, in most cases). The main results are Theorems 3.15, 3.17 and 3.19.

Introduction

If Ω is an open set in C or Rn(n ≥ 2), then we will use Ω* to denote the Alexandroff, or one-point, compactification of Ω, and will use A to denote the ideal point. Thus Ω* = Ω ∪ {A}, and a set A is open in Ω* if either A is an open subset of Ω or A = Ω*\K, where K is a compact subset of Ω. In the special case where Ω is C or Rn we continue to write ∞ for A.

If A is a subset of C, we denote by Hol(A) the collection of all functions which are holomorphic on an open set containing A. Historically the following result (essentially in [Run]; cf. [Con, pp.198, 201]) can be regarded as the starting point of the theory of holomorphic approximation.

Runge's Theorem (1885).Let Ω be an open subset ofCand K be a compact subset of Ω. The following are equivalent:

1. (a) for each f in Hol(K) and each positive number ∈, there exists g in Hol(Ω) such that |g – f| < ∈ on K;

2. (b) Ω*\K is connected.

Condition (b) above is equivalent to asserting that no component of Ω\K is relatively compact in Ω. Also, when Ω = C, this condition is clearly equivalent to saying that C\K is connected.

We record below one further important development in the theory of holomorphic approximation, which deals with approximation of a much larger class of functions on a given compact set K.

Introduction

Many of the results in the preceding chapters have superharmonic analogues, some of which we will consider in this chapter. Thus, for example, we will examine which pairs (Ω, E) have the property that every u in S(E) can be uniformly approximated on E by functions in S(Ω). However, in the case of superharmonic functions, it may be possible not only to approximate, but even to extend, functions in S(E).

For example, suppose that K is a compact subset of an open set Ω such that Ω*\K is connected. Then, as we saw in Theorem 1.7, for every u in H(K) and every positive number ∈, there exists υ in H(Ω) such that |υ – u| < ∈ on K. The corresponding fact for superharmonic functions (a special case of Theorem 6.1 below) is that, for every u in S(K) there exists v in S(Ω) such that v = u on K.

Strong Extension

We begin the chapter with some such extension results. Later we will deal with Runge and Arakelyan approximation. Throughout this chapter Ω denotes an open set in Rn and E is a relatively closed subset of Ω. By a continuous superharmonic function we mean one which is both finite-valued and continuous.

We call (Ω, E) an extension pair for superharmonic functions (resp. for continuous superharmonic functions) if, for each function (resp. each continuous function) u in S(E) there exists υ in S(Ω) (resp. in C(Ω)∩S(Ω)) such that υ = u on E.

Introduction

This book differs essentially from the existing monographs on countable Markov chains. It intends to be, on the one hand, much more constructive than books similar to, for example Chung's [Chu67] and, on the other hand, much less constructive than some elementary monographs on queueing theory, where the emphasis is mainly put on the derivation of explicit expressions. The method of generating functions, which is to be sure the most constructive approach, is not included, since the dimension of the problems it can solve is small (in general ≤ 2). Our book could equally be called Constructive use of Lyapounov functions method. Here the term constructive is taken in the sense close to the one widely accepted in constructive mathematical physics. One can say that the objects considered have a sufficiently rich structure to be concrete, although the results may not always be explicit enough, as commonly understood. Semantically, it is permissible to say that our methods are more qualitative constructive than quantitative constructive.

The main goal of the book is to provide methods allowing a complete classification (necessary and sufficient conditions) or, in other words, allowing us to say when a Markov chain is ergodic, null recurrent or transient. Moreover, it turns out that, without doing much additional work, it is possible to study the stability (continuity or even analyticity) with respect to parameters, the rate of convergence to equilibrium, …, etc. by using the same Lyapounov functions.