CHAPTER I

THEOR Y OF FELLER SEMIGROUP S

This chapter provides a brief description of the basic definitions and results about

a class of semigroups (Feller semigroups) associated with Markov processes, which

forms a functional analytic background for the proof of Theorems 1 and 2. The

results discussed here are adapted from Chapter 9 of Taira [12].

1.1 Markov Transition Functions and Feller Semigroups

Let (K, p) be a locally compact, separable metric space and B the cr-algebra of all

Borel sets in K.

A function Pt{x, E), defined for alH 0, x G K and E G #, is called a (tem-

porally homogeneous) Markov iransiiion function on K if it satisfies the following

four conditions:

(a) pt(x, •) is a non-negative measure on B and pt{x, K) 1 for each t 0 and

each x G K.

(6) pf(-, E) is a Borel measurable function for each t 0 and each E G B.

(c) po(x, {x}) = 1 for each x G K.

(d) (The Chapman-Kolmogorov equation) For any t, s 0, x G K and any

E G B, we have

(1-1) pt+s(x,E)= / pt(x,dy)p9(y,E).

JK

Here is an intuitive way of thinking about the above definition of a Markov transi-

tion function. The value pt{x, E) expresses the transition probability that a physical

particle starting at position x will be found in the set E at time t. Equation (1.1)

expresses the idea that a transition from the position x to the set E in time t + s

is composed of a transition from x to some position y in time t, followed by a

transition from y to the set E in the remaining time s; the latter transition has

probability ps(y,E) which depends only on y. Thus a particle "starts afresh"; this

property is called the Markov property.

We add a point d to K as the point at infinity if K is not compact, and as an

isolated point if K is compact; so the space KQ = K U {d} is compact.

Let C(K) be the space of real-valued, bounded continuous functions on K. The

space C(K) is a Banach space with the supremum norm

11/1 1 = sup |/(x)|.

We say that a function / G C(K) converges to zero &s x -+ d if, for each e 0,

there exists a compact subset E of K such that

|/(x) | e, x € K\E,

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