12

KAZUAKI TAIRA

and write lim*-^ /(# ) = 0. We let

Co(K) = { / € C(K); h m / ( * ) = o j .

The space CQ(K) is a closed subspace of C(K); hence it is a Banach space. Note

that Co(K) may be identified with C(K) if K is compact.

Now we introduce a useful convention:

Any real-valued function f on K is extended to the space KQ = K U {d}

by setting f(d) = 0.

From this point of view, the space CQ(K) is identified with the subspace oiC(Kd)

which consists of all functions / satisfying f(d) = 0, that is,

Co(K) = {/ € C(Kd); f(d) = 0} .

Further we can extend a Markov transition function pt on K to a Markov transition

function p't on Kg as follows:

f p't(x, E) = pt{x, E), xeK,E€B;

I p't(x,{d}) = l-Pt(x,K),x€K;

[p't(d,K) = 0,P't(d,{d}) = l.

Intuitively, this means that a Markovian particle moves in the space K until it

"dies" at which time it reaches the point d] hence the point d is called the terminal

•point.

Now we introduce some conditions on the measures pt(x, •) related to continuity

in x G K, for fixed t 0.

A Markov transition function pt is called a Feller function if the function

Ttf(x)= [ Pt{x,dy)f{y)

JK

is a continuous function of x € K whenever / is in C(K), or equivalent ly, if we

have

fec(K) =» Ttfec(K).

In other words, the Feller property is equivalent to saying that the measures pt{x, •)

depend continuously on x € K in the usual weak topology, for every fixed t 0.

We say that pt is a Co-function if the space C${K) is an invariant subspace of

C(K) for the operators Tt:

f

e

c0(K) =» Ttf e c0(K).

The Feller or Co-property deals with continuity of a Markov transition function

Pt(xjE) in x) and does not, by itself, have no concern with continuity in t. We