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
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