FELLER SEMIGROUPS WITH BOUNDARY CONDITIONS 13
give a necessary and sufficient condition on pt(x,E) in order that its associated
operators {Tt}to be strongly continuous in t on the space Co(K):
I i m | | T »
+
. / - T « / | | = 0, f€C0(K).
A Markov transition function pt on K is said to be uniformly stochastically
continuous on K if the following condition is satisfied: For each e 0 and each
compact E C K, we have
limsup[l -pt(x,Ue(x))] = 0,
where U£(x) = {y K\ p(x, y) e} is an ^-neighborhood of x.
Then we have the following (cf. [12, Theorem 9.2.3]):
Theorem 1.1. Let pt be a Co-transition function on K. Then the associated
operators {Tt}to, defined by
(12) Ttf(x) = / M M y ) / ( y ) , / e c0(K),
JK
is strongly continuous in t on CQ(K) if and only if pt is uniformly stochastically
continuous on K and satisfies the following condition (L):
(L) For each s 0 and each compact E C K, we have
lim sup pt(xyE) = 0.
X^d0t8
A family {Tt}to of bounded linear operators acting on CQ(K) is called a Feller
semigroup on K if it satisfies the following three conditions:
(i) T*+a = Tt Ts, t,s 0 ; T0 = / = the identity,
(ii) The family {Tt} is strongly continuous in t for t 0:
l i m | | T
( +
. / - T
(
/ | | = 0, f€C0(K).
(iii) The family {Tt} is non-negative and contractive on Co(K):
f C0(K) , 0 / l o n / { = 0 Ttf 1 on K.
The next theorem gives a characterization of Feller semigroups in terms of
Markov transition functions (cf. [12, Theorem 9.2.6]):
Theorem 1.2. Ifpt is a uniformly stochastically continuous Co-transition function
on K and satisfies condition (L), then its associated operators {Tt }to form a Feller
semigroup on K.
Conversely, if {Tt}to is a Feller semigroup on K, then there exists a uniformly
stochastically continuous Co-transition pt on K, satisfying condition (L), such that
formula (1.2) holds.
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