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.