With this fact in mind, we let
C0(D\M) = {ue C(D)\ u = 0 on M}.
The space Co(D\M) is a closed subspace of C{D)\ hence it is a Banach space.
A strongly continuous semigroup {Tt}to on the space CQ(D\M) is called a
Feller semigroup on D\M if it is non-negative and contractive on Co(D\M):
f G C0(D\M)y 0 / 1 on D\M = * 0 Ttf 1 on D\M.
We define a linear operator % from Co(D\M) into itself as follows:
(a) The domain of definition D(2t) of 5t is the set
(0.4) D(») = {«G C
( 5 \ M ) ; Wu C0(D\M), Lu = 0} .
(6) 2U* = Wu, w G D(2l).
The next theorem is a generalization of Theorem 1 to the non-transversal case:
T h e o r e m 2. Assume that the following conditions (A) and (H) are satisfied:
(A) fi(x') + S(x') - T(Z') 0 on 3D.
(H) There exists a transversal VentceF boundary condition Lv of second order
such that
Lu(x') = n(x')Lvu{x') + y(x')u(x') - 8{x')Wu{x'), x' dD.
Then the operator 21 defined by formula (0.4) generates a Feller semigroup
on D\M.
If Tt is a Feller semigroup on D\M, then there exists a unique Markov transition
function pt on D\M such that
Ttf(x)= [ pt{x,dy)f{y\ feC0(D\M),
and further pt is the transition function of some strong Markov process. On the
other hand, the intuitive meaning of conditions (A) and (H) is that the absorption
phenomenon occurs at each point of the set M = {xf G dD]p(x') S(x')
0}. Therefore, Theorem 2 asserts that there exists a Feller semigroup on D\M
corresponding to such a diffusion phenomenon that a Markovian particle moves
both by jumps and continuously in the state space D\M until it "dies" at which
time it reaches the set M.
We remark that Taira [13] has proved Theorem 2 under the condition that Lv
d/du and 6 = 0 on dD, by using the Lp theory of pseudo-differential operators (cf.
[13, Theorem 4]).
Finally we consider the case when all the operators 5, T and R are pseudo-
differential operators of order less than one. Then one can take o~(x, y) = 1 on
D x D, and write the operator W in the following form:
(0.1') Wu(x) = Pu{x) + Sru(x)
+ ( /
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