FELLER SEMIGROUPS WITH BOUNDARY CONDITIONS 9
This is a generalization of the classical Fredholm integral equation. It is known
(cf. [2], [5], [10]) that the operator LHa is a pseudo-differential operator of second
order on the boundary dD.
By using the Holder space theory of pseudo-differential operators, we can show
that if the boundary condition L is transversal on the boundary dD, then the
operator LHa is bijective in the framework of Holder spaces. The crucial point
in the proof is that we consider the term 8{WU\QD) of viscosity in the boundary
condition
Lu = LQU 6(Wu\dD)
as a term of "perturbation" of the boundary condition LQU.
Therefore, we find that a unique solution u of problem (*) can be expressed as
follows:
u = G°J - Ha (LH-1 LG°af).
This formula allows us to verify all the conditions of the generation theorems of
Feller semigroups discussed in Chapter I. Intuitively, this formula tells us that if
the boundary condition L is transversal on the boundary dD, then one can "piece
together" a Markov process on the boundary dD with VK-diffusion in the interior
D to construct a Markov process on the closure D = D U dD.
In Chapter IV, we prove Theorem 2. We explain the idea of the proof.
First we remark that if condition (H) is satisfied, then the boundary condition
L can be written in the following form:
Lu = \iLvu + 7 {u\9D) - 8 (Wu\dD),
where the boundary condition Lv is transversal on dD. Hence, applying Theorem 1
to the boundary condition Lv, we can solve uniquely the following boundary value
problem:
(a - W)v = / in D,
LyV^O on dD.
We let
The operator Gva is the Green operator for the boundary condition Lv. Then it
follows that a function u is a solution of the problem
( (a-W)u = f in D,
[ Lu = \iLvu + 7(u|di) - 8{Wu\dD) = 0 on dD
if and only if the function w = u v is a solution of the problem
J (a - W)w = 0 in D,
\ Lw = -Lv = (a8 - 7)(v|ajp) - 8(f\dD) on dD.
Thus, as in the proof of Theorem 1, one can reduce the study of problem (**) to
that of the equation:
LH^ = (aS -
7
) (GUlao) - S(f\9D).
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