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).