FELLER SEMIGROUPS WITH BOUNDARY CONDITIONS 7

where:

4') The integral kernel s(x) y) is the distribution kernel of a properly supported,

pseudo-differential operator S G L\~^K(R ), /c 0, which has the transmission

property with respect to the boundary dD, and s(x, y) 0 off the diagonal

{(x,x)\xeRN} i n R ^ x R * .

6') Wl(x) = c(x) 0 in D.

Similarly, the boundary condition L can be written in the following form:

(0.2') Lu{x') = Qu{x') + fi(x')^(x') - 6(x')Wu(x') + Tu(x')

5 5

( £ «

W

V)-£h^) + £W)^(*') + 7(*')«(*'))

+ Mz')^(z') - W « M +

(VM*')

+ E rt*')£:(*')

+ / r{x\ y')[u(y') - u{x')]dy' + / t{x\ y)[u(y) - u(x')]dy) ,

JdD JD J

where:

6') The integral kernel r(z', y') is the distribution kernel of a pseudo-differential

operator R € L\~0Kl(dD), KX 0, and r(x'\y') 0 off the diagonal {(x',x')\x' G

9£} in 8D x 9Z.

7') The integral kernel t{x,y) is the distribution kernel of a properly supported,

pseudo-differential operator T € L\~0K2(RN), K2 0, which has the transmis-

sion property with respect to the boundary dD, and t(x, y) 0 off the diagonal

{(x,x)]x eRN} mRN xRN.

9') n ( z ' ) = 77(2') 0 on 3D.

Then Theorems 1 and 2 may be simplified as follows:

T h e o r e m 3. Assume that the operator W and the boundary condition L are of

the forms (0.1*) and (0.2*), respectively. If the boundary condition L is transversal

on the boundary dD, then the operator 51 defined by formula (0.3) generates a

Feller semigroup {Tt}to on D.

T h e o r e m 4. Assume that the operator W and the boundary condition L are of

the forms (0.1') and (0.21), respectively. If conditions (A) and (H) are satisfied,

then the operator 21 defined by formula (0.4) generates a Feller semigroup {Tt}to

on D\M.

Theorems 1, 2, 3 and 4 solve from the viewpoint of functional analysis the

problem of construction of Feller semigroups with VentceP boundary conditions for

elliptic Waldenfels operators.

The rest of this paper is organized as follows.

In Chapter I, we present a brief description of the basic definitions and results

about a class of semigroups (Feller semigroups) associated with Markov processes,

which forms a functional analytic background for the proof of Theorems 1 and 2.

Chapter II provides a review of the basic concepts and results of the theory

of pseudo-differential operators - a modern theory of potentials - which will be