I N T R O D U C T I O N A N D R E S U L T S

Let D be a bounded domain of Euclidean space R ^ , with C°° boundary dD\ its

closure D = D U dD is an TV-dimensional, compact C°° manifold with boundary.

Let C(D) be the space of real-valued, continuous functions on D. We equip the

space C(D) with the topology of uniform convergence on the whole D\ hence it is

a Banach space with the maximum norm

||/|| = max |/(x)|.

A strongly continuous semigroup {Tt}to on the space C(D) is called a Feller

semigroup on D if it is non-negative and contractive on C(D):

f € C ( 5 ) , 0 / 1 on D = 0 Ttf 1 on D.

It is known (cf. [12]) that if Tt is a Feller semigroup on Z), then there exists a

unique Markov transition function pt on D such that

Ttf(x)= f pt{*,dy)f(v), /ec(5).

JD

It can be shown that the function pt is the transition function of some strong

Markov process; hence the value pt(x,E) expresses the transition probability that

a Markovian particle starting at position x will be found in the set E at time t.

Furthermore, it is known (cf. [1], [9], [12], [17]) that the infinitesimal generator

21 of a Feller semigroup {Tt}to is described analytically by a Waldenfels operator

W and a VentceP boundary condition L, which we formulate precisely.

Let W be a second-order elliptic integro-differential operator with real coefficients

such that

(0.1) Wu(x) = Pu(x) + Sru(x)

N ~n N

+

(/B.(..,)[«)-^„)(*)+i:(»-.,)i(.))]*),

where:

1)

a,J

€ C°°(R ),

a,J

=

a;*

and there exists a constant ao 0 such that

N

Received by the editor November 8, 1990.

1