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
Previous Page Next Page