ABSTRACT
Notice that it B is any set which does not contain [x], then 1 ( B ) I does not contain x, and
P* {Y0 B} lim0 P {Yo B} = lim0 P {X0
Therefore P* {Y0 = [x]} = l. By the definition of D* we know that ( *F) (
I) = 0, which implies that for any 0 s t,
( *F) (Ys) ds =
( *F) (Ys)ds.Now,
E*[{F (Yt) F(Ys) t
E [{ F(Yt) F(Ys) tS ( * F)(Yu) du
( * F)(Yu) du }
k=l h*k (Ytk)] .. (23)
Notice that
S T u/ 2 F (Xu) du
(Xu) (M(F )) (Xu) ) du
S T ( (M(F ))(Xu) ( * F)(Yu)
)du
Keeping this expression in mind, we use Proposition 3 about F approximating F o , the statement of the martingale problem (21), Lemma 1 with f(x, t) =
F (x) concerning the first-level averaging, and Eq. (20) to see that (23) implies (22), and this completes the proof. •
The very last step in proving Theorem 1 is to verify that the P -laws have only one limit point. Using the Hille-Yosida theorem, we can show that
uniqueness holds for the solution of the martingale problem for ( *, [x]),
and the P -laws of {Yt,t 0} converge to the law of a -valued Markov process.
