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# Fix 0 t t2 <...<t s < t and also fix functions

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Fix 0 t t2 <...<t s < t and also fix functions book

# Fix 0 t t2 <...<t s < t and also fix functions

DOI link for Fix 0 t t2 <...<t s < t and also fix functions

Fix 0 t t2 <...<t s < t and also fix functions book

## 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.