Use Lemma 7.33 and give an alternative derivation of the result of Lemma 8.10.

Data From Lemma 7.33

Data From Leema 8.10

Transcribed Image Text:

7.33 Lemma. Let (X,,F)o be a Feller process with a right continuous filtration, trans- ition semigroup (P) and generator (A, D(A)). If xo is not absorbing, then a) there exists some u D(A) such that Au(x) + 0; b) there is an open neighbourhood U = U(xo) of xo such that for V := Rd \ U the first hitting time Ty := inf(t>0: X, V) satisfies Exo Ty 0. Thus, (7.11b) shows that Pu(x) = u(xo). Since (A) is dense in Co. (Rd) we get Eu(X) = Pu(x) = u(xo) for all u (R). Using the sequence from (7.7), we