By Richard F. Bass, Krzysztof Burdzy (auth.), E. Çinlar, K. L. Chung, R. K. Getoor, P. J. Fitzsimmons, R. J. Williams (eds.)
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Additional resources for Seminar on Stochastic Processes, 1989
Wahrscheinlichkeitstheorie verw. , 67 (1984), 1-62. [GStl) R. K. Getoor and J. Steffens. The energy functional, balayage, and capacity. Ann. Inst. Henri Poincare, 23 (1987), 321-357. [GSt2] R. K. Getoor and J. Steffens. More about capacity and excessive measures. Sem. Stoch. , 1987, 135-157. Birkhauser, Boston, 1988. BLUMENTHAL-GETOOR-McKEAN THEOREM 57 [Gll) J. Glover. Representing last exit potentials as potentials of measures. Z. Wahrscheinlichkeitstheorie verw. , 61 (1982), 17-30. [Gl2) J. Glover.
J. FITZSIMMONS. R. K. GETOOR and M. J. ce for the processes (X, T) and tained by killing X (resp. t) at T (resp. it follows that H E &<. The Borel q T). Y, T) ob- Since -algebra 11. of H is the trace of £ on H and so 11. C &e . 2) (extended to BE &e) that (X, T) and (X, T) have the same hitting distributions for sets in 11.. 2]. Y, T) SeX, T) c £e(X). Y). Similarly If B E £e we write TB = E\B. It follows C &e(x). p = P' [exp (-T»). Y, T). = T(B) = P' [exp( -T)] E &e(x). 2), PX(TF < 00) =1 =1 for all x E E (equivalently for all a' E E).
J. 4) Proposition. Let X and (E,E). Suppose that PK(X,,) Then 5 = S. H X and X be right processes with the same state space = FK(X,,) for each x E E and compact K C E. are transient, then the equality 5 = S implies that PB(x,·)=FB(X,,) for all xEE and BEE. Proof. 33]. The converse for transient processes is proved on page 553 of [Fl]. (8t w) identically in t,s,and w. 19}, for example) this entails no loss of generality. We are now prepared to state BGM for transient processes. 5) Theorem. ) for each x E E and BEE.