On Two Results in the Potential Theory of Excessive Measures
Part of the Progress in Probability and Statistics book series (PRPR, volume 13)
Let (Pt) be the semigroup of a right Markov process and let m be an excessive measure for (Pt) (i.e., m is σ-finite and mPt m for t > 0). As is well known, m can be uniquely decomposed as m = mi+mp where mi is invariant (miPt= mi for t > 0), and mp is purely excessive (mpPt(f) ↓ 0 as t ↑ ∞ for f ≥ 0 with m(f) < ∞). The component mp can be decomposed further:
where (v t: t > 0) is a family of σ-finite measures satisfying v tPs = v t+s for s,t > 0. The decomposition (1) seems to be well known (cf. ; see also  for a related result). A probabilistic proof of (1) is given in  by means of the stationary process associated with (Pt) and m. In , Getoor and Glover use (1) as an important step in their construction of the aforementioned stationary process. Actually, Getoor and Glover consider the more general (and more difficult) time inhomogeneous case, but even in the time homogeneous case their proof of (1) is involved.
KeywordsMarkov Process Unique Measure Semi Group Excessive Function Auxiliary Hypothesis
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