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On Two Results in the Potential Theory of Excessive Measures

  • P. J. Fitzsimmons
Part of the Progress in Probability and Statistics book series (PRPR, volume 13)

Abstract

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:
$${{\text{m}}_{\text{p}}}=\int{_{0}^{\infty }}{{v}_{\text{t}}}\text{dt,}$$
(1)
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. [2]; see also [7] for a related result). A probabilistic proof of (1) is given in [3] by means of the stationary process associated with (Pt) and m. In [6], 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.

Keywords

Markov Process Unique Measure Semi Group Excessive Function Auxiliary Hypothesis 
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Copyright information

© Birkhäuser Boston 1987

Authors and Affiliations

  • P. J. Fitzsimmons
    • 1
  1. 1.Department of Mathematical SciencesThe University of AkronAkronUSA

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