Seminar on Stochastic Processes, 1986 pp 21-29 | Cite as

# On Two Results in the Potential Theory of Excessive Measures

Chapter

## Abstract

Let (P where (

_{t}) be the semigroup of a right Markov process and let m be an excessive measure for (P_{t}) (i.e., m is σ-finite and mP_{t}m for t > 0). As is well known, m can be uniquely decomposed as m = m_{i}+m_{p}where m_{i}is*invariant*(m_{i}P_{t}= m_{i}for t > 0), and m_{p}is*purely excessive*(m_{p}P_{t}(f) ↓ 0 as t ↑ ∞ for f ≥ 0 with m(f) < ∞). The component m_{p}can be decomposed further:$${{\text{m}}_{\text{p}}}=\int{_{0}^{\infty }}{{v}_{\text{t}}}\text{dt,}$$

(1)

*v*_{t}: t > 0) is a family of σ-finite measures satisfying*v*_{t}P_{s}=*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 (P_{t}) 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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## References

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*Pnobabit tés et Potentíe2*, Vol. I, II, IV. Hermann, Paris, 1975, 1980, 1986.Google Scholar - 2.E.B. DYNKIN. Minimal excessive measures and functions. Tnan. Am. Math.
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## Copyright information

© Birkhäuser Boston 1987