On Two Results in the Potential Theory of Excessive Measures

Abstract

Let (P_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_iP_t= m_i for t > 0), and m_p is purely excessive (m_pP_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)

where (v _t: t > 0) is a family of σ-finite measures satisfying v _tP_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.