Excessive kernels and Revuz measures

Abstract

We consider a proper submarkovian resolvent of kernels on a Lusin measurable space and a given excessive measure ξ . With every quasi bounded excessive function we associate an excessive kernel and the corresponding Revuz measure. Every finite measure charging no ξ -polar set is such a Revuz measure, provided the hypothesis (B) of Hunt holds. Under a weak duality hypothesis, we prove the Revuz formula and characterize the quasi boundedness and the regularity in terms of Revuz measures. We improve results of Azéma [2] and Getoor and Sharpe [20] for the natural additive functionals of a Borel right process.

On the Strongly Supermedian Functions and Kernels

“…Let ρ • U be the potential component of ξ . There exists a ξ -semipolar set A ∈ B such that every ξ -semipolar B-measurable subset of E\A is ρ-negligible (see, e.g., Corollary 2.4 in [3]). By [2] there exists a finite measure λ carried by A such that a subset of A is ξ -polar and ρ-negligible if and only if it is λ-negligible (λ is the so-called Dellacherie measure of A).…”

On the Tightness of Capacities Associated With Sub-Markovian Resolvents

“…See [4] and [6] for more details on regular excessive functions. It is known (see, for example, [8] Remark.…”

Sub-Markovian resolvents under weak duality hypothesis

“…which is the potential of a measure µ then s will be m-quasi bounded if and only if µ charges no m-copolar set and s will be m-regular (cf. [9], or equivalently s · m is regular in Exc m ( U)) if and only if µ charges no m-cosemipolar set i.e. every m-semipolar set.…”

“…the discontinuities of A are disjoint from those of the process X). For further connections between the natural excessive kernels and the additive functionals see the comments in [9], page 275.…”

“…Suppose that U is the resolvent associated with a Borel right process X with state space E. Then it is known (see [9]) that a U-excessive function s which is finite m-a.e. will be m-quasi bounded if and only if for every increasing sequence (T n ) n of stopping times, T n ζ , we have E x (s • X T n ) 0 m-a.e.…”