Homogeneous random measures and a weak order for the excessive measures of a Markov process

Abstract

ABSTRACT. Let X = (Xt,Px) be a right Markov process and let m be an excessive measure for X. Associated with the pair (X, ra) is a stationary strong Markov process (Yt, Qm) with random times of birth and death, with the same transition function as X, and with m as one dimensional distribution.We use (Yt,Qm) to study the cone of excessive measures for X. A "weak order" is defined on this cone: an excessive measure £ is weakly dominated by m if and only if there is a suitable homogeneous random measure re such that (Yt,Qç) is obtained by "birthing" (Yt,Qm), birth in [t,t + dt] occurring at rate re(di). Random measures such as re are studied through the use of Palm measures. We also develop aspects of the "general theory of processes" over (Yt,Qm), including the moderate Markov property of {Yt,Qm) when the arrow of time is reversed. Applications to balayage and capacity are suggested. Introduction.Let (Ps : s > 0) be a Borel right semigroup on a Lusin state space (E, £) and let (Xt, Px) denote the associated right continuous strong Markov process. Recall that a measure m on (E, £) is excessive for (Ps) if m is tr-finite and if mPs < m for all s > 0. Since (Ps) is a right semigroup, one then has mPs ] m as s J. 0, as is well known. Let Exc denote the convex cone of excessive measures for (Pa)-Given m E Exc, according to a theorem of Kuznetsov [31], one can construct a stationary process (Yt : t E R) having random birth and death times and a tr-finite governing measure Qm, such that for ii < t2 < • ■ ■ < tn (i» E R)

Potential Theory of Moderate Markov Dual Processes

“…The keys to these extensions are the strong Markov property (Proposition 2.1) and the section theorem (Proposition 2.2). Using them in place of (3.10) and (3.16)(b) in [9], the results we require are proved with only minor modifications of the arguments given in [9]. In general, we shall use the corresponding result with Y * and * without special mention.…”

“…(In [9], the truncated shift operator was denoted τ t ; here we follow [13] in using θ t .) Given m ∈ Exc, the Kuznetsov measure Q m is the unique σ -finite measure on…”

“…Many of the definitions and results in [9] involve Y and . We shall need the extensions of these results in which Y and are replaced by Y * and * .…”

“…In making this precise we follow [9]. However our notation differs slightly from that used there; in particular we use Y * in place of Y, as explained in Section 2.…”

“…In [9],θ t is denoted by τ t . It is proved in the Appendix of [9] that there exists a Borel measurable family { P x , x ∈ E } of probability measures on ( , F • ) under which { X t , t > 0} has the moderate Markov property (MMP);…”