Homogeneous Random Measures and Strongly Supermedian Kernels of a Markov Process

The potential kernel of a positive left additive functional (of a strong Markov process $X$) maps positive functions to strongly supermedian functions and satisfies a variant of the classical domination principle of potential theory. Such a kernel $V$ is called a regular strongly supermedian kernel in recent work of L. Beznea and N. Boboc. We establish the converse: Every regular strongly supermedian kernel $V$ is the potential kernel of a random measure homogeneous on $[0,\infty[$. Under additional finiteness conditions such random measures give rise to left additive functionals. We investigate such random measures, their potential kernels, and their associated characteristic measures. Given a left additive functional $A$ (not necessarily continuous), we give an explicit construction of a simple Markov process $Z$ whose resolvent has initial kernel equal to the potential kernel $U_{\!A}$. The theory we develop is the probabilistic counterpart of the work of Beznea and Boboc. Our main tool is the Kuznetsov process associated with $X$ and a given excessive measure $m$.


Introduction.
In a recent series of papers [BB00,BB01a,BB01b,BB02], L. Beznea and N. Boboc have singled out an important class of kernels for which they have developed a rich potential theory. These kernels (called regular strongly supermedian kernels) are those that map positive Borel functions to strongly supermedian functions of a strong Markov process X and that satisfy a form of the domination principle. If κ is a random measure of X, homogeneous on [0, ∞[ as in [Sh88], then the potential kernel U κ is a strongly supermedian kernel. Using entirely potential theoretic arguments, Beznea and Boboc were able to develop a theory of characteristic (Revuz) measures, uniqueness theorems, etc., for regular strongly supermedian kernels that parallels a body of results on homogeneous random measures (under various sets of hypotheses) due to J. Azéma [A73], E.B. Dynkin [Dy65,Dy75], and others. In fact, the theory developed by Beznea and Boboc goes far beyond that previously developed for homogeneous random measures. The question of the precise relationship between regular strongly supermedian kernels and homogeneous random measures poses itself. One of our goals in this paper is to show that the class of regular strongly supermedian kernels is coextensive with the class of potential kernels U κ as κ varies over the class of (optional, co-predictable) homogeneous random measures. Our examination of these matters will be from a probabilistic point of view. Before describing our work in more detail we shall attempt to provide some historical background.
Let X = (X t , P x ) be a strong Markov process with transition semigroup (P t ) and state space E. To keep things simple, in this introduction we assume that X is transient and has infinite lifetime. details can be found in [Sh88].
The story simplifies greatly in case X is in weak duality with a second strong Markov processX, with respect to an excessive measure m. In this case there is a Borel function a ≥ 0 with {a > 0} m-semipolar such that A d is P m -indistinguishable from t → 0<s≤t a(X s− ), the existence of the left limit X s− being guaranteed P m -a.s. by weak duality. See [GS84; (16.8)(i)]. Associated with A are its left potential kernel defined outside an m-polar set, and its Revuz measure The kernel U − A , which maps positive Borel functions to excessive functions, determines A up to P m -evanescence, and it is clear from the Revuz formula that µ − A determines U − A , modulo an m-polar set. See [Re70a,Re70b] in the context of standard processes in strong duality and [GS84] in the context of right processes in weak duality as above.
Can the duality hypothesis imposed above be relaxed? Given a right-continuous strong Markov process X (more precisely, a Borel right Markov process) and an excessive measure m, there is always a dual processX (essentially uniquely determined), but in general it is a moderate Markov process: the Markov property holds only at predictable times. If, in the general case, we must use the left-handed dual process, then time reversal dictates that we must trade in A for its right-handed counterpart. Thus, we are mainly concerned with additive functionals (and, more generally, homogeneous random measures) that can be expressed as where A c is a continuous additive functional and {a > 0} is m-semipolar. Observe that this additive functional is left-continuous and adapted. As we shall see, A is co-predictable, meaning (roughly) that it is predictable as a functional ofX. Such additive functionals were introduced and studied by J. Azéma in his pioneering work [A73], under the name d-fonctionelle.
The potential kernel U A associated with the additive functional defined in (1.4) is In contrast to U − A , if f ∈ pE then the function U A f is strongly supermedian and regular, but is excessive only when A d vanishes. Azéma showed, among other things, that A is uniquely determined by U A 1 provided this function is finite, and that any regular strongly supermedian function u of class (D) is equal to U A 1 for a unique A. It is a crucial observation of Beznea and Boboc, foreshadowed by a remark of Mokobodzki [Mo84;p. 463], that the analytic concept of regular strongly supermedian kernel corresponds to the probabilistic conditions (1.4) and (1.5). Beznea and Boboc develop their theory analytically; our approach to these matters is largely probabilistic.
Our setting will be a Borel right Markov process X coupled with a fixed excessive measure m. Our main tool will be the stationary Kuznetsov process Y = ((Y t ) t∈R , Q m ) associated with X and m. The theory of HRMs over Y has been developed in [Fi87,G90] and applied to the study of (continuous) additive functionals in [FG96,G99]. In section 2 we recall the basic definitions and notation concerning Y ; other facts about Y will be introduced as the need arises. In section 2 we also present a small but necessary refinement of the strong Markov property of Y proved in [Fi87], and in section 3 we record some basic facts about HRMs, drawn mainly from [Fi87]. Section 4 contains fundamentals on potential kernels and characteristic measures of optional HRMs that are either co-natural or co-predictable. We show, in particular, that the potential kernel of a suitably perfected optional co-predictable HRM satisfies the domination principle. We also prove a formula for the characteristic measure of such an HRM that provides a first link with the work of Beznea and Boboc. Section 5 (and the accompanying appendix) contains a probabilistic approach to some results in [BB01b]. Starting from a regular strongly supermedian kernel, assumed to be proper in a suitable sense, we construct the associated characteristic measure and HRM. The analog of (1.3) in this context appears in section 5 as well. We specialize, in section 6, to the situation of additive functionals.
Of particular interest are criteria, based on the characteristic measure µ κ or the potential kernel U κ of an optional co-predictable HRM κ, ensuring that A t := κ ([0, t[) defines a finite additive functional. The probabilistic approach used here extends that found in [FG96] in the context of continuous additive functionals. If a regular strongly supermedian kernel V satisfies a suitable "properness" condition (such conditions are discussed in section 6) then V is the initial kernel of a subMarkovian resolvent. Results of this type have a history going back to Hunt [Hu57], including work of Taylor [T72, T75] and Hirsch [Hi74], and culminating in [BB01a]. We give a probabilistic treatment of this topic in section 7, making use of the additive functional material from section 6, thereby obtaining an explicit expression for the resolvent associated with V . Finally, the appendix contains technical results on strongly supermedian functions as well as a proof of the main result of section 5.
We close this introduction with a few words on notation. We shall use B to denote the Borel subsets of the real line R. If (F, F, µ) is a measure space, then bF (resp. pF) denotes the class of bounded real-valued (resp. [0, ∞]-valued) F-measurable functions on F . For f ∈ pF we use µ(f ) to denote the integral F f dµ; similarly, if D ∈ F then µ(f ; D) denotes D f dµ. We write F * for the universal completion of F; that is, F * = ∩ ν F ν , where F ν is the ν-completion of F and the intersection runs over all finite measures on (F, F). If (E, E) is a second measurable space and K = K(x, dy) is a kernel from (F, F) to (E, E) (i.e., F x → K(x, A) is F-measurable for each A ∈ E and K(x, ·) is a measure on (E, E) for each x ∈ F ), then we write µK for the measure A → F µ(dx)K(x, A) and Kf for the function x → E K(x, dy)f (y).

Preliminaries.
Throughout this paper (P t : t ≥ 0) will denote a Borel right semigroup on a Lusin state space (E, E), and X = (X t , P x ) will denote a right-continuous strong Markov process realizing (P t ). We shall specify the realization shortly. Recall that a (positive) measure m on (E, E) is excessive provided mP t ≤ m for all t ≥ 0. Since (P t ) is a right semigroup, it follows that mP t ↑ m setwise as t ↓ 0. See [DM87;. Let Exc denote the cone of excessive measures. In general, we shall use the standard notation for Markov processes without special mention. See, for example, [BG68], [DM87], [Sh88], and [G90]. In particular, U q := on W : the simple shifts σ t , t ∈ R, and the truncated shifts θ t , t ∈ R, (In [Fi87], the truncated shift operator was denoted τ t ; here we follow [G90] in using θ t .) We refer the reader to [G90] for additional notation and terminology. Given m ∈ Exc, the Kuznetsov measure Q m is the unique σ-finite measure on G • not charging {[∆]} such that, for −∞ < t 1 < t 2 < · · · < t n < +∞, We let X = (X t , P x ) be the realization of (P t ) described on page 53 of [G90]. In particular, the sample space for X is X t is the restriction of Y t to Ω for t > 0, and X 0 is the restriction of Y 0+ . Moreover, Because of its crucial role in our development we recall the modified process Y * of [G90; (6.12)]. Let d be a totally bounded metric on E compatible with the topology of E, and let D be a countable uniformly dense subset of the d-uniformly continuous bounded real-valued functions on E. Given a strictly positive h ∈ bE with m(h) < ∞ define W (h) ⊂ W by the conditions: for all g ∈ D and all rationals q > 0; Evidently σ −1 t (W (h)) = W (h) for all t ∈ R, and W (h) ∈ G • α+ since E is a Lusin space. We now define The other important feature of Y * is the strong Markov property recorded in (2.5) below. For a proof of the following result see [G90; (6.15)]. The filtration (G m t ) t∈R is obtained by augmenting (G • t ) t∈R with the Q m null sets in the usual way.
We shall also require the following form of the section theorem. Define (2.6) Proposition. Let (H t ) t∈R and (K t ) t∈R be positive (G m t )-optional processes. If for all (G m t )-stopping times T , then H1 Λ * and K1 Λ * are Q m -indistinguishable.
See [FG91] for a proof of (2.6). Certain results from [Fi87] will be crucial for our development. We shall recall some definitions from [Fi87] and give precise references to the results we shall need.
In particular, Λ = {(t, w) : Y t (w) ∈ E}. Many of the definitions and results in [Fi87] involve Y and Λ. We shall need the extensions of the results in which Y and Λ are replaced by Y * and Λ * . The keys to these extensions are the strong Markov property (2.5) and the section theorem (2.6). Using them in place of (3.10) and (3.16)(b) in [Fi87], the results we require are proved with only minor modifications of the arguments given in [Fi87].
For example, using (2.6) the next result is proved exactly as (3.20) is proved in [Fi87].
The optional (resp. co-predictable) σ-algebra O m (resp.P m ) is defined on page 436 in [Fi87]. A process Z defined over (W, We adhere to the convention that a function defined on E takes the value 0 at ∆.
In general, we shall just use the corresponding result with Y * and Λ * without special mention.

Homogeneous Random Measures.
In this section we shall expand on some of the results in section 5 of [Fi87]. Our definition of random measure W × B (w, B) → κ(w, B) is exactly that of [Fi87; (5.1)].
A random measure κ is homogeneous (in abbreviation, an HRM) provided the measures B → κ(σ t w, B) and B → κ(w, B + t) coincide for Q m -a.e. w ∈ W , for each t ∈ R. This notion of HRM, which clearly depends on the choice of m, differs from Definition 8.19 in [G90]-what is defined there is essentially a perfect HRM, to be discussed later. We emphasize that for each w, κ(w, ·) is a measure on (R, B) that is carried by R∩[α(w), β(w)[. As in [Fi87], a random measure κ is σ-integrable over a class H ⊂ pM m : The class of such random measures is denoted σI(H). In keeping with our program of using Y * and Λ * systematically, we alter the definition [Fi87; (5.6)] of optional random measure by substituting Λ * for Λ there. We say that an HRM κ is by Λ * , then the dual optional projection κ o of κ is defined as in [Fi87]: κ o is the unique optional HRM carried by Λ * satisfying for all Z ∈ pM m , where o Z denotes the Q m optional projection of Z. The properties of κ → κ o elucidated in [Fi87] remain valid with the obvious modifications.
If κ is an optional HRM, its Palm measure P κ is the measure on (W, G • ) defined by Note that P κ is carried by {α = 0}. Moreover, [Fi87; (5.11)] states that if F ∈ p(B ⊗ G • ), then In particular, taking We say that κ is co-predictable provided κp = κ, up to Q m -indistinguishability.
Observe that B ∈ E is m-polar if and only if {Y ∈ B} := {(t, w) : Y t (w) ∈ B} is Q m -evanescent. As before, let m = η + ρU be the decomposition of m into harmonic and potential components. Then [FG91;(2.3)] implies that for B ∈ E, the set {Y * ∈ B} is Q m -evanescent if and only if B is both m-polar and ρ-null. It will be convenient to name this class of sets: Here is the basic existence theorem for HRMs, taken from [Fi87; (5.20)].
(3.5) Theorem. Let µ be a σ-finite measure on (E, E). Then P µ := E µ(dx)P x is the Palm measure of a (necessarily unique) optional co-predictable HRM if and only if µ charges no element of N (m).
(3.6) Remarks. (a) Of course, P µ is carried by Ω ⊂ {α = 0}, so it makes sense to think of P µ as a Palm measure. Actually, since X is a Borel right process, P µ is carried by The uniqueness assertion in (3.5) is to be understood as modulo Q m -indistinguishability.
We next define the characteristic measure µ κ (sometimes called the Revuz measure) of an optional HRM κ by where the second equality follows from (3.3), and ϕ is any positive Borel function on the real line with R ϕ(t) dt = 1. This definition differs from [Fi87; (5.21)] where Y is used in place of Y * . If κ is carried by Λ, then one may replace Y * by Y in (3.7). In view of the remarks made above (3.4), the measure µ κ charges no element of N (m). Also, if f ∈ pE The following result is drawn from [Fi87; §5]; we omit the proof.
(3.8) Proposition. (i) Let κ be an optional HRM. Then κ ∈ σI(P m ) if and only if P κ is σ-finite. If κ is carried by Λ * then P κ = P µ κ , and P κ is σ-finite if and only if µ κ is σ-finite.
Proof. With the exception of the last sentence, everything stated here follows immediately from Theorem (5.27) in [Fi87]. Using (2.7) and the fact that κ is carried by Λ * , and arguing as in the proof of [Fi87; (5.27)], one obtains a weak form of (3.12) with {j > 0} m-semipolar. To complete the proof of (3.11) we require the following lemma, which is of interest in its own right. We shall use (3.13) to complete the proof of (3.11), after which we shall prove (3.13). Thus, let (3.12) hold with {j > 0} m-semipolar. Then by (3.13) we can write {j > 0} = A ∪ B where A ∈ E ∩ N (m) and B ∈ E is semipolar. Define j := j1 B , so that j ∈ pE and {j > 0} = B is semipolar. Let κ := t∈R j (Y * t ) t + κ c . If D ∈ E and ϕ ≥ 0 with R ϕ(t) dt = 1, then But A ∈ N (m), hence {Y * ∈ A} is Q m -evanescent. Therefore µ κ = µ κ , and so (3.12) obtains with κ replacing κ. Since κ is carried by Λ * , κ and κ are Q m -indistinguishable, and we are done.
Proof of (3.13). According to a theorem of Dellacherie [De88; p. 70], an m-semipolar set B is of the form B 1 ∪ B 2 , with B 1 m-polar and B 2 semipolar. (In fact, this is the definition of "m-semipolar"; its equivalence with the P m -a.e. countability of {t : X t ∈ B} is part of the result referred to in the preceding sentence.) Moreover, using the fact that an m-polar set is contained in a Borel m-polar set it is easy to see that if B is Borel then both B 1 and B 2 may be chosen Borel. Suppose now that B ∈ E is m-polar. Let m = η + ρU be the Riesz decomposition of m into harmonic and potential parts. Then B is ρU -polar. Let D = D B := inf{t ≥ 0 : X t ∈ B} denote the début of B. Recall that for a σ-finite measure µ, the phrase "B is µ-polar" means that P µ [D < ∞] = 0. Thus, and so P ρ [D • θ t < ∞] = 0 for (Lebesgue) a.e. t > 0. It follows that {t ≥ 0 : X t ∈ B} ⊂ {0}, P ρ -a.s. Using Dellacherie's theorem again, B is ρ-semipolar, so B = B 1 ∪ B 2 with B 1 ρpolar and B 2 semipolar. Hence ρ(B 1 ) = 0 and B 1 ⊂ B, so B 1 is m-polar; that is, We find it convenient to make the following definition.
Thus (3.11) may be rephrased as follows: Each µ ∈ S # 0 (m) is the characteristic measure of a unique perfect HRM κ. It follows from (5.24) and (5.25) of [Fi87] that a perfect HRM is optional and co-predictable provided it is in σI When µ charges no m-polar set there is second HRM γ, carried by ]α, β[, with characteristic measure µ. The discussion of "co-natural" that follows is dual to that found in [GS84] concerning natural HRMs.
A co-natural HRM is carried by α, β and, if it lies in σI(O m ), then it is optional in view of [Fi87;(5.25)]. It has all of the properties listed in (3.11), except that (iii) must be replaced by (3.17) Theorem. Let µ be a σ-finite measure charging no m-polar set. Then there exists a unique co-natural HRM γ with characteristic measure µ.
Proof. If γ is co-natural then it is immediate from (3.11) that γ is of the stated form. Conversely, a γ of the stated form is clearly carried by α, β . Define κ : Then κ is carried by Λ and is a perfect HRM. Since γ = 1 α,∞ κ, γ is co-natural.

Homogeneous Random Measures and Potential Kernels.
In this section we fix a perfect HRM κ and we let µ = µ κ denote the associated characteristic measure. Recall the "jump function" j as in the representation (3.12). In what follows, named subsets of R are taken to be Borel sets, unless mention is made to the contrary.
Thus κ Ω is an optional random measure over X, homogeneous on [0, ∞[ in the sense of [Sh88], and κ Ω is perfect. For notational simplicity we now drop the subscript Ω from our notation, but it should be clear from context when we are restricting κ to Ω.
We define the potential kernel U κ of κ by setting, for f ∈ pE * , where j ∈ pE comes from (3.12). Defineκ := 1 α,∞ κ on Ω. Then and using the Markov property one sees that Uκf is an excessive function of X; indeed, Uκf is the excessive regularization of U κ f .
If T is a stopping time then the associated hitting operator P T is defined by X t ∈ B} is the début of B, and these are both stopping times. We shall write P B and H B as abbreviations of P T B and P D B .
A function f is strongly supermedian provided f ∈ pE n and P T f ≤ f for all stopping times T . This definition differs slightly from the definition in [FG96] where f was required to be measurable over the σ-algebra E e generated by the 1-excessive functions. In this paper, because we are assuming that X is a Borel right process, it is more natural to use E n . The critical point is that t → f • X t is nearly optional over (F t ); see [Sh88; (5.2)]. In the present case E e ⊂ E n , so it follows from (4.1) that U κ f is nearly Borel measurable whenever f is so. Therefore the strong Markov property implies that U κ f is strongly supermedian if f ∈ pE n . Consequently, the kernel U κ is a strongly supermedian kernel as defined in [BB01b]. The content of the next proposition is that U κ is a regular strongly supermedian kernel, provided it is proper; see (5.1) and [BB01b;§2]. The property asserted is a form of the familiar domination principle.
(4.2) Proposition. Fix a strongly supermedian function u and f ∈ pE n . If Recalling from (3.12) that κ c denotes the diffuse part of κ, we see that and, since κ c is diffuse, On the other hand, using the strong Markov property and the fact that 1 B n • X vanishes on T n,p , T B n , we find Combining these observations with the expression for U κ g n (x) obtained above, we find that lim p→∞ P T n,p U κ g n (x) = U κ g n (x). Now B n , being thin, is finely closed, these facts together gives P T n,p U κ g n (x) ≤ P T n,p u(x) ≤ u(x), the last inequality following because u is strongly supermedian. Therefore U κ g n (x) ≤ u(x). Sending n → ∞ we obtain Our next task is to express µ directly in terms of the kernel U κ . To this end we begin by extending [G90; (7. 2)] to a class of strongly supermedian functions. Let H ∈ pF, so that H is defined on Ω. Suppose that In particular, if u is E n measurable, then u is strongly supermedian. If α(w) < s < t then one can check as in section 7 of [G90] that The next result should be compared with [G90; (7.2)]. Once again we use the Riesz decomposition m = η + ρU of m into harmonic and potential components. Also, L is the energy functional defined for ξ ∈ Exc and excessive h by Recall that if ξ ∈ Exc is conservative then ξ dominates no non-zero potential, so L(ξ, h) = 0 for all excessive h.
(4.5) Lemma. Let H, H * , and u be as above, and letū :=↑ lim t↓0 P t u denote the excessive regularization of u. Then where ν represents a generic σ-finite measure on (E, E n ).
Proof. If νU ≤ η + ρU , then by [G90; (5.9),(5.23)] we can decompose νU = ν 1 U + ν 2 U with ν 1 U ≤ η and ν 2 U ≤ ρU . Moreover, (4.3) and [G90; . Therefore, it suffices to prove (4.5) when m ∈ Har (the class of harmonic elements of Exc). We may also assume that m ∈ Dis, the class of dissipative elements of Exc. Indeed, let m = m c +m d be the decomposition of m into conservative and Hence (4.5) holds if and only if it holds for m d .
In the remainder of the proof we suppose that νU ≤ m and that m ∈ Har ∩ Dis. Combining (7.10), (7.5i), and (6.20) in [G90] we find that where (T (t) : 0 ≤ t ≤ 1) is an increasing family of stationary terminal times with α ≤ T (t), Let S be a stationary time with α < S < β, Q m -a.e. Since m ∈ Dis, such an S exists according to [G90; (6.24 by monotone convergence, wherē and it is now evident thatH is excessive as defined in section 7 of [G90]. Finally, observe Therefore we may apply [G90; (7.2)] to obtain On the other hand, if ν n U ↑ m, then This establishes (4.5) when m ∈ Har ∩ Dis, hence in general by the earlier discussion.
(4.7) Theorem. If u is a strongly supermedian function, then Proof. First observe that it suffices to prove (4.8) for bounded u. Indeed, given an arbitrary strongly supermedian function u, define u k := u ∧ k for k ∈ N. By monotonicity and lim k ρ(u k ) = ρ(u). By the same token, since the limits defining the excessive regularizationsū k andū are monotone increasing, we have lim k L(η,ū k ) = L(η,ū).
Next, arguing as at the beginning of the proof of Lemma (4.5) we can assume that m is dissipative, and then, without loss of generality, that X is transient. Now let u be a bounded strongly supermedian function. By Corollary (A.7) in the appendix, there is an increasing sequence {v n } of regular strongly supermedian functions with pointwise limit u. By [Sh88; (38.2)], for each n there is a PLAF is decreasing, so Lemma (4.5) applies and we find that As in the first paragraph of the proof, monotonicity allows us to let n → ∞ in (4.9) to arrive at (4.8).
We are now in a position to express µ in terms of the kernel U κ . As before, κ is a perfect HRM and µ = µ κ . Also, m = η + ρU .
(4.10) Theorem. Suppose m ∈ Dis. If f ∈ pE, then Proof. Fix a positive Borel function ϕ on R with R ϕ(t) dt = 1. Define, for a given f ∈ pE, But m ∈ Dis and so there is a stationary time S with α < S < β, Q m -a.s. Thus, from [G90; (6.27)] we obtain , The first term on the right equals where the first equality just above comes from (2.5) and the last from (2.4). Because in the present situation. As in the proof of (4.5), with m replaced by η, this last expression equals L(η,Ū κ f ). Clearlȳ U κ f = Uκf withκ as in (4.10). Consequently, µ(f ) = ρU κ f + L(η, Uκf ) and the fact that this in turn equals sup{νU κ f : νU ≤ m} results from one final appeal to (4.5).
The most interesting case (m ∈ Dis, which occurs if X is transient) is covered by (4.10). However if m is invariant, in particular if m is conservative, then one has (4.14) Remark. As observed in Theorem (4.10), the co-natural HRMκ := 1 α,∞ κ has potential kernel Uκ =Ū κ , which is a semi-regular excessive kernel in the sense of Definition (5.1) below. By an argument used in the proof of Theorem (3.17), the characteristic measure µκ ofκ is p · µ κ , where p • Y * = as in Remark (3.18); cf. (5.11). Observe that (4.5) and the proof of (4.10) together imply that µκ(f ) = L(m,Ū κ f ). Combining (8.21) and (8.9) of [G90] we obtain the classical expressions for the characteristic measure ofκ:

Strongly Supermedian Kernels.
We shall now examine some of the relationships between our results on HRMs and the material presented in [BB01b]. Beznea and Boboc assume that the potential kernel U is proper (equivalently, that X is transient), so throughout this section and the next we shall assume that U is proper. More precisely, we assume that there is a function b ∈ E with 0 < b ≤ 1 and U b ≤ 1. Reducing b if necessary, we can (and do) assume that m(b) < ∞.
In particular, each excessive measure of X is dissipative.
It is well known that if u is a strongly supermedian function and 0 ≤ S ≤ T are stopping times, then P T u ≤ P S u everywhere on E. More generally, let us say that a σ-finite measure µ is dominated by another σ-finite measure ν in the balayage order (and write µ ν) provided µU ≤ νU . Then for strongly supermedian u we have µ(u) ≤ ν(u) whenever µ ν. (These assertions follow, for example, from Rost's theorem; see [G90; (5.23)].) It follows in turn that if T is a terminal time (namely, for each t > 0) then P T u is strongly supermedian.
We modify slightly the definition of regular strongly supermedian kernel found in [BB01b] by dropping the assumption that such a kernel is proper. The connection between regular strongly supermedian kernels and the notion of regularity for strongly supermedian functions (as in [Sh88; (36.7)] or (A.1) in the appendix to this paper) is made in Proposition (A.8) and Remark (A.11).
(5.1) Definition. (a) A strongly supermedian (resp. excessive) kernel V is a kernel on (E, E n ) such that Vf is a strongly supermedian (resp. excessive) function for each f ∈ pE n .
(b) A strongly supermedian kernel V is regular provided, for each f ∈ pE n and each Proposition (4.2) may now be restated as follows: If κ is a perfect HRM, then U κ is a regular strongly supermedian kernel.
The following result records some facts that have familiar analogs in the context of  (ii) Let V be a regular strongly supermedian kernel. If B is m-exceptional (resp. (5.3) Definition. Let ν be a measure on (E, E). A strongly supermedian kernel V is ν-proper provided there exists a strictly positive function g ∈ E n with V g < ∞, ν-a.e.
n · u on all of E by the regularity of V . We now define g : implies that V 1 B = 0 off an m-polar (resp. m-exceptional) set.
(b) We can write W =V , where V is a regular strongly supermedian kernel. Let It follows that V is m-proper, so assertion (b) follows from (a) because Wf ≤ Vf for all f ∈ pE n .
[As before, ν on the right side of (5.6) represents a generic σ-finite measure on (E, E n ).] Thus, to show that µ V is a measure, it suffices to show that This inequality is evident if either term on the right is infinite, so we assume that µ V (f 1 ) + µ V (f 2 ) is finite. Given > 0, there exist ν 1 and ν 2 with ν i U ≤ m and for i = 1, 2. Define ξ := inf{η ∈ Exc : η ≥ ν 1 U ∨ ν 2 U }. Then ξ ∈ Exc, ξ ≤ m, and (because ξ is dominated by the potential (ν 1 + ν 2 )U ) ξ is a potential, say ξ = νU . Also, Therefore As > 0 was arbitrary, it follows that µ V is a measure.  Finally, suppose thatV is m-proper. By (5.5)(b) there exists g ∈ E n with 0 < g ≤ 1 such that V g ≤ U b off an m-polar set. By [G90; (2.17)] there is a sequence (ν n ) of measures, each absolutely continuous with respect to m, such that ν n U increases setwise to m, and then ν nV g increases to L(m,V g) = µV (g). Because the set {V g > U b} is m-polar, hence m-null, it follows that µV (g) ≤ lim n ν n U b = m(b) < ∞, proving that µV is σ-finite.
The measure µ V defined in (5.6) is called the characteristic measure of V . Note that if κ is a perfect HRM then, in light of Theorem (4.10), the characteristic measure µ κ of κ (defined in (3.7)) is the characteristic measure of the regular strongly supermedian kernel U κ . Writingκ := 1 α,∞ κ as before, the potential kernel ofκ is the semi-regular excessive kernelŪ κ := U κ , and the characteristic measure ofκ is p · µ κ ; see Remark (4.14).
We come now to the main result of this development. V (x, ·) = U κ (x, ·)} ∈ N (m).
This yields the desired conclusion. Since the proof of this uniqueness theorem in [BB01b] depends on some rather deep results in potential theory, we shall give an alternate proof of Theorem (5.8) in the Appendix; this proof may be more palatable to probabilists.
(b) By (5.7) the characteristic measure µV is σ-finite and charges no m-polar set.
Theorem (3.17) guarantees the existence of a co-natural HRM with characteristic measure µV . Theorem 3.2(ii) in [BB01b] now implies thatV (x, ·) = U γ (x, ·) for all x outside an m-polar set.  We now record some corollaries of Theorem (5.8) that are of interest. The first of these is an immediate consequence of (4.8) and (5.6).
The third corollary will follow immediately from the next result about HRMs. This result-more precisely its dual-appears for diffuse κ in [G99].
(5.12) Theorem. Let κ be a perfect HRM. Then whereÛ is the potential kernel of the moderate Markov dual process (X,P x ) of X relative to m; see [Fi87; §4].
(5.14) Remark. Theorem 4.6 in [Fi87] establishes the existence of (X,P x ), and it is only stated there that the probability measuresP x , x ∈ E, are uniquely determined modulo mpolars. However, using Λ * in place of Λ and letting µ be a probability measure equivalent to m + ρ (rather than m), the proof given in the appendix of [Fi87] is readily modified to show that the family {P x } is unique up to an m-exceptional set. HenceÛf is uniquely determined up to an m-exceptional set; since µ κ charges no element of N (m), the integral on the right side of (5.13) is well defined.
Fix ϕ ≥ 0 with R ϕ(t) dt = 1. In the following computation, the first equality is (3.7) while the third holds because κ is co-predictable: The next-to-last equality depends on (3.11)(iii); the homogeneity of κ is used for the fourthfrom-the-last equality; the use of Fubini's theorem in the fourth-and sixth-from-the-last equalities is justified since κ is an HRM.
The following corollary is now evident.
(5.15) Corollary. Let V be an (m+ρ)-proper regular strongly supermedian kernel. Then whereÛ is as in (5.12). (ii) Clearly, both sides of (5.16) are σ-finite measures as functionals of f and g separately. Hence, if F ∈ p(E ⊗ E), then More generally, if F ∈ E ⊗ E and if either side of (5.18) is finite when F is replaced by |F |, then (5.18) holds.
(iii) Formula (5.16) with f ≡ 1 implies that mV µ V , and that a version of the Radon-Nikodym derivative isÛ 1. SinceÛ 1 > 0 off an m-exceptional set, we also have µ V mV . It is not too difficult to check this measure equivalence directly, but the fact that the Radon-Nikodym derivative does not depend on V is somewhat surprising.
(iv) A more general version of (5.16) appears as Theorem 5.2 in [BB02]. In that result the co-potentialÛf is replaced by (an m-fine version of) a general m-a.e. finite co-excessive functionû. Notice that the left side of (5.16) can be interpreted as L(πU, Vg), where π := f · m; the measure potential πU has Radon-Nikodym derivativeÛf with respect to m. (For an excessive measure ξ and a strongly supermedian function u, we follow [BB01a] in defining the energy L(ξ, u) as sup{ν(u) : νU ≤ ξ}.) Thus, in the general case, πU is replaced by the excessive measureû · m.
The final result of this section is a uniqueness theorem for perfect HRMs.
(5.20) Remark. Some of the implications in Theorem (5.19) follow from results in [BB01b], but we shall give direct proofs that are more probabilistic in character than those found in [BB01b].
(i)=⇒(iii). Fix f ∈ pE and note that for j = 1, 2 the Q m -optional projection of the Thus, if (i) holds then the Q m -optional processes U κ 1 f (Y * ) and U κ 2 f (Y * ) are Q m -indistinguishable, so (iii) follows by a monotone class argument because E is countably generated; see the sentence preceding (3.4).
(v)=⇒(i). Let E ⊂ {U κ 1 g = U κ 2 g < ∞} be a Borel absorbing set with m-exceptional complement, and let X denote the restriction of X to E . Then, for j = 1, 2, g * κ j (restricted to Ω) may be viewed as a RM of X , perfectly homogeneous on [0, ∞[, with left potential function U κ j g as in [Sh88]. Clearly A j t := [0,t[ g(X s ) κ j (ds) defines an optional LAF of X with left potential function v j = U κ j g. Since v 1 = v 2 < ∞ everywhere on E , [Sh88; (37.8)] implies that A 1 and A 2 are P x -indistinguishable for each starting point x ∈ E . Since g > 0, we deduce that the restrictions to Ω of κ 1 and κ 2 are P xindistinguishable for each x ∈ E . Now define Ω 0 := {w ∈ Ω : α(w) = 0, κ 1 (w, B) = κ 2 (w, B) for some B ∈ B}, and (recalling that κ 1 and κ 2 are carried by Λ * ) observe that We have, because E \ E is (m + ρ)-null, the second display following from (2.4). This proves (i).
(5.21) Remark. (a) There is an analogous uniqueness theorem for co-natural HRMs with σ-finite characteristic measures. It reads exactly like Theorem (5.19) except that "m-exceptional" is replaced by "m-polar" in (iii) and (v), and (m + ρ)-null is replaced by m-null in (iv). We leave the details to the reader. (b) One consequence of Theorem (5.19) is this: A perfect HRM κ with σ-finite characteristic measure is Q m -indistinguishable from a perfect HRM κ whose potential kernel is proper. Indeed, if µ κ is σ-finite then by (4.10) and (5.7) the potential kernel U κ is (m + ρ)proper. Thus, by (5.5)(a) there is a strictly positive Borel function g and an m-exceptional set N such that U κ g ≤ 1 off N . We can (and do) assume that N ∈ E and that E \ N is absorbing. The desired modification of κ is then κ (dt) : of E because U κ is a regular strongly supermedian kernel.

Additive Functionals.
It is often important to describe an HRM in terms of the associated distribution function, at least under appropriate finiteness conditions. Thus we are led to the concept of additive functional (AF). Our definitions are essentially those given in [Sh88], except that we allow an exceptional set of starting points; cf. [Sh71] and [FOT94; Chap. 5].
Throughout this section we work with a fixed m ∈ Exc with Riesz decomposition m = η + ρU into harmonic and potential components. As in the last section, we assume in this section that X is transient, meaning that U is a proper kernel: There exists b ∈ pE with 0 < b ≤ 1, m(b) < ∞, and U b ≤ 1.
Recall that a nearly Borel set N is m-inessential provided it is m-polar and E \ N is absorbing for X. By [GS84; (6.12)], any m-polar set is contained in a Borel m-inessential set. We shall say that a nearly Borel set N is strongly m-inessential provided N ∈ N (m) and E \ N is absorbing. As noted just before (5.2), if G is a finely open m-null set then G ∈ N (m). Using this observation, the proof of [GS84; (6.12)] is easily modified to show that any set in N (m) is contained in a Borel strongly m-inessential set. (ii) θ t Ω A ⊂ Ω A for all t ≥ 0; (iii) For all ω ∈ Ω A the mapping t → A t (ω) is left-continuous on ]0, ∞[, finite valued on [0, ζ(ω)[, with A 0 (ω) = 0; (iv) For all ω ∈ Ω A , A 0+ (ω) = a(X 0 (ω)), where a ∈ pE n ; (v) For all ω ∈ Ω A and s, t ≥ 0: A t+s (ω) = A t (ω) + A s (θ t ω);  proved that N A just defined is strongly m-inessential and that the construction just given yields the most general PLAF.
The following improvement of property (iv) above is useful.
(6.4) Lemma. Let A be a PLAF. Then the defining set Ω A and exceptional set N A for A can be modified so that the function a in (iv) satisfies: {a > 0} is semipolar.
Proof. From the definition, we have ∆A 0 = A 0+ = a(X 0 ) on Ω A , where a ∈ pE n . Therefore where A c is the continuous and A d the discontinuous part of A, we have, on Ω A , and by (3.13) we can write {a > 0} = S a ∪ N a , where S a is semipolar and N a ∈ N (m).
Because {a > 0} ∈ E n , the argument used in proving (3.13) shows that S a and N a can be chosen nearly Borel as well; this we assume done. Now let M be a Borel strongly Define Ω * A = Ω A ∩ {D = ∞} and a * = a · 1 E\N * . It is evident that N * and Ω * A serve as exceptional and defining sets for A, that {a * > 0} is semipolar, and that A 0 = a * (X 0 ) on Ω * A .
Theorem (3.18) motivates the following (6.5) Definition. A positive co-natural additive functional (PcNAF) is an (F t )-adapted increasing process A = (A t ) t≥0 with values in [0, ∞], for which there exist a defining set Ω A ∈ F and an m-inessential Borel set N A (called an exceptional set for A) such that, in addition to conditions (6.1)(i)(ii)(v)(vi), the following modifications of (6.1)(iii)(iv) hold: (iv) For all ω ∈ Ω A and all t > 0, ∆A t (ω) : where a ∈ pE n ; (6.6) Definition. Two PcNAFs A and B are m-equivalent provided P m [A t = B t ] = 0 for all t ≥ 0.
where A c is a PCAF (as in [FG96]) and a ∈ pE n with {a > 0} semipolar. We now define the potential kernel of an AF; this notion will play an important role in the sequel.
(6.8) Definition. Let A be a PLAF or a PcNAF. The potential kernel U A of A is defined for f ∈ E n by (6.9) where N A is an exceptional set for A. In particular, u A := U A 1 is the potential function of A.
A is a PLAF (resp. PcNAF). If, for all ω ∈ Ω A , (f * A) t (ω) < ∞ for all t ∈ [0, ζ(ω)[, then f * A is a PLAF (resp. PcNAF) provided A is a PLAF (resp. PcNAF). Of course the finiteness condition just mentioned is satisfied if f is bounded. Finally, notice that When A is a PLAF or a PcNAF with exceptional set N A , it will be convenient to let X A denote the restriction of X to the absorbing set E \ N A . In fact, using this device one can often reduce matters to the situation in which N A is empty. If A is a PcNAF, one checks easily that U A f is excessive for X A , and hence nearly Borel For the following definition, which is motivated by Theorem (4.10), recall the Riesz decomposition m = η + ρU .
(6.11) Definition. Let A be a PLAF or a PcNAF. The characteristic measure µ A of A (relative to m ∈ Exc) is defined by Observe that if f is such that f * A is a PLAF or a PcNAF, then Definition (6.11) requires some justification since U A f andŪ A f are only defined on E \N A . Thus, the energy functional L appearing in (6.12) should be regarded as the energy Also, because ρU ≤ m, ρ charges no set in N (m). Since N A ∈ N (m), ρU A f is well defined. Thus (6.12) is justified. If A is a PcNAF then (6.12) reduces to µ A (f ) = L(m, U A f ).
If A is a PcNAF and F is m-polar, then U A 1 F = 0, m-a.e., and so µ A (F ) = L(m, U A 1 F ) = 0. Thus µ A charges no m-polar set. If A is a PLAF, and F ∈ N (m), thenŪ A 1 F = 0, m-a.e., and hence η-a.e. Therefore L(η,Ū A 1 F ) = 0. Moreover, combining the formula [G90; (6.20)] for ρ with the strong Markov property we obtain N (m)). Thus µ A charges no m-exceptional set when A is a PLAF.
(6.13) Definition. A nest is an increasing sequence (B n ) of nearly Borel sets such that (6.14) Definition. A measure ν on (E, E) is smooth (resp. strongly smooth) provided it charges no m-polar set (resp. no element of N (m)) and there is a nest (resp. strong nest) (G n ) of finely open nearly Borel sets such that µ(G n ) < ∞ for all n.
If (G n ) is a nest (resp. strong nest), then E \ ∪ n G n is m-polar (resp. m-exceptional). Thus, a smooth or strongly smooth measure is necessarily σ-finite. If a smooth measure µ charges no m-semipolar set then it is smooth as the term was defined in [FG96], where it was proved that such a µ is the characteristic measure of a uniquely determined PCAF A.
Below we develop similar results for PLAFs and PcNAFs. We have already seen that µ A charges no m-polar set (resp. no element of N (m)) if A is a PcNAF (resp. PLAF).
We need to extend the notion of regularity, introduced in section 5 for strongly supermedian functions, as follows: If E ∈ E n is an absorbing set, then we say that f ∈ bE n is regular on E provided lim n P x [f (X T n )] = P x [f (X T )] for all x ∈ E and every increasing sequence (T n ) of stopping times with limit T .
(6.15) Theorem. If A is a PcNAF (resp. PLAF) then µ A is smooth (resp. strongly smooth). Moreover, there exists a finely lower-semicontinuous nearly Borel function g Proof. Suppose first that A is a PLAF, with exceptional set N = N A and defining set Ω A .
On Ω A let M be the Stieltjes exponential of A; that is, where A c is the continuous part of A and ∆A s : Clearly g > 0 on E \ N , and, for x / ∈ N , where the fifth equality comes from (6.17). Thus g + U A g = U b on E \ N ; because U b ≤ 1 and g ≥ 0, we have U A g ≤ 1 on E \ N . Since M 0+ = 1 + ∆A 0 = 1 + a(X 0 ), where a comes from Definition (6.1)(iv), Let L t := M 0+ M −1 t+ . Then L = (L t ) t≥0 is a right-continuous, decreasing multiplicative functional of X A with L 0 = 1. Hence L is exact and the expectation in (6.20) is excessive with respect to (X A , L)-the L subprocess of X A . In particular, g is nearly Borel measurable. If T is a stopping time then from which it follows that g is regular and finely lower-semicontinuous on E \ N (by Theorem 4.9 in [Dy65], since M 0+ ≥ 0). Thus the sets G n := {g > 1/n}, n ≥ 1, form an increasing sequence of finely open, nearly Borel, subsets of E \ N . Let τ n be the exit time of G n . Since G c n is finely closed, g(X τ n ) ≤ 1/n, P x -a.s. for x ∈ E \ N . Thus, for such x, But b > 0, and M t < ∞ on [0, ζ[ (P x -a.s.), so we must have lim n τ n = ζ, P x -a.s. for all This establishes (6.15) when A is a PLAF.
Now consider the case in which A is a PcNAF. Define a PLAF A * by A * t := A c t + 0≤s<t a(X s ), where a comes from Definition (6.5)(iv). Then A t = A * t+ − A * 0 . Define M as in (6.16) with A replaced by A * , and define g as in (6.19). As before, g is nearly Borel measurable, finely lower-semicontinuous, and regular. In the present case the computation just below (6.19) yields for x ∈ E \N . Hence g ≤ U b and U A g ≤ U b ≤ 1 on E \N . Just as for PLAFs, the sequence defined by G n := {g > 1/n} is a nest (recall that N is m-inessential) with µ A (G n ) < ∞ for each n. This completes the proof of Theorem (6.15).
The next result is the fundamental existence theorem for AFs. It is essentially the converse of Theorem (6.15). The accompanying uniqueness result is (6.29).
(6.21) Theorem. Let µ be a strongly smooth (resp. smooth) measure. Then there exists a PLAF (resp. PcNAF) A with characteristic measure µ. Moreover, in either case, there exists a Borel function j ≥ 0 with {j > 0} semipolar such that ∆A ≡ j • X.
Proof. Suppose first that µ is strongly smooth. Clearly µ ∈ S # 0 (m), so there is an HRM κ associated with µ as in (3.11). Let κ Ω denote the restriction of κ to Ω. If f ∈ pE and t ≥ 0 then In addition, from (4.10), we have Combining these estimates we have, for G ∈ E, Now let (G n ) be a strong nest with µ(G n ) < ∞ for all n. Then (6.22) implies (6.23) Define (See the second paragraph of section 4 for notation.) Let S := inf{t : A t = ∞}. It is evident that S is an (F * t )-stopping time and that S = t+S • θ t on {S > t}. Now (6.23) implies that We need the following lemma, which we shall prove after using it to complete the proof of the theorem. Suppose next that µ is smooth. Because µ is a σ-finite measure charging no m-polar set, Theorem (3.16) guarantees the existence of a co-natural HRM κ with characteristic measure µ. In this case κ Ω ({0}) = 0. Let (G n ) be a nest of finely open sets with µ(G n ) < ∞. Dropping the subscript Ω from κ, it follows just as before that This time we define A t := lim n→∞ κ]0, t + 1/n], and S := inf{t : A t = ∞} = inf{t : κ]0, t] = ∞}. Arguing exactly as in the proof of [G95; Prop. 4.3], one see that A is an adapted, right-continuous increasing process with A t+s = A t + A s • θ t for s, t ≥ 0, and that A is exact in the sense that All of these statements hold identically on Ω.  [G95] shows that h( where E e is the σ-algebra generated by the 1-excessive functions of X. Since X is a Borel right process, we have E e ⊂ E n . It is evident that P T h ≤ h for all stopping times T , and so h is strongly supermedian. (6.26) Remarks. The AFs constructed in the proof of (6.21) have better properties than required by the definitions. The shift property (6.1)(v) holds for all ω ∈ Ω and the "jump" function j is Borel measurable with {j > 0} semipolar. Also, the PcNAF produced is exact. The PLAF constructed is (F * t− )-adapted and the PcNAF is (F * t+ )-adapted.
Here is the analog of (6.21) for regular strongly supermedian kernels.
(6.27) Theorem. (a) Let V be a regular strongly supermedian kernel and suppose there are a strongly m-inessential set N and a finely lower-semicontinuous function g : E \ N → ]0, 1] that is regular on E \ N , such that V g ≤ 1 on E \ N . Then there is a unique PLAF (b) Let V be a regular strongly supermedian kernel and define a semi-regular excessive kernel W by W f := V f (excessive regularization). Suppose there are an m-inessential set N and a finely lower-semicontinuous function g : E \ N →]0, 1] that is regular on E \ N , such that W g ≤ 1 on E \ N . Then there is a unique PcNAF A with exceptional set Proof. (a) It is clear that V is (m+ρ)-proper, so by Theorem (5.8) there is a unique perfect HRM κ such that {x ∈ E \ N : U κ (x, ·) = V (x, ·)} is contained in a strongly m-inessential ≥ 0, and S := inf{t : A t = ∞}. As in the proof of Theorem (6.21), we will be done once we show that P m+ρ [S < ζ] = 0. To this end observe that because g is regular and strictly positive on E \ N 0 , we have inf 0≤s≤t g(X s ) > 0 on {t < ζ}, P x -a.s.
for each x ∈ E \ N 0 and each t > 0. But for x ∈ E \ N 0 , from which it follows that κ[0, t[< ∞, P x -a.s. on {t < ζ} for each x ∈ E \ N 0 and each t > 0. This is more than enough to imply that P m+ρ [S < ζ] = 0.
(b) The proof of this assertion is quite similar to that of part (a), so we omit it.
(6.28) Remark. The smoothness conditions appearing in Theorems (6.21) and (6.27) are comparable. Thus, if µ is a strongly smooth element of S # 0 (m) then by (6.15) and (6.21) there is a strictly positive function g that is finely lower semicontinuous and regular on an On the other hand, if V is a regular strongly supermedian kernel satisfying the hypothesis of part (a) of Theorem (6.27), then G n := {g > 1/n}, n ≥ 1, defines a strong nest such that, for each n, V (1 G n ) ≤ n off an m-exceptional set. This should be compared to the notion smooth kernel used in [BB01a]; see especially Theorem 2.1 in [BB01a].
(6.29) Theorem. Let A and B be PLAFs (resp. PcNAFs) with characteristic measures µ A and µ B , and potential kernels U A and U B . The following are equivalent: (i) For all t ≥ 0, P m+ρ (A t = B t ) = 0 (resp. P m (A t = B t ) = 0); (v) There exists a strictly positive function g ∈ pE n such that U A g = U B g < ∞ off an m-exceptional (resp. m-polar) set. Theorem (6.29) is a direct consequence of Theorem (5.19) and the following result that links the notions of HRM and PLAF (or PcNAF). The reader is invited to extract a proof of Proposition (6.31) from the proof of Theorem (5.8) found in the appendix.
(6.31) Proposition. Let A be a PLAF (resp. PcNAF) with characteristic measure µ A and potential kernel U A . Then there is a unique perfect HRM (resp. co-natural HRM) κ with µ κ = µ A and U κ (x, ·) = U A (x, ·) for all x outside an m-exceptional (resp. m-polar) set.
We end this section with a brief discussion of the fine support of a PLAF and of its associated potential function. Let V be a regular strongly supermedian kernel such that v := V 1 < ∞. Then by Theorem (6.27) and its proof there is a (unique) PLAF A with empty exceptional set such that V = U A . Recall from Section 5 that ν µ (balayage order) provided νU ≤ µU . Because X is transient, we have ν µ if and only if ν(u) ≤ µ(u) for every excessive function u. We follow Feyel [Fe83] in defining the fine support δ(v) of the regular strongly supermedian function v as in the theory of Choquet boundaries: δ(v) is the set of points x ∈ E such that the only measure ν on E with ν (6.32) Proposition. (a) δ(v) is a finely closed element of E n ; An analytic proof of this proposition can be found in [Fe83] or in [BB02]. In our setting the proposition is an immediate consequence of the following description of δ(v) and the subsequent discussion. Proof. Suppose x ∈ δ(v) and define ν := x P S . Clearly ν x while . By Rost's theorem ( [Ro71] or [G90; (5.23)]) there is a (randomized) stopping time T with ν = x P T . Then so P x [A T = 0] = 1. When coupled with the fact that A t > 0 for all small t > 0, P x -a.s.
where a ∈ pE n with J := {a > 0} semipolar, then clearly S = S c ∧ D J , where S c := inf{t : A c t > 0}. It is well known that S c = T F c almost surely, where F c is the fine support (in the usual sense) of the CAF A c ; see [BG68;p. 213]. Also, since the début of a nearly Borel set is almost surely equal to the début of its fine closure, we have F = F c ∪J f , whereJ f is the fine closure of J.
Conversely, suppose that F is a given finely closed nearly Borel set. Recall that b ∈ pE is strictly positive and U b ≤ 1. Consider the function v := H F U b. It is easy to check that v is a (bounded) regular strongly supermedian function, so by [Sh88;(38.2)] there is a PLAF A with empty exceptional set such that U A 1 = v. Furthermore, using Rost's theorem as in the proof of Proposition (6.33), we can show that δ(v) = F . Thus, each finely closed nearly Borel set is the fine support of a PLAF.
The fine support F A of a positive CAF A is finely perfect: F A is finely closed and each point of F A is regular for F A . The known converses to this assertion (e.g. [Az72,FG95]) are more involved than the construction suggested in the preceding paragraph. The paper [DG71], especially Example (4.4) on pp. 543-544, provides an instructive discussion of these matters.

Resolvents.
One of the mains results of [BB01a] is that if V is a regular strongly supermedian kernel, then V satisfies the hypothesis of Theorem (6.27) (equivalently, V agrees off an m-exceptional set with the potential kernel of a PLAF) if and only if V is, off an mexceptional set, the initial kernel of a subMarkovian resolvent. Because the regularity of a strongly supermedian kernel amounts to a form of the domination principle, this assertion is closely related to the work of Hunt [Hu57], Taylor [T72,T75], and Hirsch [Hi74] on the existence of subMarkovian resolvents with given initial kernel.
Our aim in this section is to give an explicit representation of the resolvent (V q ) q≥0 such that with V 0 = U A for a given PLAF A. We even construct a (simple) Markov process possessing the given resolvent.
Throughout this section we suppose that A satisfies the conditions listed in (6.26).
For q ≥ 0 define (1 + q∆A s ), with the convention that M q 0− = 1. Then M 0 t = 1 for all t ≥ 0, t → M q t is right-continuous, increasing, and finite valued on [0, ζ[, and M q 0 = 1 + q∆A 0 = 1 + qA 0+ . Moreover, for t ≥ 0, (7.3) Theorem. With the above notation, define In proving Theorem (7.3), by the device of restricting X to E \ N A , it suffices to suppose that N A is empty, and this we shall do. The key computation is contained in the Proof. If q > 0, then by (7.2)(ii) Now suppose r > 0 and q = r. Then where (7.2)(i) was used for the second equality. Because of (7.2)(ii), the final integral above is equal to Consequently, (1 − q/r)I(q, r) = r −1 (M q s ) −1 M r s − 1 , which implies (7.4).
We now prove Theorem (7.3). First note that V q f ∈ pE n provided f ∈ pE n , and that Also, if 0 < g ≤ 1 with U A g ≤ 1, then V q g ≤ 1. Now suppose that q = r and f ∈ pE n with U A f < ∞. Then, for x ∈ E, This proves that (V q ) q≥0 is a subMarkovian resolvent of proper kernels, with V 0 = U A .
(7.5) Question. Suppose that u is a bounded strongly supermedian function, and define f := u − qV q u. Using the resolvent equation for (V q ) we find that V f = V u − qV V q u = V q u < q −1 u on {f > 0}\N A . Therefore V q u ≤ q −1 u on all of E \N A . A routine truncation argument now establishes the following assertion: If u is strongly supermedian then u is supermedian with respect to the resolvent (V q ) q≥0 . Is the converse true? That is, suppose that a nearly Borel function u is supermedian with respect to the resolvent associated with each PLAF of X. Must u then be strongly supermedian?
If A is continuous, then (V q ) is the well-known resolvent of the strong Markov process obtained by time-changing X using the strictly increasing right-continuous inverse of A.
In the general case one must, in addition to the time change, make each x in the semipolar set {j > 0} (see (6.21)) an exponentially distributed holding point with mean holding time equal to j(x). This will be made more precise in the next result.
Let A be a PLAF satisfying the conditions in (6.26). Since J := {j > 0} is semipolar, there is a sequence (T n ) n≥1 of stopping times with disjoint graphs such that ∪ n T n is j(X T n )U n , t ≥ 0.
Note that conditional on F, the random variable j(X T n )U n has the exponential distribution with mean j(X T n ); since n:T n <t j(X T n ) ≤ A t , it follows that t → B t is finite on [0, ζ[ and left-continuous on ]0, ζ[, P x -a.s. for all x ∈ E \ N A . Also, B 0 = 0. Let (τ t ) t≥0 , the right-continuous process inverse to B, be defined by (7.7) Theorem. Let (V q ) q≥0 be as in (7.3) and define Z t = X τ (t) , t ≥ 0. Then for x ∈ E \ N A and f ∈ pE, exponential (mean j(x)) time. Also of interest is the special case in which A c = 0 and {t : X t ∈ J} is almost surely dense in [0, ζ[.

A. Appendix.
In this appendix we collect some important properties of strongly supermedian functions, and we give a direct proof of Theorem (5.8). See [Mr73,Fe81,Fe83,BB99] for background on strongly supermedian functions.
First note that if f is strongly supermedian and if (T n ) is a monotone sequence of stopping times, then lim n P x [f (X T n )] exists. Consequently, the process t → f (X t ) has left limits on ]0, ∞] and right limits on [0, ∞[, almost surely. Letf := lim t↓0 P t f denote the excessive regularization of f . We claim that the processes (f (X) t+ ) t≥0 and (f (X t )) t≥0 are indistinguishable. It suffices to prove this for bounded f since f ∧ c =f ∧ c for c ∈ R + .
Next, since both processes are optional, we need only check that for all stopping times T . Because f is bounded, These facts will be used without special mention in the sequel.
(A.1) Definition. Let E ∈ E n be an absorbing set. A strongly supermedian function f is regular on E if f is finite on E and for every increasing sequence (T n ) of stopping times we have where T := lim n T n . When E = E we simply say that f is regular.
The next theorem is a fundamental result of J.-F. Mertens [Mr73]. The proof in complete generality is rather complicated; we present a simpler proof for the special case of bounded strongly supermedian functions, which is the only case we shall be using. A strongly supermedian function u dominates another strongly supermedian function v in the specific order provided there is a strongly supermedian function w such that u = v + w.
(A.2) Theorem. [Mertens] A strongly supermedian function f may be decomposed uniquely as h+g, where h is the largest (in the specific order) excessive function specifically dominated by f , and g is a regular strongly supermedian function.
Proof. We give a proof only in the special case of bounded f . As before,f denotes the excessive regularization of f . The process (f (X t )) t≥0 is a bounded optional strong supermartingale (under each measure P x , x ∈ E), so by Theorem 20 on page 429 of [DM80] there exists, for each x ∈ E, an increasing predictable process (A x t ) t≥0 such that for each stopping time T . It follows from the proof given in [DM80] that A x t = A x,− t +A x,+ t− , with A x,− increasing and predictable, and A x,+ increasing and optional, both processes being right-continuous. Moreover, the process A x,+ is purely discontinuous. In view of the footnote on page 430 of [DM80], up to P x evanescence, for each x ∈ E. Defining B t to be the sum on the far right of (A.4), noting that B 0 = 0, equation (A.3) may be re-written as Since A x,− is increasing, it follows that (f (X t ) + B t ) t≥0 is a strong supermartingale under each measure P x , x ∈ E. Following Mertens we now define g(x) := P x [B ∞ ]. Then , and if T n ↑ T then P T n g(x) ↓ P T g(x) since B is leftcontinuous. Thus g is a regular strongly supermedian function. Also, Moreover, since B 0 = 0 and (f (X t ) + B t ) t≥0 is a supermartingale. Thus h is excessive.
Now suppose that f = h 0 + g 0 is a second decomposition of f into excessive and strongly supermedian components. Because g 0 is strongly supermedian, by what has already been proved we can write g 0 = h 1 + g 1 with h 1 excessive and g 1 a regular strongly supermedian function. Because h 0 is excessive, g 0 − g 0 = f −f = g −ḡ. From this and the construction of the "purely discontinuous" component g 1 it follows that g 1 = g. Therefore, g 0 = h 1 + g, so h = h 0 + h 1 specifically dominates h 0 . This establishes the case of (A.2) that we shall need.
(A.6) Remark. This proof works just as well if (f (X t )) t≥0 is of class (D) relative to each P x , and this holds if and only if P {f >n} f (x) → 0 as n → ∞ for each x ∈ E; see [Sh88;(33.3)].
The following is an analog of the classical approximation of excessive functions by potentials; it seems to have gone unnoticed in the literature.
(A.7) Corollary. If X is transient, then a strongly supermedian function f is the increasing limit of a sequence of regular strongly supermedian functions.
Proof. Use Mertens' theorem to write f as h+g, where h is an excessive function and g is a regular strongly supermedian function. Since X is transient, there is an increasing sequence (U b n ) of potentials with U b n ↑ h. Now each potential U b n is regular as is f n := U b n + g, which increases pointwise to f .
We come now to a key fact concerning strongly supermedian kernels.
(A.8) Proposition. Let V be a regular strongly supermedian kernel. Let f ∈ pE n with h = Vf bounded. If X is transient, then h is a regular strongly supermedian function.
Clearly Rg 1 ≤ Rg 2 if g 1 ≤ g 2 . Replacing V by the kernel g → V (f g), we may suppose that f = 1 in the proof. We begin by proving the following assertion, in which h := V 1.
(A.9). If (h n ) is an increasing sequence of strongly supermedian functions with h n ↑ h, This assertion is proved in [BB01b], and we repeat that proof here for the convenience of the reader. Given > 0 let A n, := {h < h n + }. Then A n, ↑ E as n → ∞ for each > 0. Now V 1 A n, ≤ h, so V 1 A n, ≤ h n + on A n, , hence everywhere by the regularity of V . Consequently, ↓ 0 as n → ∞ because h < ∞, so lim n R(h − h n ) ≤ for all > 0. This establishes (A.9). Now suppose that X is transient. By (A.7) there exists an increasing sequence (h n ) of regular strongly supermedian functions with h n ↑ h and, by (A.9), R(h − h n ) ↓ 0. Let (T k ) be an increasing sequence of stopping times with limit T . Let g be a strongly supermedian function dominating h − h n . Then P T h ≤ P T k h ≤ P T k g + P T k h n ≤ g + P T k h n , and, since h n is regular, lim k P T k h ≤ g + P T h n . Therefore where the infimum is taken over all strongly supermedian majorants g of h − h n . Letting n → ∞ in (A.10) we see that lim k P T k h = P T h, proving that h is regular. It can be shown that the two definitions are in fact equivalent.
As a final bit of preparation for the proof of Theorem (5.8), we record the following result. A parallel result for regular strongly supermedian kernels appears as Theorem 2.2 in [BB01b]; see also [A73; Thm. 5.2, p. 509].
(A.12) Proposition. Let A 1 and A 2 be PLAFs with exceptional sets N 1 and N 2 , and define B := A 1 + A 2 . Then there exist g 1 ∈ pE n and g 2 ∈ pE n with g 1 + g 2 ≤ 1 such that A j and g j * B are P x -indistinguishable for j = 1, 2 and all x / ∈ N 1 ∪ N 2 .
Proof. We can write A j t = A j,c t + 0≤s<t a j (X s ), where A j,c t is a PCAF, a j ∈ pE n , and {a j > 0} is semipolar. Then B c := A 1,c + A 2,c is the continuous part of B, and it is well known that this implies the existence of f 1 , f 2 ∈ pE n with f 1 + f 2 ≤ 1 such that A j,c and f j * B are P x -indistinguishable for all x / ∈ N 1 ∪ N 2 , where N j is an exceptional set for A j .
Proof of Theorem (5.8). As in the statement of the theorem, X is transient and V is a regular strongly supermedian kernel that is (m + ρ)-proper. Suppose, for the moment, that V 1 is bounded. If f ∈ bpE then Vf is a bounded regular strongly supermedian function by (A.8). Hence [Sh88; Thm. 38.2] implies that there is a unique PLAF A f , with empty exceptional set, such that Vf (x) = P x [A f ∞ ] for all x ∈ E. In the present context, "unique" means up to P x -indistinguishability for all x ∈ E. Theorem (38.2) in [Sh88] states that the PLAF A f is perfect as defined in [Sh88], but the proof only shows that it is "almost perfect" as defined there. In our terminology, this means that the exceptional set is empty although the defining set need not be all of Ω. The uniqueness implies that f → A f is additive and positive-homogeneous. In particular, if (f n ) ⊂ bpE is an increasing sequence with limit f ∈ bpE, then A f n ↑ A f . Define A := A 1 . If f ∈ bpE satisfies 0 ≤ f ≤ 1, then A = A f + A 1−f . These PLAFs have empty exceptional sets, so (A.12) implies that there existsf ∈ bpE n such that A f =f * A. Define an operator T by Tf :=f , so that A f = (T f ) * A. If c j ∈ R + and f j ∈ bpE for j = 1, 2, then (by uniqueness) T (c 1 f 1 + c 2 f 2 ) = c 1 T f 1 + c 2 T f 2 , U A -a.e.; that is, off a set H with U A (x, H) = 0 for all x ∈ E. Also, if f n ↑ f then T f n ↑ T f , U A -a.e. Now regard T as a map from bpE into equivalence classes of bpE n functions agreeing U A -a.e., and extend T to bE by linearity. Then T is a pseudo-kernel from (E, E n ) to (E, E) as defined in [DM83; IX.11], and by the theorem of IX.11, there exists a kernel K from (E, E n ) to (E, E) such that T f = Kf , U A -a.e., for all f ∈ bE. Thus, for f ∈ bpE, we have A f = (Kf ) * A.
We are now going to show that Kf = f , U A -a.e. To this end let f = 1 B where B ∈ E, and write A B for A 1 B and k B for K1 B . Then A B = k B * A, so V 1 B = U A k B . Recall that H B f (x) := P x [f (X D B )] for x ∈ E. Then, since V is regular and H B V 1 B is strongly supermedian, We claim that (A.14) P x Given > 0, let g 1 < g 2 < · · · be the left endpoints of the successive excursion intervals exceeding in length. Then T n := g n + is a stopping time for each n; see [De72; VI-T2]. Also, T n + D B • θ T n is the right endpoint of the n th such excursion interval. From (A.13), Recalling the dependence on and summing over = 1/k for k = 1, 2, . . ., we obtain In order to complete the proof it remains to show that (A.15) P x s∈G 1 B c (X s )k B (X s ) ∆A s = 0. Now G = G i ∪ G r , almost surely, where G i is a countable union of graphs of stopping times and G r meets the graph of no stopping time. Using the strong Markov property as before, the portion of the sum in (A.15) corresponding to G i vanishes. Finally, the process (∆A t ) t≥0 is indistinguishable from (a(X t )) t≥0 , where {a > 0} is semipolar. In fact, if u = U A 1, then one may take a := u −ū, by [Sh88; (37.7)]. But the set {t : a(X t ) > 0} is also the countable union of graphs of stopping times, so the portion of the sum in (A.15) corresponding to G r also vanishes. This establishes the claim (A.14). Now fix x ∈ E. Then U A (x, ·) is a finite measure. Let B be an open subset of E with U A (x, ∂B) = 0. Formula (A.14) implies that B c k B (y) U A (x, dy) = 0, and so k B (y) ≤ k B (y) = 0 on B c , U A (x, ·)-a.e. Hence K(·, B) = 0, U A (x, ·)-a.e. on B c since U A (x, ∂B) = 0. Using (A.14) with B replaced by B c we obtain B k B c (y) U A (x, dy) = 0, and so K(·, B c ) = 0 on B, U A (x, ·)-a.e. Recalling the meaning of K, we see that K(·, B) + K(·, B c ) = 1, U A (x, ·)-a.e. Therefore K(·, B) = 1 B , U A (x, ·)-a.e. The class of open sets B ⊂ E with U A (x, ∂B) = 0 contains a countable subcollection that generates the topology of E and is closed under finite intersections, hence Kf = f , U A (x, ·)-a.e., for each Then γ is perfectly homogeneous, optional, and is carried by Λ * . Also, it is not hard to check that γ satisfies property (3.11)(iii); consequently γ is Q m co-predictable. Since γ| Ω = λ, we have U γ (x, ·) = U A (x, ·) for all x ∈ E . By Theorem (6.15) the characteristic measure µ γ = µ A is σ-finite and charges no m-exceptional set. Thus, by (3.11) and (3.12) there is a perfect HRM κ that is Q m -indistinguishable from γ. Since κ is perfect and γ is perfectly homogeneous, an application of the strong Markov property (2.5) and the section theorem (2.6) (as in the proof of the implication (i)=⇒(iii) of (5.19)) show that the set {x ∈ E : U κ (x, ·) = U γ (x, ·)} is m-exceptional. It follows that U κ (x, ·) = U A (x, ·) for all x outside an m-exceptional set. Since V (x, ·) = U A (x, ·) for x ∈ E , this proves Theorem (5.8) when V 1 is bounded off an m-exceptional set.
In the general case we use (5.5) to find g ∈ pE n with 0 < g ≤ 1, such that V g ≤ 1 off an m-exceptional set. Define V g f := V (gf ) for f ∈ pE. Then we can apply what is proved above to find a perfect HRM κ g such that V g = U κ g off and m-exceptional set. The perfect HRM κ defined by κ(dt) := g(Y * t ) −1 κ g (dt) evidently has potential kernel equal to V off an m-exceptional set. This establishes Theorem (5.8) in full generality.
Added Note. L. Beznea and N. Boboc have recently given a positive answer to Question (7.5). Their paper "On the strongly supermedian functions and kernels" also contains new analytic proofs of (4.7) and (5.11).