Potential Theory of Moderate Markov Dual Processes P.J. Fitzsimmons and R.K. Getoor

November 12, 2008

ABSTRACT

Let X be a Borel right Markov process, let m be an excessive measure for X, and let X̂ be the moderate Markov dual process associated with X and m. The potential theory of co-excessive measures (i.e., measures that are excessive for X̂) is developed with special emphasis on the Riesz decomposition. This is then applied to obtain the Riesz decomposition of excessive functions (of X) by exploiting the correspondence between such functions and co-excessive measures. The potential theory of co-excessive measures also enables us to discuss Walsh’s interior réduite under minimal conditions. Many of the tools of the theory of Markov processes are employed in this development. For example, Kuznetsov measures, Ray compactifications, h-transforms, and duality theory for Borel right processes.

Key Words and Phrases: potential theory, excessive measure, Kuznetsov measure, duality, h-transform, Riesz decomposition, réduite, balayage

2000 Mathematics Subject Classification: Primary 60J40, secondary 60J45 1. Introduction.

The potential theory of the excessive measures of a Borel right Markov process, with a distinguished excessive measure m serving as background measure, has been well understood since the 1986 paper [FM86] of Bernard Maisonneuve and the first-named author of this paper. An expository account appears in [G90]. Slightly earlier, Joe Glover and the second-named author treated in [GG84] some aspects of the general case and obtained complete results under a weak duality hypothesis. The potential theory of the excessive functions of a general Borel right Markov process has not been treated as systematically heretofore. Of course, under weak duality there is a one-to-one correspondence between excessive functions and co-excessive measures, and in [GG84] this correspondence was exploited to treat the Riesz decomposition of excessive functions. For a general Borel right process X with distinguished excessive measure m there is always a left continuous moderate Markov dual process X̂, and there is a one-to-one correspondence between excessive functions of X and co-excessive measures (i.e., excessive measures for X̂). Thus to study excessive functions of X it is natural to develop the potential theory of the excessive measures of a left-continuous moderate Markov process. This study is carried out in the present paper when the moderate Markov process is the dual of a Borel right process X. At the end of section 7, this is applied to obtain results about the excessive functions of X.

As one might expect, the theory is quite similar to that for excessive measures. But there are important differences, and certain technical difficulties. As for excessive measures, one of the main tools is the Kuznetsov measure Q_{m} governing the Kuznetsov process Y = (Y_{t})_{t∈R} associated with X and m. In fact, X̂ is most conveniently defined in terms of Y. A key tool in the present theory, which did not arise in the study of excessive measures, is the notion of h-transforms for X. This is not surprising since such transforms play a key role in [GG84] and in Chapter 13 on duality in [CW05]. If h is an excessive function (of X) that is finite m-a.e., then ξ := h · m is a co-excessive measure. The class of co-excessive measures that can be so represented coincides with the class of those that are absolutely continuous with respect to m. In the remainder of this introductory section the co-excessive measure ξ will be taken from this class, with representation ξ = h · m, h being excessive.

In analogy with excessive measures, the Kuznetsov measure Q̂_{ξ} associated with X̂ and the co-excessive measure ξ will be a key tool. But it is well known that ξ is excessive for X^{h} (the h-transform of X) and Q̂_{ξ} is just the time reversal of Q_{m}^{h}, the Kuznetsov measure associated with X^{h} and ξ. It turns out to be more convenient to use Q_{m}^{h} rather than Q̂_{ξ} in most instances; this notation emphasizes the excessive function h, which is the eventual object of our development. We shall also make use of the Ray compactifications of X and X̂. For X this is a standard tool in the theory of Markov processes. In the case of a moderate Markov processes this material is not so well known, but it receives an excellent treatment in [CW05].

Precise definitions are given in section 2, where the basic properties of the Kuznetsov process are reviewed. The Ray-Knight compactification of X is also summarized there. Section 3 introduces the moderate Markov dual process X̂. For the most part this section is also a review. The elementary properties of h-transforms and their relationship with the Kuznetsov measure Q_{m}^{h} are recorded at the beginning of section 4. Proposition 4.3 extends to Q_{m}^{h} a standard property of h-transforms. Theorem 4.9 is the key technical fact that makes Q_{m}^{h} a useful tool for the remainder of the paper. The Ray-Knight compactification of X̂ is contained in section 5. Here we follow [CW05], although it necessary to adapt their hypotheses to our situation. We prove the key separation result of Walsh under our hypotheses in Theorem 5.3. In section 6 we come to one of the places where the potential theory of co-excessive measures differs from that of excessive measures. Namely, we are unable to prove that a purely excessive measure may be written uniquely as the integral of an entrance law. However, by relaxing the customary σ-finiteness requirement in the definition of the notion “entrance law”, we do obtain a unique representation in terms of what we call “loose” entrance laws. This is recorded as Theorem 6.3. It is somewhat surprising that one can prove the uniqueness of the representation without assuming that the measures involved are σ-finite. This allows us in Theorem 6.10 to extend the moderate Markov property to the death time β of Y in a manner analogous to the familiar extension of the strong Markov property to the birth time α of Y. See, for example, [G90; (6.15)]. Our proof is patterned on that used for the extended strong Markov property in [FM86]. The results of the preceding sections are used in section 7 to pursue our original objective of obtaining potential theoretic results for co-excessive measures (Theorem 7.9) and for excessive functions (Corollary 7.11).

Walsh has defined the “interior réduite” of an excessive function h on a set A, denoted by p_{A}h, and has obtained various properties and characterizations of p_{A}h under strong duality hypotheses and a strong Feller condition on the dual process. In section 8, p_{A}h is investigated in the framework of this paper. It turns out that the co-excessive measure p_{A}h · m is a co-balayage of the co-excessive measure ξ = h · m, similar to Hunt’s balayage for excessive measures, as re-interpreted in [FM86]. See also [G90]. Theorems 8.12 and 8.14 are direct generalizations under our minimal hypotheses of results found in section 13.12 of [CW05]. It is also observed in section 8 that the cone of co-potential measures is not a solid subcone of the cone of co-excessive measures in general. Thus a basic property of excessive measures fails for co-excessive measures.

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 shall 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).

2. 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; XII.36-37]. 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} := ∫_{0}^{∞} e^{−qt}P_{t}dt, q ≥ 0, denotes the resolvent of (P_{t}).

We are going to need the Kuznetsov process (or measure) Q_{m} associated with (P_{t}) and a given m ∈ Exc. We refer the reader to Section 6 of [G90] for notation and definitions. See also [DMM92]. For the convenience of the reader we shall review some of the basic notation here. Thus W denotes the space of all paths w : R → E_{Δ} := E ∪ {Δ} that are right continuous and E-valued on an open interval ]α(w), β(w)[ and take the value Δ outside of this interval. Here Δ is a point adjoined to E as an isolated point. The dead path [Δ], constantly equal to Δ, corresponds to the interval being empty; by convention α([Δ]) = +∞, β([Δ]) = −∞. The σ-algebra G^{∘} on W is generated by the coordinate maps Y_{t}(w) = w(t), t ∈ R, and G_{t}^{∘} := σ(Y_{s} : s ≤ t). Two families of shift operators are defined on W: the simple shifts σ_{t}, t ∈ R,

σ_{t}w(s) = [σ_{t}w](s) := w(t + s), s ∈ R,

and the truncated shifts θ_{t}, t ∈ R,

θ_{t}w(s) = [θ_{t}w](s) := \{ \begin{matrix}w(t + s), & s > 0, \\ Δ, & s ≤ 0. \end{matrix}

(In [F87], the truncated shift operator was denoted τ_{t}; here we follow [G90] in using θ_{t}.) Given m ∈ Exc, the Kuznetsov measure Q_{m} is the unique σ-finite measure on G^{∘}, not charging {[Δ]}, such that, for −∞ < t_{1} < t_{2} < … < t_{n} < +∞,

(2.1)  \begin{aligned}Q_{m}[Y_{t_{1}} ∈ dx_{1}, Y_{t_{2}} ∈ dx_{2}, & …, Y_{t_{n}} ∈ dx_{n}] \\ & = m(dx_{1})P_{t_{2}−t_{1}}(x_{1}, dx_{2}) … P_{t_{n}−t_{n−1}}(x_{n−1}, dx_{n}). \end{aligned}

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 b ∈ bE with m(b) < ∞ define W (b) ⊂ W by the conditions:

(2.2)

(i) α ∈ R,
(ii) Y_{α+} := lim_{t↓α} Y_{t} exists in E,
(iii) U^{q}g(Y_{α+1/n}) → U^{q}g(Y_{α+}) as n → ∞, for all g ∈ D and all rationals q > 0,
(iv) Ub(Y_{α+1/n}) → Ub(Y_{α+}) as n → ∞.

Evidently σ_{t}^{−1}(W (b)) = W (b) for all t ∈ R, and W (b) ∈ G_{α+}^{0} since E is a Lusin space. We now define

(2.3)  Y_{t}^{∗}(w) = \{ \begin{matrix}Y_{α+}(w), & if t = α(w) and w ∈ W (b), \\ Y_{t}(w), & otherwise. \end{matrix}

Fix m ∈ Exc and b as above. If m = η + π = η + ρU is the Riesz decomposition [G90; (5.33),(6.19)] of m into harmonic and potential parts, then Q_{m} = Q_{η} +Q_{π}, Q_{η}(W (b)) = 0, and Q_{m}( ·; W (b)) = Q_{π}. See [G90; (6.19)]. In particular, if b′ is another function with the properties of b then Q_{m}[W (b)ΔW (b′)] = 0.

As in [G90; p. 53], we shall realize X on the path space W of the Kuznetsov process. More precisely,

Ω := {w ∈ W : α(w) = 0, Y_{α+}(w) exists in E} ∪ {[Δ]},

and X_{t} is the restriction of Y_{t} to Ω if t > 0, X_{0}(ω) := Y_{0+}(ω) for ω ∈ Ω. The σ-algebras generated by X are F_{t}^{∘} := σ{X_{s} : 0 ≤ s ≤ t} and F^{∘} := σ{X_{s} : s ≥ 0}. For a proof of the following result, the strong Markov property of Y^{∗}, see [G90; (6.15)]. The filtration (G_{t}^{m})_{t∈R} is obtained by augmenting (G_{t}^{0})_{t∈R} with the Q_{m} null sets in the usual way.

(2.4) Proposition. Let T be a (G_{t}^{m})-stopping time. Then Q_{m} restricted to G_{T}^{m} ∩ {Y_{T}^{∗} ∈ E} is a σ-finite measure and

Q_{m}[F^{∘} θ_{T} |G_{T}^{m}] = P^{Y_{T}^{∗}}[F], Q_{m}-a.e. on {Y_{T}^{∗} ∈ E},

for all F ∈ pF^{∘}.

We shall also require the following form of the section theorem. Define

Γ^{∗} := {(t, w) ∈ R × W : Y_{t}^{∗}(w) ∈ E};

evidently Γ^{∗} is (G_{t+}^{∘})-optional.

(2.5) Proposition. Let (H_{t})_{t∈R} and (K_{t})_{t∈R} be positive (G_{t}^{m})-optional processes. If

Q_{m}(H_{T}; Y_{T}^{∗} ∈ E) = Q_{m}(K_{T}; Y_{T}^{∗} ∈ E)

for all (G_{t}^{m})-stopping times T, then H1_{Γ^{∗}} and K1_{Γ^{∗}} are Q_{m}-indistinguishable.

See [FG91] for a proof of (2.5).

Certain results from [F87] will be crucial for our development. We recall some definitions from [F87] and give precise references to the results we shall need. As in [F87; (3.3)] we define ℓ := _{p}(1 _{α,β} ), the co-predictable projection of 1 _{α,β} , and then Λ := {ℓ > 0}. One readily checks using the argument in [F87; (3.6)], that Λ ⊂ Λ^{∗}, modulo I^{m}, the class of Q_{m}-evanescent processes. See page 436 of [F87] . It follows that the process \overline{Y} defined in [F87; (3.8)] is related to Y^{∗} defined above in (2.3) as follows:

\overline{Y}_{t}(w) = \{ \begin{matrix}Y_{t}^{∗}(w), & if (t, w) ∈ Λ, \\ Δ, & if (t, w) ∉ Λ. \end{matrix}

In particular, Λ = \{(t, w) : \overline{Y}_{t}(w) ∈ E\}. Many of the definitions and results in [F87] involve \overline{Y} and Λ. We shall need the extensions of these results in which \overline{Y} and Λ are replaced by Y^{∗} and Λ^{∗}. The keys to these extensions are the strong Markov property (2.4) and the section theorem (2.5). Using them in place of (3.10) and (3.16)(b) in [F87], the results we require are proved with only minor modifications of the arguments given in [F87]. In general, we shall use the corresponding result with Y^{∗} and Λ^{∗} without special mention.

An arbitrary subset A ⊂ E is m-exceptional provided A is contained in a Borel mpolar subset B ⊂ E with ρ(B) = 0, where m = η + ρU is Riesz decomposition of m into its harmonic and potential parts. It is known that B ∈ E is m-polar (resp. m-exceptional) if and only if {(t, w) : Y_{t}(w) ∈ E} (resp. {(t, w) : Y_{t}^{∗}(w) ∈ E}) is Q_{m}-evanescent. See [FG91]. The collection of m-exceptional sets is denoted by N (m). A property that holds off an m-polar set is said to hold m-quasi-everywhere or just quasi-everywhere if m is understood; in symbols m-q.e. or just q.e.

Another key ingredient in what follows is a Ray-Knight compactification \overline{E} of E_{Δ}. See [Sh88; §17-18] or [G75]. Since this is a Ray-Knight compactification of E_{Δ}, Δ is regarded as an ordinary trap in the definition of \overline{E}. Thus U^{q}(Δ, ·) = ε_{Δ} and U^{q}(x, (\{Δ\})) = q^{−1}P^{x}[e^{−qζ}] for x ∈ E, q > 0. In particular \overline{E} is a compact metric space in which E_{Δ} sits as a dense Borel subset, since E is Lusinian. The topology E_{Δ} inherits as a subspace of \overline{E} is the Ray topology of E_{Δ}. Relative to this topology, t ↦ X_{t} is right continuous (rc) on [0, ∞[ and has left limits in \overline{E} on ]0, ∞[ (denoted by X_{t−}^{r}, t > 0), almost surely. We emphasize that X_{t} ∈ E_{Δ} for t ≥ 0 while X_{t−}^{r} ∈ \overline{E} for t > 0. Of course since X is rc in the (original) topology of the Lusin space E, t ↦ X_{t}(ω) can have at most a countable number of discontinuities in the original topology. In particular the left limit X_{t−}, in the original topology, exists in E for all but a countable number of t, the exceptional t-set depending on ω. For the relationship between X_{t−}^{r} and X_{t−} see [Sh88; §46]. Using (2.4) one checks that Q_{m}-a.s., t ↦ Y_{t} is Ray rc (i.e., right continuous in the Ray topology) on ]α, ∞[ and has Ray left limits in \overline{E} on all of R. Since m ∈ Exc is arbitrary, we may re-define W to incorporate this behavior. Therefore in the remainder of the paper we shall suppose that:

(2.6) Definition. W consists of all maps w : R → E_{Δ} which are E-valued on a non-empty open interval ]α(w), β(w)[, take the value Δ outside this interval and are such that

(i) w is rc on ]α(w), ∞[ in the topology of E_{Δ};

(ii) w is Ray rc on ]α(w), ∞[ and has left limits, Y_{t−}^{r}(w) ∈ \overline{E} on R.

Note that if w ∈ W (b), then t ↦ Y_{t}^{∗}(w) is right continuous on R, in both the original topology of E and in the Ray topology.

(2.7) Remark. Since X is Borel right process in the Ray topology, one may define W^{r}(b) analogously to W (b) when E is given the Ray topology. In this case (iii) of (2.2) is automatically satisfied and so W^{r}(b) is characterized by (i), (ii) and (iv) of (2.2) with Y_{α+} replaced by Y_{α+}^{r} := lim_{t↓α} Y_{t} where the limit is taken in the Ray topology. The decomposition m = η + π into harmonic and potential parts depends only on the resolvent and so just as in the discussion below (2.3), Q_{m}(W (b)ΔW^{r}(b)) = 0.

In general, we adhere to the usual convention that a function, f, defined on E is extended to Δ by declaring f(Δ) = 0 unless explicitly stated otherwise, the construction of \overline{E} being such an exception. We also write X(t) for X_{t} and Y (t) for Y_{t} when convenient.

Similarly, X(t, ω) = X_{t}(ω), etc.

3. The Moderate Markov Dual Process.

Because of the time-symmetry of the Markov Property, the process (Y_{t}, Q_{m}) is a Markov process with respect to the reverse filtration G_{≥t}^{0} := σ{Y_{s}; s ≥ t}, t ∈ R. Unlike the situation in “forward” time, this process need not be a strong Markov process, but it is a moderate Markov process. In making this precise we follow [F87]. However our notation differs slightly from that used there; in particular we use Y^{∗} in place of \overline{Y}, as explained in section 2.

Let G^{m} be the Q_{m} completion of G^{∘}, and let G_{≥t}^{m} denote the σ-algebra generated by G_{≥t}^{∘} and the Q_{m} null sets in G^{m}. The co-predictable σ-algebra P̂^{∘} of subsets of R × W is defined by

(3.1) P̂^{∘} := σ{Z ∈ B(R) × G^{∘} : Z is (G_{≥t}^{∘}) adapted, t ↦ Z_{t} is rc and Z_{t} = 0 on [β, ∞[}.

The Q_{m}-co-predictable σ-algebra P̂^{m} consists of sets which differ from a co-predictable set by a Q_{m}-evanescent set. One checks that Y^{∗} ∈ P̂^{∘}; see [G99; p. 106]. A map T : W → [−∞, ∞] is a co-stopping time (resp. m-co-stopping time) provided {T ≥ t} ∈ G_{≥t}^{∘} (resp. {T ≥ t} ∈ G_{≥t}^{m}) for each t ∈ R. A map T : W → [−∞, ∞] is co-predictable (resp. m-co-predictable) provided 1 _{−∞,T} ∈ P̂^{∘} (resp. P̂^{m}). Clearly a co-predictable (resp. m-co-predictable) T is a co-stopping time (resp. m-co-stopping time). Associated with a co-predictable (resp. m-co-predictable) T is the σ-algebra G_{>T}^{∘} (resp. G_{>T}^{m}) generated by sets of the form Γ ∩ {T < t} with Γ ∈ G_{≥t}^{0} (resp. G_{≥t}^{m}), t ∈ R. In the sequel we shall just make statements in the unaugmented case and leave the obvious extension to the reader.

Of course co-predictable means predictable “with time reversed”. In order to make this precise define ŵ(t) := w(−t) for w ∈ W and Ŵ := {ŵ : w ∈ W }. Define, for t ∈ R, the coordinate maps Ŷ_{t}(ŵ) := ŵ(t) and the σ-algebras Ĝ^{∘} := σ{Ŷ_{t}; t ∈ R}, Ĝ_{t}^{∘} := σ{Ŷ_{s}; s ≤ t}. Define the reversal operator r : W → Ŵ by rw(t) := ŵ(t). Then r^{−1} : Ŵ → W is given by r^{−1}ŵ(t) = ŵ(−t). Observe that r is an isomorphism between (W, G^{∘}) and (Ŵ, Ĝ^{∘}) with the property that r(G_{≥t}^{∘}) = Ĝ_{−t}^{∘} and r^{−1}Ĝ_{t}^{∘} = G_{≥−t}^{∘}. One readily checks that T is copredictable if and only if T̂ := −T^{∘}r^{−1} is (Ĝ_{t}^{∘})-predictable, in which case G_{>T}^{∘} = r^{−1}Ĝ_{_{T}−}^{∘}. This enables one to translate statements about predictable times into the corresponding statements about co-predictable times and conversely. For example, since Y^{∗} ∈ P̂^{0}, if T is co-predictable and we set Y_{T}^{∗} = Δ on {| T |= ∞}, then Y_{T}^{∗} ∈ G_{>T}^{0} . The corresponding statement in the predictable case is [DM75; IV-67]. (The reader is cautioned that in the 1978 English translation of [DM75], the statement of this result contains an unfortunate misprint.) Similarly T is a co-stopping time if and only if T̂ := −T^{∘}r^{−1} is a (Ĝ_{t}^{∘}) stopping time. Define G_{≥T}^{∘} to consist of all Γ ∈ G^{∘} such that Γ ∩ {T ≥ t} ∈ G_{≥t}^{∘} for all t ∈ R. Then

G_{≥T}^{∘} = r^{−1}Ĝ∘ ; _{T}.

We are now prepared to describe the moderate Markov process X̂ associated with X and m ∈ Exc. Define

(3.2)  Ω̂ := {β = 0} ∪ {[Δ]} ⊂ W;
(3.3)  X̂_{t}(ω̂) := Y_{−t}^{∗}(ω̂), t > 0, ω̂ ∈ Ω̂;
(3.4)  F̂_{t}^{∘} := σ{X̂_{s}; 0 < s ≤ t}, F̂^{∘} = σ{X̂_{s}; s > 0},

and for t ∈ R

(3.5)  θ̌_{t}w(s) = \{ \begin{matrix}w(t + s), & s < 0, \\ Δ, & s ≥ 0; \end{matrix}

and

(3.6)  θ̂_{t} = θ̌_{−t}.

Note that θ̌_{t}({t ≤ β}) ⊂ Ω̂ and that X̂_{s}^{∘} θ̂_{t} = X̂_{s+t} for s > 0, t ≥ 0. In [F87], θ̌_{t} is denoted by τ̂_{t}. It is proved in the Appendix of [F87] that there exists a Borel measurable family {P̂^{x}, x ∈ E_{Δ}} of probability measures on (Ω̂, F̂^{∘}) under which {X̂_{t}, t > 0} has the moderate Markov property (MMP);

(3.7)  P̂^{x}[F^{∘} θ̂_{T} | F̂_{T −}] = P̂^{_{X}(T)}(F) on {T̂ < ∞},

for an (F̂_{t})-predictable time T and F ∈ pF̂, where F̂ is the intersection over all finite μ of the P̂^{μ} completion of F̂^{∘} and F̂_{t} is the usual augmentation of F̂_{t}^{∘} in F̂. (P̂^{Δ} is unit mass at {[Δ]}.) The relationship between Q_{m} and (P̂^{x}) is expressed as follows: For each co-predictable T, G ∈ pĜ_{>T}^{∘} and F ∈ pF̂^{∘},

(3.8)  Q_{m}[F^{∘} θ̌_{T} · G; −∞ < T < β] = Q_{m}[P̂^{Y^{∗}(T)}[F] · G; −∞ < T < β].

The family {P̂^{x}} is uniquely determined modulo m-exceptional sets by (3.7) and (3.8).

Note that F̂^{∘} = G^{∘} |_{Ω} and ζ̂ = (−α) ∨ 0. Clearly X̂ is left continuous on ]0, ζ̂[ in both the original and Ray topologies and has Ray right limits in \overline{E} on [0, ∞[. In fact, on Ω̂ ∩ W (b), X̂ is left continuous on ]0, ∞[. However it is important to note that X̂ is (F̂_{t}^{∘})-predictable because Y^{∗} is co-predictable. See the discussion following Lemma 2.9 in [G90].

The moderate Markov dual semigroup (P̂_{t}, t > 0) and resolvent (Û^{q}, q ≥ 0) are defined by

P̂_{t}f(x) := P̂^{x}[f^{∘}X̂_{t}] = P̂^{x}[f^{∘}X̂_{t}; t < ζ̂]

Û^{q}f(x) := ∫_{0}^{∞} e^{−qt}P̂_{t}f(x)dt = P̂^{x} ∫ _{ζ} ; 0e^{−qt}f^{∘}X̂_{t}dt,

with the usual convention f(Δ) = 0. If f, g ∈ pE^{∗} one has the duality relationships

(3.9)  (P_{t}f, g) = (f, P̂_{t}g) and (U^{q}f, g) = (f, Û^{q}g),

where (f, g) = ∫ fg dm whenever the integral makes sense. Of course P̂_{t}(x, ·) and Û^{q}(x, ·) depend on m and are uniquely determined modulo N (m). In what follows we shall usually omit the hat “ ” where it is obviously required. For example we shall write P̂^{x}[f^{∘}X_{t}] in place of P̂^{x}[f^{∘}X̂_{t}] and P̂^{x} ∫_{0}^{∞} e^{−qt}f^{∘}X_{t} dt for P̂^{x} ∫_{0}^{∞} e^{−qt}f^{∘}X̂_{t} dt.

Since X is a right process P^{x}(ζ > 0) = 1 for all x ∈ E. But this may fail for the moderate Markov process X̂. Indeed, P̂^{x}(ζ = 0) = 1 is possible, but the set of such x is m-polar according to the next result.

(3.10) Lemma. Define N̂ := {x ∈ E : P̂^{x}(ζ = 0) = 1}. Then N̂ is m-polar.

Proof. Observe that N̂ = {x : Û^{q}1(x) = 0} for some, and hence every, q ≥ 0. Let φ(x) := P̂^{x}(ζ = 0). For each x ∈ E

\begin{aligned}Û^{q}φ(x) & = P̂^{x} ∫_{0}^{∞} e^{−qt}P̂^{X(t)}(ζ = 0) dt \\ & = P̂^{x} ∫_{0}^{∞} e^{−qt}1_{\{ζ=t\}} dt = 0. \end{aligned}

Hence 0 = qmÛ^{q}φ ↑ m(φ) as q → ∞. But 1_{N} ≤ φ, so m(N̂) = 0. Therefore P̂^{m}(X_{t} ∈ N̂) = mP̂_{t}(N̂) ≤ m(N̂) = 0 for t > 0. In other words P̂^{m}[Û^{q}1 ∘ X_{t} = 0] = 0 for each t > 0. But t ↦ Û^{q}1^{∘}X̂_{t} is left continuous on ]0, ∞[, almost surely; see, for example, [G99; (2.6)] and the paragraph immediately following it. Consequently P̂^{m}(Û^{q}1 ∘ X_{t} = 0 for some t > 0) = 0. Combining this with (3.8) we find that {Y ∈ N̂} is Q_{m}-evanescent, and so N̂ is m-polar.

Remark. It is not the case that N̂ is m-exceptional, in general, as the following example shows. Let X be translation to the right at unit speed on [0, ∞[ and let m = ε_{0}U. Then P̂^{0}(ζ = 0) = 1 and {0} is m-polar but not m-exceptional.

4. h-Transforms.

We now fix a version of the moderate Markov kernels {P̂^{x}, x ∈ E_{Δ}} and the corresponding semigroup (P̂_{t}, t > 0). Let S denote the class of excessive functions (for X) and S(m) the class of h ∈ S such that h is Borel measurable and h < ∞, m-a.e. If h ∈ S(m), ξ := h · m is σ-finite and if f ∈ pE,

ξP̂_{t}f = (h, P̂_{t}f) = (P_{t}h, f) ↑ ξ(f) as t ↓ 0.

Thus ξ ∈ Êxc, the class of excessive measures relative to (P̂_{t}). It is known and relatively easy to check that if ξ ∈ Êxc and ξ ≪ m, then ξ = h · m with h ∈ S(m). See, for example, [GS84; §6].

If h ∈ S(m), define E_{h} := {0 < h < ∞} ∈ E. Recall that the h-transform semigroup (P_{t}^{h}, t ≥ 0) is defined on E_{h} by

(4.1)  P_{t}^{h}f(x) := h(x)^{−1}P_{t}(hf)(x), x ∈ E_{h},

and for definiteness set P_{t}^{h}(x, ·) = 0 for x ∈ E \ E_{h}. However for our purposes the precise value on E \ E_{h} is not particularly relevant. We refer the reader to the excellent discussion of h-transforms in Chapter 11 of the recent book [CW 05]. If h ∈ S(m), ξ = h · m is carried by E_{h} and

ξP_{t}^{h}f = mP_{t}(fh) ↑ ξ(f) as t ↓ 0.

Hence, using a self-explanatory notation, ξ ∈ Exc^{h}. Associated with the pair (ξ, P_{t}^{h}) there exists a Kuznetsov measure on (W, G^{∘}) that we denote by Q_{m}^{h}. The finite-dimensional distributions of Q_{m}^{h} are given by

(4.2)  \begin{matrix}Q_{m}^{h}(Y_{t_{1}} ∈ dx_{1}, …, & Y_{t_{n}} ∈ dx_{n}) \\ = m(dx_{1})P_{t_{2}−t_{1}}(x_{1}, dx_{2}) … P_{t_{n}−t_{n−1}}(x_{n−1}, x_{n})h(x_{n}), \end{matrix}

for −∞ < t_{1} < t_{2} < … < t_{n} < ∞. This also is the Kuznetsov measure associated with the stationary entrance law m and stationary exit law h. See [K73].

The following proposition extends a familiar result on h-transforms to the current situation. Using the pattern of notation in section 3, (G_{t}^{h·m}) is the filtration obtained by augmenting (G_{t}^{∘}) by all Q_{m}^{h} null sets in the Q_{m}^{h} completion G^{∘}. Note that G_{t}^{m} ⊂ G_{t}^{h·m}.

(4.3) Proposition. Let T be a (G_{t}^{m}) stopping time. Then for Γ ∈ G_{T}^{m},

(4.4)  Q_{m}^{h} [Γ ∩ {−∞ < T < β}] = Q_{m} [h^{∘}Y_{T}; Γ ∩ {−∞ < T < β}] .

Proof. Decomposing m = m_{c} + m_{d} into its conservative and dissipative components it suffices to prove (4.3) in the two cases separately. Also it suffices to suppose that T is a (G_{t}^{∘}) stopping time and that Γ ∈ G_{T}^{∘}. It is easy to see that (4.4) is valid if T ≡ t ∈ R by noting that it holds for Γ = ⋂_{j=1}^{n}{Y_{t_{j}} ∈ B_{j}}, t_{1} < ⋯ < t_{n} ≤ t, B_{j} ∈ E. For the general case

let a(k, n) := k2^{−n} and define T_{n} := a(k, n) on Γ(k, n) := {a(k −1, n) ≤ T < a(k, n)}, n ≥ 1, k = 0, ±1, … and T_{n} := T on {| T |= ∞}. Then T_{n} ↓↓ T on {−∞ < T < β} and

Q_{m}^{h}(Γ; −∞ < T < β) =↑ lim_{n} ∑_{k} Q_{m}^{h}(Γ ∩ Γ(k, n); a(k, n) < β)

=↑ lim_{n} ∑_{k} Q_{m}^{h}(Γ; T_{n} = a(k, n) < β).

Since Γ ∈ G_{T}^{0} , {a(k − 1, n) ≤ T } ∩ (Γ ∩ {T < a(k, n)}) ∈ G_{a(k,n)}^{∘} and so for each n and k using (4.4) for the constant time a(n, k),

Q_{m}^{h}(Γ; T_{n} = a(k, n) < β) = Q_{m}(Γh ∘ Y_{a(k,n)}; T_{n} = a(k, n) < β).

Therefore setting Y_{±∞} = Δ for convenience

(4.5)  Q_{m}^{h}[Γ; −∞ < T < β] =↑ lim_{n→∞} Q_{m}[h ∘ Y_{T_{n}}; Γ].

Now suppose that m is dissipative. Then according to the argument leading to [G90; (2.14)] there exist a Borel absorbing set, A, carrying m and an increasing sequence (Uf_{k}) of potentials with f_{k} ≥ 0 and Uf_{k} ↑ h on A. Therefore g := ↑ lim Uf_{k} is excessive and g = h, m-a.e. Hence {g ≠ h} is m-polar and Uf_{k} ↑ h except on an m-polar set. This in turn implies that if T is a (G_{T}^{∘}) stopping time, then Uf_{k} ∘ Y_{T} ↑ h ∘ Y_{T}, Q_{m}-a.e. Consequently,

Q_{m}[h ∘ Y_{T_{n}}; Γ] =↑ lim_{k→∞} Q_{m}[Uf_{k}^{∘}Y_{T_{n}}; Γ].

Also Γ ∈ G_{T}^{∘} ⊂ G_{T_{n}}^{∘} since T ≤ T_{n} and so

Q_{m}(Uf_{k} ∘ Y_{T_{n}}; Γ) = Q_{m} (∫_{T_{n}}^{∞} f_{k} ∘ Y_{t} dt; Γ) .

Therefore each of these Q_{m} integrals increases with both n and k. Hence

Q_{m}^{h}(Γ ∩ {−∞ < T < β}) = lim_{n} lim_{k} Q_{m} (∫_{T_{n}}^{∞} f_{k} ∘ Y_{t} dt; Γ)

= lim_{k} Q_{m}[Uf_{k} ∘ Y_{T}; Γ] = Q_{m}[h^{∘}Y_{T}; Γ],

establishing (4.4) for m ∈ Dis.

It remains to prove (4.4) for m ∈ Con. Suppose m ∈ Con. Then Q_{m}-a.e., α = −∞ and β = ∞. Also it follows from [G90; (2.9)] or [G90; (2.15)] that Q_{m}-a.e., t ↦ h ∘ Y_{t} is constant on R. Let H := sup_{t} h ∘ Y_{t}. Then {t : h ∘ Y_{t} < H} is Q_{m}-evanescent and H < ∞, Q_{m}-a.e., since h < ∞, m-a.e. It now follows from (4.5) that

Q_{m}^{h}(Γ ∩ {−∞ < T < β}) = Q_{m}(H; Γ) = Q_{m}(h ∘ Y_{T}; Γ),

establishing (4.4) for m ∈ Con.

One may use the results of section 3 for ξ = h · m, P_{t}^{h} and Q_{m}^{h} to obtain a moderate Markov family (Π̂^{x}, x ∈ E_{Δ}) depending on h and m for which the analogs of (3.7) and (3.8) hold. But for what follows we shall need the fact that it is not necessary to introduce a new kernel; the kernel P̂^{x} from section 3 suffices for all h ∈ S(m). This will be made precise in Theorem 4.9. However we shall need some preliminaries before coming to its statement. Because Q_{m}^{h} is σ-finite and stationary it is immediate that for each t ∈ R,

(4.6)  Q_{m}^{h}(α = t) = 0 = Q_{m}^{h}(β = t).

In particular Q_{m}^{h}(Y_{t}^{∗} ≠ Y_{t}) ≤ Q_{m}^{h}(α = t) = 0.

(4.7) Lemma. If F ∈ pF̂^{∘}, G ∈ pG_{≥t}^{0} and t ∈ R, then

\begin{aligned}Q_{m}^{h}[F ∘ θ̌_{t} · G; t < β] & = Q_{m}^{h}[P̂^{Y^{∗}(t)}(F) · G; t < β] \\ & = Q_{m}^{h}[P̂^{Y (t)}(F) · G; t < β]. \end{aligned}

Proof. The second equality follows immediately because Q_{m}^{h}(Y_{t}^{∗} ≠ Y_{t}) = 0. To see the first equality fix t ∈ R. Suppose F ∈ pF_{t}^{∘}, t ≤ u_{1} < ⋯ < u_{n}, and G = ∏_{i=1}^{n} g_{i} ∘ Y_{u_{i}} where g_{i} ∈ pE. Then, using (3.8) for the second equality below,

\begin{aligned}Q_{m}^{h}[F ∘ θ̌_{t} · G; t < β] & = Q_{m}[F ∘ θ̌_{t} · G · h^{∘}Y_{u_{n}}] \\ & = Q_{m}[P̂^{Y^{∗}(t)}(F) · G · h^{∘}Y_{u_{n}}] = Q_{m}^{h}[P̂^{Y^{∗}(t)}(F) · G]. \end{aligned}

Let 0 < g ≤ 1, g ∈ E with m(gh) < ∞. Fix F ∈ pbF̂^{∘}, and for G ∈ pbG_{≥t}^{∘} define

\begin{aligned}μ(G) & := Q_{m}^{h}[F ∘ θ̌_{t}g ∘ Y_{t} · G] \\ ν(G) & := Q_{m}^{h}[P̂^{Y^{∗}(t)}(F)g ∘ Y_{t} · G]. \end{aligned}

Then μ and ν are finite measures on F̂^{∘} which agree on G of the form G = ∏_{i=1}^{n} g_{i} ∘ Y_{u_{i}} and hence for all G ∈ bpG_{≥t}^{∘}. Finally let g ↑ 1 through a sequence (g_{k}) with 0 < g_{k} ≤ 1 and m(hg_{k}) < ∞ for each k to complete the proof of (4.7).

Before coming to the main result of this section we need to introduce a convenient metric on Ω̂, following the appendix of [F87]. Since E is Lusin there exists a totally bounded metric d, say bounded by 1, on E compatible with the topology of E and this is extended to E_{Δ} by setting d(x, Δ) = 2 for x ∈ E and d(Δ, Δ) = 0. Next, define a metric ρ on Ω̂ = {w : β(w) = 0} ∪ {[Δ]} by

ρ(w, w′) = ∫_{−∞}^{0} e^{t}d[w(t), w′(t)]dt.

Note that Ω̂ consists of functions from R to E_{Δ} that are right continuous except at α < 0 and are constantly equal to Δ on [0, ∞[. Thus the elements of Ω̂ may be thought of as functions on ] − ∞, 0[ and this is convenient at times. Clearly ρ is a metric on Ω̂ bounded by 2 and the topology induced by ρ is the topology of convergence in measure relative to η(dt) := e^{t}dt on ] − ∞, 0[. The next lemma contains the properties of the metric space (Ω̂, ρ) that we shall need.

(4.8) Lemma.

(i) (Ω̂, ρ) is separable.

(ii) If B(Ω̂) denotes the Borel σ-algebra associated with the metric space (Ω̂, ρ), then

B(Ω̂) = F̂^{0}. (iii) There exists a countable class C(ρ) of ρ-uniformly continuous functions from Ω̂

to [0, 1], closed under finite products, such that F̂^{∘} = B(Ω̂) = σ[C(ρ)].

Proof. Property (i) is easily checked. For example let D be a countable dense subset of E_{Δ} with Δ ∈ D and let D be the D-valued elements of Ω̂ that are constant on each dyadic interval I_{n,k} := [(k − 1)2^{−n}, k2^{−n}[, 1 ≤ n ≤ N, −N ≤ k ≤ 0, N ≥ 1. Then D is easily seen to be dense in Ω̂. For (ii) let \overline{E_{Δ}} be the compact completion of (E_{Δ}, d), and fix f ∈ pC(\overline{E_{Δ}}). Then f is bounded and d-uniformly continuous on E_{Δ}. For w ∈ Ω̂, set for n ≥ 1, t ∈] − ∞, 0[,

ϕ_{n,t,f}(w) := ne^{−t} ∫_{t}^{t+1/n} e^{s}f ∘ w(s) ds.

Note that ϕ_{n,t,f}(w) → f ∘ w(t) as n → ∞ provided t ≠ α(w). If w_{k} → w in Ω̂, then f ∘ w_{k} → f ∘ w in η measure, and so ϕ_{n,t,f}(w_{k}) → ϕ_{n,t,f}(w) by the bounded convergence theorem. Therefore ϕ_{n,t,f} is continuous on Ω̂, and hence B(Ω̂) measurable. Since Δ is isolated in \overline{E_{Δ}}, f := 1 on \overline{E_{Δ}} \ \{Δ\} and f(Δ) = 0 is uniformly continuous on \overline{E_{Δ}}. Then ϕ_{t}(w) := ∫_{−∞}^{t} e^{s}f ∘ w(s) ds is continuous on Ω̂ for t < 0. But {α ≥ t} = {ϕ_{t} = 0} and so α ∈ B(Ω̂). Combining these observations, 1_{{α≠t}}f ∘ Y_{t} is B(Ω̂) measurable and since 1_{{α=t}}f ∘ Y_{t} = 1_{{α=t}}f(Δ), it follows that f ∘ Y_{t} |_{Ω} is B(Ω̂) measurable for t < 0 and f ∈ C(\overline{E_{Δ}}). Consequently F̂^{∘} = G^{∘} |_{Ω} ⊂ B(Ω̂).

For the opposite inclusion suppose that G ⊂ Ω̂ is open. Then there exists an increasing sequence (G_{n}) of open sets with G_{n} ⊂ \overline{G}_{n} ⊂ G_{n+1} ⊂ G with G_{n} ↑ G. Let f_{n}(w) := [nρ(w, \overline{G}_{n}^{c})] ∧ 1 and note that f_{n} ↑ 1_{G}. If w′ ∈ G is fixed, w ↦ ρ(w, w′) is F̂^{∘} measurable, and since \overline{G}_{n}^{c} is open, if D is a countable dense subset of Ω̂, then

w ↦ ρ(w, \overline{G}_{n}^{c}) = inf\{ρ(w, w′) : w′ ∈ D ∩ \overline{G}_{n}^{c}\}

is F̂^{∘} measurable. Therefore G ∈ F̂^{∘} and this establishes (ii).

For (iii) let (G_{n}) be a countable base for the topology of Ω̂. For each n, there exists a sequence (F_{n,k}) of ρ-uniformly continuous functions such that 0 ≤ F_{n,k} ↑ 1_{G_{n}} as k → ∞. Then the closure under finite products of {F_{n,k}; n ≥ 1, k ≥ 1} has the required properties.

We now come to the main result of this section.

(4.9) Theorem. . Let T be Q_{m}^{h} co-predictable, F ∈ pF̂^{∗}, G ∈ G_{>T}^{h·m}. Then

Q_{m}^{h}[F^{∘} θ̌_{T}; G ∩ {−∞ < T < β}] = Q_{m}^{h}[P̂^{Y^{∗}(T)}(F); G ∩ {−∞ < T < β}].

Note that {Y_{T}^{∗} ∈ E} ⊂ {−∞ < α ≤ T < β}.

Proof. As usual it suffices to consider T co-predictable, G ∈ bpG_{>T}^{∘} and F ∈ pF̂^{∘}. There exists a sequence (T_{n}) of co-predictable times each of which takes dyadic rational values such that T_{n} > T on {T < ∞} and T_{n} ↓ T, Q_{m}^{h}-a.e. See [DM; IV.77] for the corresponding result for predictable times. Let D denote the set of dyadic rationals. For each n

Q_{m}^{h}[F ∘ θ̌_{T_{n}} · G; T_{n} < β] = ∑_{s∈D} Q_{m}^{h}[F ∘ θ̌_{s} · G; T_{n} = s < β].

For the moment fix n and suppose that G ∈ bpG_{>T_{n}}^{∘}. If p ≥ n, then T_{p} ≤ T_{n} and G ∈ G_{>T_{n}}^{∘} ⊂ G_{>T_{p}}^{∘}. Hence G ∩ {T_{p} = s} ∈ G_{≥s}^{∘}; [DM; IV-56]. Bringing in Lemma 4.7 we find that for p ≥ n

(4.10)  \begin{aligned}Q_{m}^{h}[F ∘ θ̌_{T_{p}} · G; T_{p} < β] & = ∑_{s∈D} Q_{m}^{h}[P̂^{Y^{∗}(s)}(F) · G; T_{p} = s < β] \\ & = Q_{m}^{h}[P̂^{Y^{∗}(T_{p})}(F) · G; T_{p} < β]. \end{aligned}

We now adapt the argument in the appendix of [F87]. Let g ∈ pE, g > 0 with m(gh) < ∞. Write (4.10) with G replaced by g∘Y_{t}·G1_{{T_{n}<t}} ∈ pbG_{>T_{n}}^{∘} and F ∈ C(ρ) (defined (4.8)(iii)). For each w ∈ W, t ↦ θ̌_{t}w is rc as a map from ] − ∞, β(w)[ to Ω̂. Thus letting p → ∞ the left side of (4.10), with the above replacement, approaches

Q_{m}^{h}[F ∘ θ̌_{T} · G · g ∘ Y_{t}1_{{T_{n}<t}}; T < β]

for F ∈ C(ρ), G ∈ pbG_{>T_{n}}, g as above and t ∈ R. It follows from (3.8) that t ↦ Z_{t} := P̂^{Y^{∗}(t)}(F) is a Q_{m} version of the co-predictable projection of the rc process t ↦ 1_{]−∞,β[}(t)F ∘ θ̌_{t}, F ∈ C(ρ). Consequently Z is rc on ] − ∞, β[, Q_{m}-a.e.. See Theorem 47 and the footnote on p. 120 of [DM80]. Moreover Z is (G_{t}^{∘}) adapted since F ∈ C(ρ). Now let (W, G_{t}^{Q}, G^{Q}) denote the usual right continuous completion of (W, G_{t}^{∘}, G^{∘}) with respect to the σ-finite measure Q := Q_{m} + Q_{m}^{h}. According to [DM75; IV-33], \overline{Z}_{t} := lim sup_{s↓↓t} Z_{s} is

(G_{t}^{Q}) progressive. For t ∈ R define

Γ := \{(s, w) : s ≤ t < β(w), \overline{Z}_{s}(w) ≠ Z_{s}(w)\},

and let Γ_{t} be the projection of Γ on W. Then Γ_{t} ∈ G_{t}^{Q} and Q_{m}(Γ_{t}) = 0. If 0 < f ≤ 1 with m(hf) < ∞ then from (4.3),

Q_{m}^{h}(Γ_{t} · f ∘ Y_{t}) = Q_{m}(Γ_{t}(fh) ∘ Y_{t}) = 0,

and hence Q_{m}^{h}(Γ_{t}) = 0. Combining this with the same argument in which \overline{Z} is replaced by \underline{Z}_{t} := lim inf_{s↓↓t} Z_{s}, we see that s ↦ Z_{s} is rc on ] − ∞ < t[, t < β, Q_{m}^{h}-a.e., and since t ∈ R is arbitrary Z is rc on ] − ∞, β[, Q_{m}^{h}-a.e. Consequently the right side of (4.10) with G replaced by G · g(Y_{t}) 1_{{T_{n}<t}} and F ∈ C(ρ) approaches

Q_{m}^{h}[P̂^{Y^{∗}(T)}(F) · g ∘ Y_{t} · G; T_{n} < t, T < β]

as p → ∞. Hence

Q_{m}^{h}[F^{∘} θ̌_{T} g^{∘}Y_{t} · G; T_{n} < t, T < β] = Q_{m}^{h}[P̂^{Y^{∗}(T)}[F] · g^{∘}Y_{t} · G; T_{n} < t, T < β]

for F ∈ C(ρ), G ∈ bpG_{>T_{n}}, g as before, and t ∈ R. Using the defining property of C(ρ), the above is then seen to hold for all F ∈ pF̂_{0}. Finally let g ↑ 1 through a sequence and then t ↑ ∞ to complete the proof of (4.9), since T_{n} < ∞ on {T < β}.

5. Ray-Knight Compactification for X̂.

In this section we are going to establish the existence of a Ray-Knight compactification for the dual process X̂. We would like to apply Theorems 8.30 and 8.45 in [CW05] to obtain such a compactification. However the hypotheses of those theorems are not quite satisfied. The first issue is that hypothesis (MMP) on p. 272 of [CW05] assumes that t ↦ X̂_{t} has right limits on [0, ∞[. This is easily overcome by looking at X in its Ray topology. Recall from section 2 that \overline{E} is a Ray-Knight compactification of E_{Δ} for X, and as pointed out in section 3, X̂ is left continuous on ]0, ∞[ with right limits in \overline{E} on [0, ∞[ in the Ray topology of X. It is not difficult to see that this suffices for the construction. The second issue is that the key separation Theorem 8.45 requires that the resolvent be “compatible” [CW05; Def. 8.48], and (Û^{q}) need not be compatible. Chung and Walsh show (in [CW05; Lem. 8.49]) that under their hypotheses the assumption of compatibility entails no essential loss of generality. But the proof uses the fact that X̂ has right limits in E, whereas in our situation these right limits are only known to exist in \overline{E}. However it turns out that the proof of Theorem 8.45 goes through under our hypotheses with some minor modifications. Since this is the key technical fact for the construction and involves a beautiful argument due originally to Walsh [W71], we shall give it here for the convenience of the reader. In addition the argument is somewhat simpler in the present situation. In the remainder of this section topological statements refer to the Ray topology (2.6).

Remark. If it is known a priori that X̂ has right limits in E in the original topology then one may use the original topology rather than the Ray topology in the construction of this section. For example this will be the case if X has left limits in E in the original topology.

Recall that C(\overline{E}) denotes the set of continuous functions on \overline{E}. The set of provisional co-branch points B̂_{0} ⊂ E is defined as follows:

(5.1)  E \ B̂_{0} := \{x ∈ E : qÛ^{q}f(x) → f(x) as q → ∞ for all f ∈ C(\overline{E})\}.

Since C(\overline{E}) maybe replaced by a countable dense subset of C(\overline{E}) without altering B̂_{0}, E \ B̂_{0} and B̂_{0} are Borel subsets of E. Of course, in (5.1) we may replace C(\overline{E}) by C_{r}(E) := C(\overline{E}) |_{E} since Û^{q}(x, ·) is a measure on E for x ∈ E.

(5.2) Proposition. A necessary and sufficient condition that x ∈ E\B̂_{0} is that P̂^{x}(X_{0+} = x) = 1. Of course X̂_{0+} exists in \overline{E}.

Proof. If f ∈ C(\overline{E}), then f ∘ X̂_{t} → f ∘ X̂_{0+} as t → 0. Therefore if x ∈ E

qÛ^{q}f(x) = P̂^{x} ∫_{0}^{∞} e^{−t}f(X_{t/q}) dt → P̂^{x}[f ∘ X_{0+}]

as q → ∞, which establishes (5.2) since this holds for all f ∈ C(\overline{E}).

It is evident that the resolvent (Û^{q}; q > 0) separates the points of E \ B̂_{0}. Hence if D(E) denotes the restrictions to E of a countable dense subset of C(\overline{E}), \{Û^{1}f : f ∈ D(E)\} is a countable set of 1-co-potentials separating the points of E\B̂_{0} such that t ↦ Û^{1}f ∘X̂_{t} is left continuous on ]0, ∞[ a.s. Recall that f ∈ pE is q-co-supermedian provided pÛ^{p+q}f ≤ f for all p > 0. We say that f is co-regular provided t ↦ f ∘ X̂_{t} is left continuous on ]0, ∞[ a.s. A co-potential, Û^{q}f, q ≥ 0, f ∈ pE is co-regular.

(5.3) Theorem. (Walsh) There exists a countable family, H of bounded co-regular 1-co-supermedian Borel functions on E separating the points of E.

For the proof we need a lemma whose proof we lift from [CW05].

(5.4) Lemma. Almost surely {t : X̂_{t} ∈ B̂_{0}} has Lebesgue measure zero.

Proof. Fix x ∈ E. Almost surely P̂^{x}, X̂ has at most a countable number of discontinuities, and so by the Fubini theorem, for Lebesgue a.e. t, X̂ is continuous at t, P̂^{x}-a.s.. Fix such a t > 0 and f ∈ C(\overline{E}). Then using the bounded convergence theorem for conditional expectations, P̂^{x}-a.s.

f ∘ X̂_{t} = lim_{n} P̂^{x}[n ∫_{0}^{∞} e^{−ns}f ∘ X̂_{t+s} ds | F̂_{t}]

= lim_{n} P̂^{x}[nÛ^{n}f ∘ X̂_{t} | F̂_{t}] = lim_{n} nÛ^{n}f ∘ X̂_{t}.

Letting f run through a countable dense subset of C(\overline{E}), it follows that X̂_{t} ∈ E \ B̂_{0}, P̂^{x}-a.s.

Proof of (5.3) Suppose G ⊂ E is open. Note that {T̂_{G} < t} is the union of {X̂_{r} ∈ G} over all rationals r, 0 < r < t. Hence T̂_{G} ∈ F̂^{∘} and so P̂^{·}(e^{−T_{G}}) ∈ E. (Recall the one hat convention from section 3.) Define ϕ_{G}(x) = P̂^{x}(e^{−T_{G}}) if x ∈ E \ (B̂_{0} ∩ G) and ϕ_{G}(x) = 1 if x ∈ B̂_{0} ∩ G. Since B̂_{0} ∈ E, ϕ_{G} ∈ E. Now (5.4) implies that Û^{q}(·, B̂_{0}) = 0 on E, and this in turn implies that qÛ^{q+1}ϕ_{G} ≤ P̂^{·}(e^{−T_{G}}) ≤ ϕ_{G}; hence ϕ_{G} is 1-co-supermedian. Also if x ∈ G \ B̂_{0}, then P̂^{x}[lim_{t↓0} X̂_{t} = x] = 1, and so ϕ_{G}(x) = P̂^{x}(e^{−T_{G}}) = 1. Therefore ϕ_{G} = 1 on G. If A ∈ E, let D̂_{A} = inf{t ≥ 0 : X̂_{t} ∈ A} denote the début of A. Note that D̂_{G}^{∘} θ̂_{t} = T̂_{G}^{∘} θ̂_{t} on {X̂_{t} ∉ G}. Therefore, if T is an (F̂_{t}) predictable time,

(5.5)  P̂^{x}[exp(−D_{G} ∘ θ_{T})|F̂_{T −}] = ϕ_{G}(X̂_{T}) on {X̂_{T} ∉ G}.

But if X̂_{T} ∈ G, D̂_{G} ∘ θ̂_{T} = 0, so (5.5) also holds on {X̂_{T} ∈ G}. Now suppose F ⊂ E is closed and choose open sets G_{n} with G_{n} ⊃ \overline{G}_{n+1} ⊃ F and ∩ G_{n} = F. Define ψ_{F} = ↓ lim_{n} ϕ_{G_{n}}. Then ψ_{F} is a Borel 1-co-supermedian function since it is the decreasing limit of a sequence of such functions. Also if x ∈ B̂_{0} ∩ F ⊂ B̂_{0} ∩ G_{n}, each ϕ_{G_{n}}(x) = 1 and so ψ_{F} (x) = 1. On the other hand if x ∈ E \ B̂_{0} ∩ F, then for some n, x ∈ E \ B̂_{0} ∩ G_{n} and so ψ_{F} (x) ≤ P̂^{x}[exp(−T_{G_{n}})]. Moreover if T is (F̂_{t}) predictable, then from (5.5)

(5.6)  ψ_{F} ∘ X̂_{T} = lim_{n} P̂^{x}[exp(−D_{G_{n}} ∘ θ_{T}) | F̂_{T −}] = P̂^{x}[exp(−D_{F} ∘ θ_{T}) | F̂_{T −}]

since X̂ being left continuous implies that D̂_{G_{n}} ↑ D̂_{F}.

To check that ψ_{F} is co-regular, let (T_{n}) be an increasing sequence of uniformly bounded (F̂_{t})-predictable times with limit T. Then using (5.6)

lim_{n} P̂^{x}[ψ_{F} ∘ X_{T_{n}}] = lim_{n} P̂^{x}[exp(−D_{F} ∘ θ_{T_{n}})]
= P̂^{x}[exp(−D_{F} ∘ θ_{T})] = P̂^{x}[ψ_{F} ∘ X_{T}].

Hence by [DM80; VI-49], ψ_{F} ∘ X̂ is P̂^{x}-a.s. left continuous and so ψ_{F} is co-regular.

If x and y are distinct points in E and Û^{1}(x, ·) ≠ Û^{1}(y, ·), then the set {Û^{1}f; f ∈ D(E)} is a countable set of 1-co-potentials separating x and y. If Û^{1}(x, ·) = Û^{1}(y, ·), the resolvent equation shows that Û^{q}(x, ·) = Û^{q}(y, ·) for all q > 0. Therefore either x ∈ B̂_{0} or y ∈ B̂_{0}, say x ∈ B̂_{0}. Let G be a countable base of open sets for the (Ray) topology of E. By (5.2), P̂^{x}(X_{0+} = x) < 1, and so there exists G ∈ G with x ∈ G, y ∉ \overline{G} such that with positive P̂^{x} probability X̂_{t} ∉ G for 0 < t < ε for some ε > 0 depending on ω̂; that is P̂^{x}(T_{G} > 0) > 0. Then g(x) := P̂^{x}(e^{−T_{G}}) < 1. But g is 1-co-excessive. Thus there exists (f_{n}) ⊂ pE with Û^{1}f_{n} ↑ g, and since Û^{1}f_{n}(x) = Û^{1}f_{n}(y), g(y) = g(x) < 1. Now x ∈ G ∩ B̂_{0}, so ϕ_{G}(x) = 1 while y ∈ E \ G ∩ B̂_{0}, so ϕ_{G}(y) = g(y) < 1. If H ∈ G with x ∈ H ⊂ \overline{H} ⊂ G, then ψ\overline{H}(x) = 1 > ψ\overline{H}(y). Therefore

(5.7)  H := \{ψ\overline{H}; H ∈ G\} ∪ \{Û^{1}f : f ∈ D(E)\}

is a family with the desired properties.

Remark. The reason that we could avoid passing to a quotient space and using Lebesgue penetration times as in [CW05] is that there they are considering a moderate Markov process under a fixed law, P, whereas we consider a family {P̂^{x}, x ∈ E_{Δ}} and X̂ is moderate Markov under each P̂^{x}, x ∈ E_{Δ}.

Armed with this separation theorem (5.3) we may now appeal to Theorem 8.30 in [CW05] to obtain:

(5.8) Theorem. There exists a Ray-Knight compactification Ẽ of E_{Δ} and a Ray process X̃ = ((X̃_{t})_{t≥0}; P̃^{x}, x ∈ Ẽ) on Ẽ such that for each x ∈ E_{Δ} the law of (X̃_{t−})_{t>0} under P̃^{x} is the same as the law of (X̂_{t})_{t>0} under P̂^{x}.

As in the construction of \overline{E} in section 2, Δ is regarded as an ordinary trap in this construction. Therefore in applying Theorem 8.30 in [CW05] we need a family separating E_{Δ} and so in Theorem 5.3 we are considering B̂_{0} ⊂ E_{Δ}. In particular Δ ∈ E_{Δ} \ B̂_{0} since Û^{q}f(Δ) = f(Δ) for q ≥ 0.

Remark. Theorem 5.8 implies that in the topology of Ẽ, the right-limit X̂_{t+} exists in Ẽ for each t ≥ 0, P̂^{x}-a.s., for all x ∈ E_{Δ}. Moreover, for each x ∈ E_{Δ} the Ẽ-valued processes (X̂_{t+}, P̂^{x}) and (X̃_{t}, P̃^{x}) are equivalent. In particular, (X̂_{t+}, P̂^{x}) is a right continuous strong Markov process.

We refer the reader to Chapter 8 of [CW05] for the properties of the Ray process X̃ and their relationship to X̂. For the convenience of the reader and ease of reference we list a few of the basic properties we shall need. We employ a self-explanatory notation. Objects relative to X̃ are designated by a “ ” above the corresponding symbol.

(5.9) P̂_{t}(x, ·) = P̃_{t}(x, ·) and Û^{q}(x, ·) = Ũ^{q}(x, ·) for all x ∈ E_{Δ}, t > 0, and q ≥ 0.

(5.10) Ũ^{q}(x, ·) is carried by E for all x ∈ Ẽ \ {Δ}.

(5.11) X̂ is left continuous on ]0, ∞[ with right limits in Ẽ on [0, ∞[ in the (Ray) topology of Ẽ, P̂^{x}-a.s. for all x ∈ E_{Δ}.

(5.12) If C̃_{b}(E) denotes the bounded co-Ray continuous functions on E, then Û^{q}C̃_{b}(E) ⊂ C̃_{b}(E) for all q > 0.

There are now three, generally distinct, topologies on E; the original topology and the topologies E inherits from \overline{E} and Ẽ. But the three topological Borel σ-algebras on E are the same and E is Borel in \overline{E} and in Ẽ. We shall call the topology E inherits from \overline{E} (resp. Ẽ) the Ray (resp. co-Ray) topology on E. The Ray topology on E as defined in [G75] maybe characterized as the weakest topology τ on E satisfying U^{q}C_{b}(E) ⊂ C_{b}(E, τ) and U^{q}C_{b}(E, τ) ⊂ C_{b}(E, τ) for each q > 0 where C_{b}(E), resp. C_{b}(E, τ), denotes the bounded continuous functions on E in the original, resp. τ, topology. See [G75; (15.3)]. The dependence of the co-Ray topology on the choice of the separating family H of (5.3) seems to be an open question. We shall fix a family H as in (5.3) and call the resulting topology the co-Ray topology.

Define B̂ ⊂ E by

(5.13)  E \ B̂ := {x ∈ E : qÛ^{q}f(x) → f(x) as q → ∞ for all f ∈ C(Ẽ)}

and call B̂ the set of co-branch points. Note the difference between B̂ and B̂_{0} (defined in (5.1)). Arguing exactly as for B̂_{0} we obtain

(5.14) Proposition. B̂ ∈ E and x ∈ E \ B̂ if and only if P̂^{x}(X̂_{0+}^{cr} = x) = 1 where X̂_{0+}^{cr} denotes the right limit at 0 in Ẽ. Almost surely {t : X̂_{t} ∈ B̂} has Lebesgue measure zero.

It is easy to give an example in which B̂_{0} is a proper subset of B̂. We do not know if it is possible for B̂ to be a proper subset of B̂_{0}.

Let D̃ denote the set of non-branch points for the Ray process X̃ of (5.8). It is wellknown that D̃ is absorbing for X̃ and so one may restrict X̃ to D̃. Then the restricted process, X̃_{D} is a Borel right process on the Lusin space D̃. Since E \ B̂ ⊂ D̃ and the resolvents agree on E and do not charge B̂, one may apply the known results for the potential theory of X̃_{D} to obtain results about X̂. For example if ξ ∈ Êxc and ξ(B̂) = 0, then ξ ∈ Ẽxc. Hence ξ = μŨ + ρ where ρ ∈ H̃ar and μ is a measure on D̃. Here Ẽxc and H̃ar refer to the excessive and harmonic measures for the Borel right process X̃(D̃). But ρ ≤ ξ so ρ is carried by E \ B̂, and it follows that ρ is harmonic for X̂. Note, however, that we cannot assert that μ is carried by E \ B̂, and so the result is not satisfactory. This will be improved in section 7.

6. Entrance Laws and the Extension of (4.9).

As in section 4 let h ∈ S(m) and ξ := h · m. Since ξ ∈ Êxc(m), ξ has a unique decomposition ξ = ξ_{i} + ξ_{p} where ξ ∈ Înv(m) and ξ_{p} ∈ P̂ur(m) using the obvious notation. Since both ξ_{p} and ξ_{i} are absolutely continuous with respect to m, there exist h_{p} and h_{i} in S(m) such that ξ_{p} = h_{p}m and ξ_{i} = h_{i}m. It is immediate that m-a.e., P_{t}h_{i} = h_{i}, P_{t}h_{p} ↓ 0 as t → ∞ and h = h_{i} + h_{p}. Let S_{p}(m) (resp. S_{i}(m)) denote the set of h ∈ S(m) such that P_{t}h ↓ 0, m-a.e. as t → ∞ (resp. P_{t}h = h, m-a.e. for each t). Each h ∈ S(m) may be decomposed as h = h_{p} + h_{i}, m-a.e., and this decomposition is unique modulo m null sets. Of course two excessive functions which agree m-a.e. agree off an m-exceptional set. The elements of S_{p}(m) (resp. S_{i}(m)) are called m-purely excessive (resp. m-invariant).

(6.1) Proposition. Suppose h ∈ S(m). Then h ∈ S_{p}(m) (resp. S_{i}(m)) if and only if Q_{m}^{h}(β = ∞) = 0 (resp. Q_{m}^{h}(β < ∞) = 0). Moreover Q_{m}^{h_{p}} = Q_{m}^{h} |_{{β<∞}} and Q_{m}^{h_{i}} = Q_{m}^{h} |_{{β=∞}}. In particular ξ_{p}(f) = Q_{m}^{h}[f ∘ Y_{0}; β < ∞] and ξ_{i}(f) = Q_{m}^{h}[f(Y_{0}) : β = ∞]. Finally, h_{p} = h · P^{·/h}(ζ < ∞) and h_{i} = h · P^{·/h}(ζ = ∞), m-q.e.

Proof. Let f > 0 with m(fh) < ∞. Then using Lemma 4.7 for the fourth equality below,

(P_{t}h, f) = m(hP̂_{t}f) = Q_{m}^{h}[P̂_{t}f ∘ Y_{0}; α < 0 < β]
= Q_{m}^{h}[P̂^{Y (0)}(f ∘ X̂_{t}); α < 0 < β] = Q_{m}^{h}[f ∘ Y_{−t}; α < 0 < β]
= Q_{m}^{h}[f ∘ Y_{0}; α < t < β].

Letting t → ∞ establishes the first assertion in (6.1). The second follows from the first because Q_{m}^{h} = Q_{m}^{h_{p}} + Q_{m}^{h_{i}}. Since β = ζ^{∘} θ_{0} on {Y_{0} ∈ E}, the assertions in the last sentence of (6.1) follow from those in the third sentence.

We come now to one of the main results of this section. For its statement we need a definition that is not standard but is convenient for what follows. Recall that a measure is Σ-finite provided that it is a countable sum of finite measures — called s-finite in [G90].

(6.2) Definition. A loose entrance law, ν̂, for (P̂_{t}) is a family (ν̂_{t}, t > 0) of Σ-finite measures on (E, E) such that ν̂_{t+s} = ν̂_{t}P̂_{s} for s, t > 0.

If each ν̂_{t} is σ-finite in the above definition then ν̂ is an entrance law for (P̂_{t}).

Remark. If ν̂ is a loose entrance law for (P̂_{t}), then t ↦ ν̂_{t}(f) is Borel measurable for f ∈ pE.

We remind the reader that the Fubini theorem is valid for Σ-finite measures.

(6.3) Theorem. Let h ∈ S(m). Then

ν̂_{t}(f) := Q_{m}^{h}[f ∘ Y_{β−t}^{∗}; 0 < β < 1], t > 0, f ∈ pE,

defines the unique loose entrance law for (P̂_{t}) such that ξ := h_{p}m = ∫_{0}^{∞} ν̂_{t} dt. Moreover ν̂_{t} is σ-finite for Lebesgue a.e. t and each ν̂_{t} is σ-finite on E \ N̂, where N̂ := {x : P̂^{x}(ξ = 0) = 1} is m-polar.

Proof. Let f ∈ pE. Since β is a co-stopping time, it is G_{>β}^{∘} measurable. See [DM75; 56a]. Now β − t is co-predictable and {0 < β < 1} ∈ G_{>β}^{∘} ⊂ G_{>β−t}^{∘}. Therefore from (4.9),

ν̂_{t}P̂_{s}(f) = Q_{m}^{h}[f ∘ Y_{β−t−s}^{∗}; 0 < β < 1] = ν̂_{t+s}(f).

Also,

∫_{0}^{∞} ν̂_{t}(f) dt = ∫_{0}^{∞} Q_{m}^{h}[f ∘ Y_{β−t}^{∗}; 0 < β < 1] dt
= ∫_{−∞}^{∞} Q_{m}^{h}[f ∘ Y_{t}, 0 < β < 1] dt
= ∫_{−∞}^{∞} Q_{m}^{h}[f ∘ Y_{0}; 0 < β + t < 1, β < ∞] dt = m(h_{p}f),

where f ∘ Y_{t} = 0 if t ≥ β is used for the second equality and (6.1) for the last equality. Clearly each ν̂_{t} is Σ-finite and if f > 0 with m(h_{p}f) < ∞, ∫_{0}^{∞} ν̂_{t}(f) dt = m(h_{p}f) < ∞ which implies that ν̂_{t} is σ-finite for Lebesgue a.e. t. Moreover

ν̂_{t}Ûf = ∫_{t}^{∞} ν̂_{s}(f) ds ≤ m(h_{p}f) < ∞.

Since {Ûf > 0} = E \ N̂, ν̂_{t} is σ-finite on E \ N̂ for each t > 0, and by (3.10), N̂ is m-polar.

It remains to show that ν̂ is unique. Suppose μ̂ = (μ̂_{t}, t > 0) is an arbitrary loose entrance law for (P̂_{t}) such that ξ = h_{p}m = ∫_{0}^{∞} μ̂_{t} dt. Let f ≥ 0 with ξ(f) < ∞. Then for q > 0,

\begin{aligned}∫_{0}^{∞} e^{−qt} ∫_{t}^{∞} μ̂_{s}(f) dsdt & = ∫_{0}^{∞} e^{−qt} ∫_{0}^{∞} μ̂_{s}P̂_{t}f dsdt \\ & = ∫_{0}^{∞} e^{−qt}ξ(P̂_{t}f) dt = ξÛ^{q}f. \end{aligned}

But also,

∫_{0}^{∞} e^{−qt} ∫_{t}^{∞} μ̂_{s}(f) dsdt = ∫_{0}^{∞} μ̂_{s}(f) ∫_{0}^{s} e^{−qt} dtds
= q^{−1} ∫_{0}^{∞}(1 − e^{−qs})μ̂_{s}(f) ds
= q^{−1}[ξ(f) − ∫_{0}^{∞} e^{−qs}μ̂_{s}(f) ds]

Therefore

(6.4)  ∫_{0}^{∞} e^{−qs}μ̂_{s}(f) ds = ξ(f) − qξÛ^{q}f,

the subtractions being justified since ξ(f) < ∞ implies that all the terms involved are finite. Since (6.4) also holds for ν̂_{s}(f), we obtain μ̂_{t}(f) = ν̂_{t}(f) for Lebesgue a.e. t from the uniqueness theorem for Laplace transforms. But E is countably generated, so μ̂_{t} = ν̂_{t} for a.e. t. If for some fixed t > 0, μ̂_{t} = ν̂_{t} then the entrance law property implies that μ̂_{s} = ν̂_{s} for all s ≥ t. Consequently μ̂_{t} = ν̂_{t} for all t > 0, completing the proof of (6.3).

Remark. It is known under very general conditions that a purely excessive measure may be represented uniquely as the integral of an entrance law. See, for example, [J87]. However with one exception all proofs known to us of both the existence and the uniqueness use the “uniqueness of charges”; that is, if V is the potential kernel of the underlying semigroup and μ, ν measures such that μV = νV σ-finite, then μ = ν. The one exception is the existence proof in [FM86]; uniqueness is not discussed there. Our existence proof is dual to that of [FM86]. It is easy to construct examples of processes X̂ of the type under consideration for which the uniqueness of charges does not hold. Nevertheless Theorem 6.3 gives the existence and the uniqueness of such a representation as the integral of a loose entrance law for ξ ∈ P̂ur(m). Note that the uniqueness argument is valid under minimal hypotheses.

The next result may be viewed as an extension of (6.3) from E to W.

(6.5) Theorem. Let h ∈ S(m). Then there exists a unique σ-finite measure Q̂_{ν} on (W, G^{∘}) carried by {β = 0} such that:

(i) Q_{m}^{h}[Φ(θ̌_{β}, β)] = ∫ Φ(w, t)Q̂_{ν}(dw)dt for Φ ∈ p(G^{∘} × B(R));

(ii) ∫_{−∞}^{∞} σ_{t}Q̂_{ν} dt = Q_{m}^{h_{p}};

(iii) Q̂_{ν}(X̂_{t_{1}} ∈ dx_{1}, …, X̂_{t_{n}} ∈ dx_{n}) = ν̂_{t_{1}}(dx_{1})P̂_{t_{2}−t_{1}}(x_{1}, dx_{2}) … P̂_{t_{n}−t_{n−1}}(x_{n−1}, dx_{n}) for 0 < t_{1} < … t_{n}, where ν̂ = (ν̂_{t}, t > 0) is the loose entrance law defined in (6.3).

Proof. Since β ∘ σ_{t} = β − t, θ̌_{β} ∘ σ_{t} = θ̌_{β}. Therefore the σ_{t} invariance of Q_{m}^{h} implies that

Q_{m}^{h}[Φ(θ̌_{β}, β)] = Q_{m}^{h}[Φ(θ̌_{β}, β − t)]

for t ∈ R. Consequently the measure λ(Φ) := Q_{m}^{h}[Φ(θ̌_{β}, β)] on W × R is translation invariant along R and is clearly Σ-finite. Hence there exists a unique Σ-finite measure Q̂_{ν} on (W, G^{∘}) such that

Q_{m}^{h}[Φ(θ̌_{β}, β)] = ∫ Φ(w, t) dt Q̂_{ν}(dw)

for Φ ∈ p(G^{∘} × B(R)). See [G90; (8.23)], for example. In particular if F ∈ p(G^{∘}), then

(6.6)  Q̂_{ν}(F) = Q_{m}^{h}[F ∘ θ̌_{β}, 0 < β < 1].

Noting that β ∘ θ̌_{β} = 0 if β ∈ R one sees that Q̂_{ν} is carried by {β = 0} ⊂ Ω̂, and for f ∈ pE, t > 0,

Q̂_{ν}[f ∘ X̂_{t}] = Q_{m}^{h}[f ∘ Y_{β−t}^{∗}; 0 < β < 1] = ν̂_{t}(f)

where ν̂_{t} is defined in (6.3). Fix f ∈ pE with f > 0 and m(fh_{p}) < ∞, define F := ∫_{0}^{∞} f^{∘}X̂_{t} dt. Now Q̂_{ν} is clearly Σ-finite, which justifies the interchange of integrals in computing

Q̂_{ν}(F) = ∫_{0}^{∞} ν̂_{t}(f) dt = m(fh_{p}) < ∞.

Since F > 0 on Ω̂ \ {[Δ]} = {β = 0}, Q̂_{ν} is σ-finite.

For F ∈ pG^{∘},
∫_{−∞}^{∞} σ_{t}Q̂_{ν}(F) dt = ∫_{−∞}^{∞} Q̂_{ν}(F ∘ σ_{t}) dt = ∫_{−∞}^{∞} Q_{m}^{h}[F ∘ σ_{t} ∘ θ̌_{β}; 0 < β < 1] dt.

Recall that k_{t} denotes the killing operator defined on W by (k_{t}w)(s) = w(s) if s < t and (k_{t}w)(s) = Δ if s ≥ t. One verifies that θ̌_{β} = k_{0}σ_{β} and σ_{t}k_{0} = k_{−t}σ_{t}. Thus σ_{t}θ̌_{β} = k_{−t}σ_{β+t}. Hence, using β − t = β^{∘}σ_{t} for the third equality below,

∫_{−∞}^{∞} σ_{t}Q̂_{ν}(F) dt = Q_{m}^{h}[∫_{−∞}^{∞} F [k_{−t}(σ_{β+t})] dt; 0 < β < 1]
= Q_{m}^{h}[∫_{−∞}^{∞} F [k_{β−t}(σ_{t})] dt; 0 < β < 1]
= Q_{m}^{h}[∫_{−∞}^{∞}[F^{∘}k_{β}^{∘}σ_{t}; 0 < β < 1] dt
= Q_{m}^{h}[∫_{−∞}^{∞} F^{∘}k_{β}; 0 < β + t < 1] dt
= Q_{m}^{h}[F^{∘}k_{β}; β ∈ R] = Q_{m}^{h}[F; β ∈ R],

since F ∈ pG^{∘}, implying that F^{∘}k_{β} = F on {β ∈ R} and . This establishes (6.5)(ii) in view of (6.1). If 0 < t_{1} < … < t_{n}, f_{j} ∈ pE, 1 ≤ j ≤ n, then

Q̂_{ν}[∏_{j=1}^{n} f_{j}^{∘}X̂_{t_{j}}] = Q_{m}^{h}[∏_{j=1}^{n} f_{j}^{∘}Y_{β−t_{j}}^{∗}; 0 < β < 1]

= Q_{m}^{h}[f_{1}^{∘}Y_{β−t_{1}}^{∗}(∏_{j=2}^{n} f_{j}^{∘}X̂_{t_{j}−t_{1}})^{∘} θ̌_{β−t_{1}}; 0 < β < 1]

= Q_{m}^{h}[f_{1}^{∘}Y_{β−t_{1}}^{∗} P̂^{Y^{∗}(β−t_{1})}(∏_{j=2}^{n} f_{j}^{∘}X̂_{t_{j}−t_{1}}); 0 < β < 1]

= ∫ ν_{t_{1}} (dx)f_{1}(x)P̂^{x} (∏_{j=2}^{n} f_{j}^{∘}X̂_{t_{j}−t_{1}})

where the third equality follows from (4.8) since β − t_{1} is co-predictable and β ∈ G_{>β−t_{1}}^{∘}. Using the fact that (X̂_{t}, t > 0) is Markov with transition function P̂_{t} under P̂^{x}, this establishes (6.5)(iii).

(6.7) Remarks. It is evident that (6.5)(iii) is equivalent to the statement that under Q̂_{ν}, (X̂_{t}, t > 0) is Markov with semigroup (P̂_{t}) and one dimensional distributions ν̂_{t}, t > 0.

We are now going to obtain a substitute for (4.9) when T = β, in analogy with (2.4). Recall the Ray-Knight compactification Ẽ of E_{Δ} and the discussion following (5.8). If b ∈ E with 0 < b ≤ 1, define Ŵ(b) by the following conditions:

(6.8)(i)  β ∈ R and Y_{β−}^{cr} ∈ E,

(6.8)(ii)  Ûb^{∘}Y_{β−1/n} → Ûb^{∘}Y_{β−}^{cr} as n → ∞.

We remind the reader that Y_{β−}^{cr} denotes the left limit of t ↦ Y_{t} at β in the co-Ray topology. Observe that σ_{t}^{−1}Ŵ(b) = Ŵ(b) for t ∈ R. Define

(6.9)  Y_{t}^{#}(w) = \{ \begin{matrix}Y_{β−}^{cr}(w) & if t = β(w) and w ∈ Ŵ(b), \\ Y_{t}^{∗}(w) & otherwise. \end{matrix}

Note that {β > t} ∈ G_{>t}^{∘} and that {β ≥ t} ∈ G_{>t−}^{∘}, where G_{>t−}^{∘} := ∩_{s<t}G_{>s}^{∘} = ∩_{s<t}G_{≥s}^{∘}. Define the left germ field at β by

G_{>β−}^{∘} := {G ∈ G^{∘} : G ∩ {β ≥ t} ∈ G_{>t}^{∘} for all t ∈ R},

and if h ∈ S(m) let G_{>β−}^{h·m} be the usual augmentation of G_{>β−}^{∘} by all Q_{m}^{h}-null sets.

(6.10) Theorem. Let h ∈ S(m) and fix b as above such that m(bh_{p}) < ∞. If G ∈ G_{>β−}^{h·m} and F ∈ pF̂, then

(6.11)  Q_{m}^{h}[F^{∘} θ̌_{β}; G ∩ Ŵ(b)] = Q_{m}^{h}[P̂^{Y^{#}(β)}(F); G ∩ Ŵ(b)].

Proof. Since Q_{m}^{h_{i}}(β < ∞) = 0, it suffices to prove (6.11) for h_{p}; in the remainder of the proof we assume that h ∈ S_{p}(m). As usual it suffices the establish (6.11) for G ∈ G_{>β−}^{∘} and F ∈ bpF̂^{∘}. Also observe that Ŵ(b) = {Y_{β}^{#} ∈ E} ∈ G_{>β−}^{∘}. From (6.5), Q_{m}^{h} = ∫_{R} σ_{t}Q̂_{ν} dt and Q̂_{ν} is carried by {β = 0} ⊂ Ω̂. Define Ŵ_{0}(b) := Ŵ(b)∩{β = 0} and Γ := G∩{β = 0}. Then both Ŵ_{0}(b) and Γ are in G_{>0−}^{∘}|_{Ω} = F̂_{0+}^{∘} and Γ ∩ Ŵ_{0}(b) = Γ ∩ Ŵ(b). We shall begin by showing

(6.12)  Q̂_{ν}[F^{∘} θ̌_{0}; Γ ∩ Ŵ_{0}(b)] = Q̂_{ν} [P̂^{Y^{#}(0)}[F]; Γ ∩ Ŵ(b)] .

Now Y_{0}^{#} = Y_{0−}^{cr} = X̂_{0+}^{cr} ∈ E on Ŵ_{0}(b), and so defining X̂_{0} := X̂_{0+}^{cr} on Ŵ_{0}(b), (6.12) becomes

(6.13)  Q̂_{ν}[F^{∘} θ̌_{0}; Γ ∩ Ŵ(b)] = Q̂_{ν} [P̂^{_{X}(0)}[F]; Γ ∩ Ŵ(b)] .

From the definition (6.9) of Y^{#}, Y_{0}^{#} = Δ on Ω̂ \ Ŵ(b). Therefore, using Fatou’s lemma,

Q̂_{ν}[Ûb^{∘}Y_{0}^{#}] = Q̂_{ν}[Ûb^{∘}Y_{0−}^{cr}; Ŵ(b)] ≤ lim inf_{k→∞} Q̂_{ν}[Ûb^{∘}Y_{−1/k}]

= lim inf_{k→∞} ν_{1/k}Ûb ≤ m(hb) < ∞,

by the choice of b. In particular Ûb^{∘}Y_{0}^{#} < ∞, Q̂_{ν}-a.e. If 0 ≤ f ∈ bE, define a finite measure on F̂^{∘} = G_{<0}^{∘}|_{Ω} by Q^{f}(H) = Q̂_{ν}[H · (fÛb)^{∘}Y_{0}^{#}; Γ ∩ Ŵ_{0}(b)], H ∈ F̂^{∘}, and note that Q^{f} is carried by Ω̂ ∩ Ŵ(b) since Y_{0}^{#} = Δ on Ω̂ \ Ŵ(b). If q > 0 and g ∈ C̃_{b}(E), where C̃_{b}(E) is defined in (5.12), then using (6.7) and the fact that Γ ∈ F̂_{0+}^{∘} ⊂ F̂_{1/k}^{∘} for all k ∈ N,

(6.14)

∫_{0}^{∞} e^{−qt}Q^{f}(g^{∘}X̂_{t}) dt = lim_{k→∞} ∫_{1/k}^{∞} e^{−qt}Q^{f}(g^{∘}X̂_{t−1/k}^{∘}θ̂_{1/k}) dt

= lim_{k→∞} e^{−q/k}Q^{f}(Û^{q}g^{∘}X̂_{1/k}) = Q̂^{f}(Û^{q}g^{∘}X̂_{0}),

where the last equality comes from the bounded convergence theorem and the fact that Q̂^{f}-a.e., X̂_{1/k} → X̂_{0} on Ŵ_{0}(b). Hence for q > 0

∫_{0}^{∞} e^{−qt}Q^{f}(g^{∘}X̂_{t}) dt = ∫_{0}^{∞} e^{−qt}Q^{f}[P̂_{t}g^{∘}X̂_{0}] dt.

Moreover P̂_{t}g(x) = P̂^{x}[g^{∘}X̂_{t}] is left continuous on ]0, ∞[ by (5.11). Also it follows from (6.6) that t ↦ g^{∘}X̂_{t} is left continuous on ]0, ∞[, Q̂_{ν}-a.e., and so Q^{f}-a.e. Therefore Q^{f}[g^{∘}X̂_{t}] = Q^{f}[P̂_{t}g^{∘}X̂_{0}] for all t > 0, first for g ∈ C̃_{b}(E) and then by a monotone class argument for all g ∈ pE. Putting back the definition of Q^{f} in terms of Q̂_{ν}, if g ∈ pE.

Q̂_{ν}[g^{∘}X̂_{t}(fÛb)^{∘}X̂_{0}; Γ ∩ Ŵ_{0}(b)] = Q̂_{ν}[P̂_{t}g^{∘}X̂_{0}(fÛb)^{∘}X̂_{0}; Γ ∩ Ŵ_{0}(b)]

first for f ∈ pbE and then by monotone convergence for f ∈ pE. Recall {Ûb = 0} = N̂ = {P̂^{·}(ζ = 0) = 1}. Taking f = (Ûb)^{−1} we find that Q̂_{ν}[g^{∘}X̂_{t}; Γ ∩ Ŵ_{0}(b)] = Q̂_{ν}[P̂_{t}g^{∘}X̂_{0}; Γ ∩ Ŵ_{0}(b)] since on {X̂_{0} ∈ N̂} both g^{∘}X̂_{t} and P̂_{t}g^{∘}X̂_{0} vanish for all t > 0. Note also that Q̂_{ν} is σ-finite on F̂_{0}^{∘} := σ(X̂_{0}), since Q̂_{ν}[Ûb^{∘}X̂_{0} < ∞]. Q̂_{ν} is carried by {β = 0}, so that ζ̂ = −α > 0, Q̂_{ν}-a.s. Now (6.13) and hence (6.12) follow by a standard argument using (6.7).

Recalling that h ∈ S_{p}(m), and using (6.5)(ii),

(6.15)  Q_{m}^{h}[F^{∘} θ̌_{β}; G ∩ Ŵ(b)] = ∫_{R} Q̂_{ν}[F^{∘} θ̌_{β}^{∘}σ_{t}; σ_{t}^{−1}(G ∩ Ŵ(b))] dt.

But θ̌_{β}^{∘}σ_{t} = θ̌_{β} and σ_{t}^{−1}Ŵ(b) = Ŵ(b). Since G ∈ G_{>β−}^{∘},

σ_{t}^{−1}G ∩ {β > s} = σ_{t}^{−1}(G ∩ {β > s − t}) ∈ σ_{t}^{−1}G_{>s−t}^{∘} = G_{>s}^{∘}

for s ∈ R. Therefore σ_{t}^{−1}G ∈ G_{>β−}^{∘}. Thus, since β = 0, Q̂_{ν}-a.e., and Y_{β}^{#}^{∘}σ_{t} = Y_{β}^{#}, the right hand side of (6.15) reduces to

∫_{R} Q̂_{ν} [P̂^{Y^{#}(β)}(F); σ_{t}^{−1}(G ∩ Ŵ(b))] dt = Q_{m}^{h} [P̂^{Y^{#}(β)}(F); G ∩ Ŵ(b)] ,

in view of (6.12). This completes the proof of Theorem 6.10.

(6.16) Remark. Note that (6.13) holds for any Γ ∈ F̂_{0+}^{∘}. Combining this with (6.7), it follows that if Q̂_{ν} is carried by Ŵ_{0}(b), then (X̂_{t} : t ≥ 0) under Q̂_{ν} is Markov with semigroup (P̂_{t}), where of course X̂_{0} = X̂_{0+}^{cr} ∈ E, Q̂_{ν}-a.s.

7. Potentials.

Recall the definition (5.13) of B̂, the set of co-branch points, and note that N̂ ⊂ B̂, where N̂ := {x ∈ E : P̂^{x}(ζ = 0) = 1} is the set of points in E from which X̂ branches to Δ with probability one. Also N̂ ⊂ B̂_{0}, the set of provisional co-branch points defined in (5.1). Obviously the resolvent (Û^{q}) separates the points of E \ (B̂ ∩ B̂_{0}). Therefore the following uniqueness result is proved exactly as [G90; Thm. 2.12].

(7.1) Proposition. Let μ and ν be measures on E which do not charge B̂ ∩ B̂_{0}, and let q ≥ 0. If μÛ^{q} and νÛ^{q} are σ-finite and μÛ^{q} = νÛ^{q}, then μ = ν.

Before coming to the main result of this section, we need to develop some preliminary results. Recall the definition of Ŵ(b) from (6.8) and, as in section 6, let Ŵ_{0}(b) = Ŵ(b) ∩ {β = 0}.

(7.2) Lemma. Let h ∈ S_{p}(m) and ξ := h · m. Choose 0 < b ≤ 1 with m(hb) < ∞. Then:

(i) Q_{m}^{h} is carried by Ŵ(b) if and only if Q̂_{ν} is carried by Ŵ_{0}(b), where Q̂_{ν} is defined in (6.5).

(ii) X̂_{0} ∈ E \ B̂, Q̂_{ν}-a.e. on Ŵ_{0}(b) and Y_{β}^{#} ∈ E \ B̂, Q_{m}^{h}-a.e. on Ŵ(b).

Proof. The first assertion is immediate since Q_{m}^{h} = ∫ σ_{t}Q̂_{ν} dt and Q̂_{ν} is carried by {β = 0}. For the second let g ∈ C̃_{b}(E), 0 ≤ f ≤ 1, and let Q^{f} be as in the proof (6.10). Then from the definition of Q^{f} in terms of Q̂_{ν} and the fact that Y_{0}^{#} = X̂_{0} on Ŵ_{0}(b), (6.14) implies that

Q̂_{ν}[∫_{0}^{∞} e^{−qt}g^{∘}X̂_{t} dt(fÛb)^{∘}X̂_{0}; Ŵ_{0}(b)]= Q̂_{ν}[Û^{q}g^{∘}X̂_{0}(fÛb)^{∘}X̂_{0}; Ŵ_{0}(b)].

But X̂_{t} → X̂_{0} = Y_{0−}^{cr} on Ŵ_{0}(b) as t → 0, and so multiplying by q and letting q → ∞, we obtain using the bounded convergence theorem

Q̂_{ν}[g^{∘}X̂_{0}(fÛb)^{∘}X̂_{0}; Ŵ_{0}(b)] = Q̂_{ν}[ lim_{q→∞} qÛ^{q}g^{∘}X̂_{0}(fÛb)^{∘}X̂_{0}; Ŵ_{0}(b)].

Since f ∈ E with 0 ≤ f ≤ 1 is arbitrary and Ŵ_{0}(b) ∈ F̂_{0+}^{∘}, lim_{q→∞} qÛ^{q}g^{∘}X̂_{0} = g^{∘}X̂_{0}, Q̂_{ν}-a.e. on Ŵ_{0}(b) ∩ {Ûb^{∘}X̂_{0} > 0}. But g^{∘}X̂_{0} and Ûb^{∘}X̂_{0} vanish Q̂_{ν}-a.e. on {Ûb^{∘}X̂_{0} = 0} = {X̂_{0} ∈ N̂}. This establishes the first assertion in (ii). Finally using the relation between Q_{m}^{h} and Q̂_{ν} one has

Q_{m}^{h}[Y_{β}^{#} ∈ B̂; Ŵ(b)] = ∫_{R} Q̂_{ν}[X̂_{0} ∈ B, Ŵ_{0}(b)] dt = 0,

completing the proof of (7.2).

Remark. Since Q̂_{ν} is carried by {β = 0}, Q̂_{ν}[Ŵ_{0}(b)△Ŵ(b)] = 0.

The next lemma is, perhaps, of some independent interest. Define X̂_{0+} := X̂_{0+}^{cr} ∈ Ẽ, see (5.14) for notation. Note that on Ŵ_{0}(b), X̂_{0+} = X̂_{0} ∈ E.

(7.3) Lemma. If x ∈ E \ B̂ and f ∈ pE, then for q ≥ 0

P̂^{x}[∫_{0}^{∞} e^{−qt}f^{∘}X_{t}dt | F̂_{0+}]= Û^{q}f^{∘}X̂_{0+},  P̂^{x}-a.s.

Proof. By monotone convergence it suffices to prove this for q > 0 and f ∈ pbE. In view of (5.14), X̂_{0+} ∈ E \ B̂, P̂^{x}-a.s. Let Γ ∈ F̂_{0+} and f ∈ C̃_{b}(E), see (5.12) for notation. Then

P̂^{x}[∫_{0}^{∞} e^{−qt}f^{∘}X_{t} dt; Γ] = lim_{k→∞} ∫_{1/k}^{∞} e^{−qt}P̂^{x}[f^{∘}X_{t−1/k}^{∘} θ_{1/k}; Γ] dt

= lim_{k} e^{−q/k}P̂^{x}[Û^{q}f^{∘}X_{1/k}; Γ] = P̂^{x}[Û^{q}f^{∘}X_{0+}; Γ].

Now a familiar monotone class argument shows that the extreme members of this last display are equal for f in pE or bE.

We come now to the main result of this section.

(7.4) Theorem. Let h ∈ S(m) and suppose that m(h_{p}b) < ∞. Define a measure μ on E by

(7.5)  μ(f) := Q_{m}^{h}[f^{∘}Y_{β}^{#}; 0 < β < 1, Ŵ(b)],  f ∈ pE.

Then μ(B̂) = 0, μ is σ-finite and μÛ ≤ h_{p}m. Moreover μ is the unique measure not charging B̂ such that μÛ = h · m if and only if Q_{m}^{h} is carried by Ŵ(b).

Proof. Since β ∈ R on Ŵ(b), (6.1) implies that μ(f) = Q_{m}^{h_{p}}[f^{∘}Y_{β}^{#}; 0 < β < 1, Ŵ(b)]. Thus we shall assume that h = h_{p} for the remainder of this first paragraph of the proof. By (7.2)(ii), μ(B̂) = 0. Using the relationship between Q_{m}^{h} and Q̂_{ν} in (6.5 ii) and the invariance of Y_{β}^{#} and Ŵ(b), one finds

\begin{aligned}μ(f) & = ∫_{R} Q̂_{ν}[f^{∘}Y_{β}^{#}; 0 < β − t < 1, Ŵ(b)] dt \\ & = Q̂_{ν}[f^{∘}Y_{β}^{#}, Ŵ(b)] = Q̂_{ν}[f^{∘}X̂_{0}, Ŵ(b)], \end{aligned}

since Q_{ν}[β ≠ 0] = 0, and X̂_{0} = Y_{0}^{#} on Ŵ(b) ∩ {β = 0}. Therefore using (6.16) for the second equality

\begin{aligned}μÛf & = ∫_{0}^{∞} Q̂_{ν}[P̂_{t}f^{∘}X̂_{0}; Ŵ(b)] dt = ∫_{0}^{∞} Q̂_{ν}[f^{∘}X̂_{t}^{∘} θ̌_{0}; Ŵ(b)] dt \\ & = ∫_{−∞}^{∞} Q̂_{ν}[f^{∘}Y_{t}; Ŵ(b)] dt = Q_{m}^{h}[f^{∘}Y_{0}; Ŵ(b)] ≤ Q_{m}^{h}[f^{∘}Y_{0}] = (h · m)(f), \end{aligned}

where the third equality follows because Q̂_{ν}(β ≠ 0) = 0. If f > 0 with m(hf) < ∞, this shows that μ is σ-finite since Ûf > 0 on E \ N̂ ⊃ E \ B̂.

Returning to the case of general h ∈ S(m) we see that μÛ ≤ h_{p}m and that μÛ = h·m if Q_{m}^{h} is carried by Ŵ(b). Since μ doesn’t charge B̂ it is the unique such measure according to (7.1). It remains to prove the converse; that is, if h · m = μÛ for some measure μ not charging B̂, then Q_{m}^{h} is carried by Ŵ(b). But μÛ = h · m forces μ to be σ-finite and h ∈ S_{p}(m). Define μ̂_{t} := μP̂_{t} for t > 0. Clearly μ̂_{t} is a loose entrance law for (P̂_{t}), and ∫_{0}^{∞} μ̂_{t} dt = μÛ = h · m. Consequently μ̂_{t} = ν̂_{t} for t > 0 by the uniqueness assertion in Theorem 6.3. It is now evident that (X̂_{t}, t > 0) has the same law under Q̂_{ν} as it has under P̂^{μ}. Since μ(B̂) = 0, X̂_{0+} = X̂_{0+}^{cr} ∈ E, P̂^{μ}-a.s. Moreover

P̂^{μ} ∫_{0}^{∞} b^{∘}X_{s} ds = μÛb = m(hb) < ∞,

and this suffices to justify the use of Hunt’s lemma (Th. 9.4.8 of [C01]) to conclude that

Ûb^{∘}X̂_{1/k} = P̂^{μ}[∫_{1/k}^{∞} b^{∘}X̂_{s}ds|F̂_{1/k}]→ P̂^{μ}[∫_{0}^{∞} b^{∘}X̂_{s}ds|F̂_{0+}]

as k → ∞. Now (7.3) implies that Ûb^{∘}X̂_{1/k} → Ûb^{∘}X̂_{0+}, P̂^{μ}-a.s., and so Q̂_{ν}-a.e. Consequently Q̂_{ν} is carried by Ŵ_{0}(b). This in turn implies that Q_{m}^{h} is carried by Ŵ(b) in view of (7.2), completing the proof of Theorem 7.4.

(7.6) Remark. The proof actually shows that {X̂_{t}; t ≥ 0} has the same law under Q̂_{ν} as it does under P̂^{μ}, when ν̂_{t} = μP̂_{t} for t ≥ 0.

We are now going to describe the decomposition of ξ ∈ Êxc(m) into potential and harmonic pieces. We need several definitions which are appropriate modifications of the corresponding concepts in the strong Markov case. See, for example, [G90].

(7.7) Definition. Let ξ, η ∈ Êxc(m). Then ξ strongly dominates η provided ξ = η + γ with γ ∈ Êxc(m).

(7.8) Definition. Let ξ ∈ Êxc(m). Then ξ is an m-co-potential (in symbols ξ ∈ P̂ot(m)) provided ξ = μÛ with μ not charging B̂, and ξ is m-co-harmonic (in symbols ξ ∈ Ĥar(m)) provided ξ strongly dominates no non-zero m-co-potential.

Remark. We remind the reader that the semigroup (P̂_{t}) relative to which ξ ∈ Êxc(m) is excessive depends on m. Also if ξ ∈ P̂ot(m), then the μ not charging B̂ with ξ = μÛ is uniquely determined by (7.1), and is given by (7.5).

(7.9) Theorem. Let ξ ∈ Êxc(m); that is, ξ = h · m with h ∈ S(m). Let 0 < b ≤ 1 with m(hb) < ∞. Then ξ ∈ P̂ot(m), resp. Ĥar(m), if and only if Q_{m}^{h} is carried by Ŵ(b), resp. Ŵ(b)^{c}. Finally ξ has a unique decomposition ξ = η + π with η ∈ Ĥar(m), π ∈ P̂ot(m) and π = μÛ where μ(f) = Q_{m}^{h}[f^{∘}Y_{β}^{#}; 0 < β < 1; Ŵ(b)].

Proof. Clearly Theorem 7.4 implies that ξ ∈ P̂ot(m) if and only if Q_{m}^{h} is carried by Ŵ(b). Next suppose that Q_{m}^{h}(Ŵ(b)) = 0 and ξ = η + π with η ∈ Êxc(m) and π ∈ P̂ot(m). Then η = gm and π = pm with g, p ∈ S(m). Therefore Q_{m}^{h} = Q_{m}^{g} + Q_{m}^{p}. Hence Q_{m}^{p}(Ŵ(b)) = 0. But Q_{m}^{p} is carried by Ŵ(b) since π ∈ P̂ot(m), and so Q_{m}^{p} = 0. Thus π = 0; that is, ξ ∈ Ĥar(m).

For the converse suppose at first that ξ ∈ Êxc(m) is arbitrary, ξ = h·m with h ∈ S(m). Define measures π and η on E by

(7.10)  π(f) = Q_{m}^{h}[f^{∘}Y_{0}; Ŵ(b)];  η(f) = Q_{m}^{h}[f^{∘}Y_{0}; Ŵ(b)^{c}].

Clearly π ≤ ξ = h · m and so π ≪ m. If t > 0, then since Ŵ(b) ∩ {0 < β} ∈ G_{>0} one has

\begin{aligned}πP̂_{t}f & = Q_{m}^{h}[P̂_{t}f^{∘}Y_{0}; Ŵ(b), 0 < β] = Q_{m}^{h}[f^{∘}Y_{−t}; Ŵ(b), 0 < β] \\ & = Q_{m}^{h}[f^{∘}Y_{0}; Ŵ(b), t < β] ≤ π(f). \end{aligned}

Therefore π ∈ Êxc(m), so we have π = p · m with p ∈ S(m). Then for each t ∈ R, Q_{m}^{p}(f^{∘}Y_{t}) = π(f) = Q_{m}^{h}[f^{∘}Y_{t}; Ŵ(b)]. Now using the simple Markov property (4.7) it follows that Q_{m}^{p} and Q_{m}^{h}(· ; Ŵ(b)) have the same finite dimensional distributions and hence Q_{m}^{p} = Q_{m}^{h}(· ; Ŵ(b)). A similar argument shows that η ∈ Êxc(m), so η = g · m with g ∈ S(m), and then Q_{m}^{g} = Q_{m}^{h}(· ; Ŵ(b)^{c}). From what has already been established we find π ∈ P̂ot(m), η ∈ Ĥar(m) and ξ = η + π. If ξ ∈ Ĥar(m), then π = 0 showing that Q_{m}^{h}(Ŵ(b)) = 0. In the general case we have the existence of the decomposition ξ = π + η and, in the light of (7.5), π = μÛ with μ as claimed.

It remains to prove the uniqueness of this decomposition. Suppose ξ = π′ + η′ is another such decomposition. Then using the obvious notation

Q_{m}^{p} = Q_{m}^{h}(· , Ŵ(b)) = Q_{m}^{p′}(· , Ŵ(b)) + Q_{m}^{g′}(· , Ŵ(b)).

But η′ = g′ · m ∈ Ĥar(m) implies that Q_{m}^{g′} is carried by Ŵ(b)^{c}. Therefore Q_{m}^{p} = Q_{m}^{p′}(· , Ŵ(b)) = Q_{m}^{p′}, since Q_{m}^{p′} is carried by Ŵ(b). Hence π = π′ which forces η = η′.

We shall say that an excessive function h is the m-potential of a measure μ provided h · m ∈ P̂ot(m) with h · m = μÛ, where μ(B̂) = 0. We write h = U(μ) in this case and, of course, μ is unique. Similarly h is m-harmonic provided h · m ∈ Ĥar(m). With these definitions we may translate Theorem 7.9 into results about excessive functions. These results should be compared with those available in the weak duality context, as found in section 7 of [GG84] and in sections 13.11 and 13.12 of [CW05].

(7.11) Corollary. If u ∈ S(m), then u may be decomposed uniquely as u = p + h where p = U(μ) is an m-potential and h is m-harmonic. Here uniqueness is up to m-polars. If 0 < b ≤ 1 and m(ub) < ∞, and

Γ := {0 < ζ < ∞ : X_{ζ−}^{cr} ∈ E, Ûb^{∘}X_{ζ−1/n} → Ûb^{∘}X_{ζ−}^{cr}},

then the decomposition u = p + h of u into m-potential and m-harmonic parts is given by p = uP^{^{•}/u}(Γ) and h = uP^{^{•}/u}(Γ^{c}). Moreover u is an m-potential (resp. m-harmonic) if and only if P^{^{•}/u}[Γ] = 1 (resp. P^{^{•}/u}[Γ^{c}] = 1), m-a.e. on {u > 0}. Finally, p = U(μ), where

(7.12)

μ(f) = lim_{t↓0} t^{−1} ∫_{E} u(x)P^{x/u}[f^{∘}X_{ζ−}^{cr}; Γ ∩ {ζ ≤ t}] m(dx)

= lim_{q↑∞} q ∫_{E} u(x)P^{x/u}[f^{∘}X_{ζ−}^{cr}e^{−qζ}; Γ] m(dx)

Proof. First note that Γ ∈ F^{∘} and that Γ ∩ {ζ > t} = θ_{t}^{−1}Γ. Therefore x ↦ P^{x/u}(Γ) is excessive for (P_{t}^{u}) and so there exists p ∈ S with p = uP^{^{•}/u}(Γ) on {u < ∞}; see [CW05; 11.17]. Since p ≤ u, m-a.e., we have p ∈ S(m). Similarly, there exists h ∈ S(m) with h = uP^{^{•}/u}[Γ^{c}] on {u < ∞}.

By Theorem 7.9, the potential part of ξ := um is μÛ, where from (7.5)

μ(f) = Q_{m}^{u}(f^{∘}Y_{β−}^{cr}; {0 < β < 1} ∩ Ŵ(b)),

since Y_{β}^{#} = Y_{β−}^{cr} on Ŵ(b). From the second display in the proof of (7.4)

μÛ(f) = Q_{m}^{u}(f^{∘}Y_{0}; Ŵ(b)).

But for each t ∈ R, Ŵ(b) = θ_{t}^{−1}Γ on {α < t < β}. Thus

\begin{aligned}μÛ(f) & = Q_{m}^{u}[f^{∘}Y_{0}; θ_{0}^{−1}Γ] \\ & = Q_{m}^{u} [f^{∘}Y_{0}P^{Y (0)/u}[Γ]] = m(pf); \end{aligned}

that is, μÛ = pm. It follows that p = U(μ) is the potential part of u. From (7.9), u = p+g, where g is m-harmonic. But u = p + h, and since u < ∞, m-a.e., one has g = h, m-a.e., hence m-q.e. It remains to prove (7.12). To this end define N(dt) := 1_{Γ}f^{∘}X_{ζ−}^{cr} ϵ_{ζ}(dt), where f ∈ bpE is fixed, and note that N is an HRM on Ω as defined in section 8 of [G90]. Using the fact that θ_{t}^{−1}Γ = Ŵ(b) on {α < t < β}, one readily checks that the extension N^{∗} of N to W defined in [G90; (8.18)] is given by N^{∗}(dt) = 1_{_{W} (b)}f^{∘}Y_{β−}^{cr} ϵ_{β}(dt). Let ν_{N}^{ξ} denote the characteristic measure of N relative to the u-transform of X and ξ = um ∈ Exc^{u}. Using Theorem (8.21) of [G90] with φ = 1_{]0,1[}, gives

ν_{N}^{ξ}(1) = Q_{m}^{u}[f^{∘}Y_{β−}^{cr}; Ŵ(b) ∩ {0 < β < 1}] = μ(f).

But from [G90; (8.9)] one also has

ν_{N}^{ξ}(1) = lim_{t↓0} t^{−1} ∫_{E} u(x)P^{x/u}[f^{∘}X_{ζ−}^{cr}; Γ ∩ {ζ ≤ t}] m(dx)

= lim_{q↑∞} q ∫_{E} u(x)P^{x/u}[f^{∘}X_{ζ−}^{cr}e^{−qζ}; Γ] m(dx),

which establishes (7.12).

8. Co-balayage and Walsh’s Interior Réduite.

In [FM86] the authors introduced a balayage operation associated with a stationary stopping time. This is also discussed in section 7 of [G90]. We are going to define a co-balayage operation associated with a class of stationary co-stopping times. Applying this to a specific stationary co-stopping time will lead to a generalization of the interior réduite discussed in section 13.12 of [CW05].

Let Q be a finite (or σ- finite) measure on (W, G^{∘}) and let G_{≥t}^{Q} be the “co-filtration” obtained by augmenting (G_{≥t}^{∘}) by all Q null sets in G^{Q}, the Q-completion of G^{∘}. Define G_{≥t}^{∗} := ∩_{Q}G_{≥t}^{Q} where the intersection is over all finite measures on Q on G^{∘}. G_{>t}^{∗}, G_{t}^{∗}, etc. are defined similarly.

(8.1) Definition. A universal co-stopping time S is a map S : W → [−∞, ∞] such that {S > t} ∈ Ĝ_{>t}^{∗} for each t ∈ R. If, in addition, t+S^{∘}σ_{t} = S, then S is a stationary universal co-stopping time. The class of such times is denoted by S(G_{>t}^{∗}).

Remarks. Since \{S > t\} = ∪_{n}\{S ≥ t + \frac{1}{n}\} one may replace G_{>t}^{∗} by G_{≥t}^{∗} and obtain the same class of universal co-stopping times. Recall the definition of killing operators on W, k_{t}w(s) = w(s) if s < t and k_{t}w(s) = Δ if s ≥ t.

We now define the co-balayage, R̂_{S}ξ, of ξ = h·m ∈ Êxc(m) on S a stationary universal co-stopping time,

(8.2)  R̂_{S}ξ(f) = R̂_{S}(h · m)(f) := Q_{m}^{h}[f^{∘}Y_{t}; S > t] = Q_{m}^{h}[f^{∘}Y_{t}^{∘}k_{S}].

Clearly R̂_{S}ξ does not depend on the choice of t in (8.2) and R̂_{S}ξ is a measure with R̂_{S}ξ ≤ ξ, in particular R̂_{S}ξ ≪ m. Also

\begin{aligned}R̂_{S}ξ(P̂_{t}f) & = Q_{m}^{h}[P̂_{t}f^{∘}Y_{0}; S > 0] = Q_{m}^{h}[f^{∘}Y_{−t}; S > 0] \\ & = Q_{m}^{h}[f^{∘}Y_{0}; S > t] ↑ R̂_{S}ξ(f) \end{aligned}

as t ↓ 0. Therefore R̂_{S}ξ ∈ Êxc(m) and so R̂_{S}(h · m) = p_{S}h · m with p_{S}h ∈ S(m) where p_{S}h ≤ h q.e.. Of course, p_{S}h is only determined q.e. Moreover, p_{S}(g + h) = p_{S}g + p_{S}h q.e. if g, h ∈ S(m).

(8.3) Proposition. With the above notation Q_{m}^{p_{S}h}(F) = Q_{m}^{h}[F^{∘}k_{S} : S > α] for F ∈ pG_{0}^{∘}.

Proof. Suppose F = ∏_{j=1}^{n} f_{j}^{∘}Y_{t_{j}}, where t_{1} < ⋯ < t_{n} and f_{j} ∈ pE for j ≥ 1. Then F^{∘}k_{S} = F

or 0 according as t_{n} < S or t_{n} ≥ S. Writing G := ∏_{j=1}^{n−1} f_{j}^{∘}X̂_{t_{n}−t_{j}}, we compute

Q_{m}^{h}[F^{∘}k_{S}; S > α] = Q_{m}^{h}[∏_{j=1}^{n} f_{j}^{∘}Y_{t_{j}}; t_{n} < S]
= Q_{m}^{h}[f_{n}^{∘}Y_{t_{n}} P̂^{Y (t_{n})}(G); t_{n} < S]
= ∫ R̂_{S}(h · m)(dx)f_{n}(x)P̂^{x}(G)
= (f_{n}p_{S}h, P̂^{•}(G))
= (f_{n}p_{S}h, P̂_{t_{n}−t_{n−1}}f_{n−1} ⋯ P̂_{t_{2}−t_{1}}f_{1})
= (f_{1}, P_{t_{2}−t_{1}}f_{2} ⋯ P_{t_{n}−t_{n−1}}(f_{n}p_{S}h))
= Q_{m}^{p_{S}h}(F).

Thus Q_{m}^{p_{S}h} and k_{S}Q_{m}^{h}[ ·; S > α] have the same finite dimensional distributions and both vanish on {[Δ]}. Hence they agree since a Kuznetsov measure is uniquely determined by its finite dimensional distributions and the fact that it does not charge {[Δ]}. Remark. If F ([Δ]) = 0, then Q_{n}^{p_{S}h}[F] = Q_{m}^{h}[F^{∘}k_{S}].

Proposition 8.3 and the next result are analogous to (7.5) in [G90]. (8.4) Proposition. Let S ∈ S(G_{>t}^{∗}) and suppose that S ≤ β. If ξ = h · m ∈ Êxc(m), then

(i) (R̂_{S}ξ)_{p}(f) = Q_{m}^{h}[f^{∘}Y_{0}; 0 < S < ∞] = ∫_{0}^{∞} ν_{t}^{S}(f)dt, where ν_{t}^{S}(f) =
Q_{m}^{h}[f^{∘}Y_{S−t}^{∗}; 0 < S < 1].

(ii) The potential part of R̂_{S}ξ equals μ_{S}Û where μ_{S}(f) = Q_{m}^{h}[f^{∘}Y_{S−}^{cr}; 0 < S <

1, k_{S}^{−1}Ŵ(b)].

Proof. Since β^{∘}k_{S} = β ∧ S = S under the hypothesis, (i) follows from (6.1), (6.3) and (8.3) and (ii) from (7.9) and (8.3).

We are now going to consider a specific example. Let A ∈ E and define (sup ϕ = −∞ by convention)

(8.5)  S_{A} := sup{t : Y_{t−}^{cr} ∈ A} = sup{t > α : Y_{t−}^{cr} ∈ A}.

Note that S_{A} ≤ β, S_{A} > α on {S_{A} > −∞}, and that Y_{t−}^{cr} = Y_{t−}^{∗cr} = Δ if t ≤ α. Clearly t + S_{A}^{∘}σ_{t} = S_{A}. Since A ∈ E, A is Borel in Ẽ. See (5.8) for notation. Therefore

{(s, w) : s > t, Y_{s−}^{cr}(w) ∈ A} ∈ B_{>t} ⊗ G_{>t}^{∘}

where B_{>t} denotes the Borel σ-algebra of ]t, ∞[. But {S_{A} > t} is the projection of this set on W and so {S_{A} > t} ∈ G_{>t}^{∗}. See III-82b of [DM75]. Therefore S_{A} ∈ S(G_{>t}^{∗}). If ξ = h · m ∈ Êxc(m) we write R̂_{A}ξ = R̂_{S_{A}}ξ and p_{A}h = p_{S_{A}}h.

(8.6) Proposition. Let T_{A}^{−} := inf{t > 0 : X_{t−}^{cr} ∈ A}. Then using the above notation p_{A}h = hP^{·/h}(T_{A}^{−} < ∞) m-q.e.

Proof. First note that on {Y_{0} ∈ E}, {S_{A} > 0} = {S_{A}^{∘} θ_{0} > 0} = {T_{A}^{−}^{∘} θ_{0} < ∞}. Therefore if f ∈ pE

\begin{aligned}R̂_{A}(h · m)(f) & = Q_{m}^{h}[f^{∘}Y_{0}; S_{A} > 0] \\ & = Q_{m}^{h}[f^{∘}Y_{0}P^{Y (0)/h}(T_{A}^{−} < ∞)] = m[hfP^{·/h}(T_{A}^{−} < ∞)]. \end{aligned}

But R̂_{A}(h · m)(f) = m(fp_{A}h) and since this holds for all f ∈ pE, p_{A}h = hP^{·/h}(T_{A}^{−} < ∞), m-a.e. One checks easily that P^{·/h}(T_{A}^{−} < ∞) is h-excessive and so there exists a unique excessive function u_{A} with u_{A} = hP^{·/h}(T_{A}^{−} < ∞) on {h < ∞}. [CW05; Prop. 11.7]. Now h < ∞, m-a.e., hence p_{A}h = u_{A}, m-a.e., and then m-q.e.; in particular p_{A}h = hP^{·/h}(T_{A}^{−} < ∞) m-q.e. since {h = ∞} is m-polar.

Remarks. What has been proved so far for S_{A} is also valid if we replace Y_{t−}^{cr} in the definition of S_{A} by Y_{t−}^{r}, which exists in \overline{E}, or if Y has left limits in E in the original topology by Y_{t−}. When comparing our results with those in section 13.12 of [CW05], one should note that it is assumed there that X and X̂ are in strong duality with respect to m, although branch points are allowed. Hence X and Y have left limits in the original topology. In particular if we use Y_{t−} in the definition of S_{A} it follows from (8.6) that p_{A}h defined above agrees with p_{A}h as defined in [CW05].

Recall the Ray compactification Ẽ and the Ray process X̃ of Theorem 5.8. We change the notation slightly and let X̃ denote the restriction of this Ray process to its set of nonbranch points D̃—this was denoted by X̃_{D} in the last paragraph of section 5. Then X̃ is a Borel right process in the co-Ray topology, taking values in D̃. In what follows it is important to note that (i) E \ B̂ ⊂ D̃, and (ii) if μ is a measure on E \ B̂ then the laws of (X̂_{t} : t > 0) under P̂^{μ} and of (X̃_{t−} : t > 0) under P̃^{μ} are the same. Of course, X̃_{t−} denotes the limit limit taken in the co-Ray topology. Since m(B̂) = 0, if ξ ∈ Êxc(m) then ξ ∈ Ẽxc, as was pointed out at the end of section 5.

Although P̂ot(m) need not be a solid subcone of Êxc(m), the next result shows that R̂_{A}ξ ∈ P̂ot(m) if ξ = h · m ∈ P̂ot(m), under a mild condition on A. This generalizes [CW05; Prop. 13.62]. To state this result we need to introduce some notation. If A ∈ E with A ⊂ E \B̂, define Ã to be the fine closure of A in D̃ relative to the Borel right process X̃. For t ≥ 0 define X̂_{t}^{+} := X̂_{t+}^{cr}, which exists in D̃, P̂^{x}-a.s. for x ∈ E \ B̂. Note that (X̂_{t}^{+} : t ≥ 0) under P̂^{μ} has the same law as (X̃_{t} : t ≥ 0) under P̃^{μ} provided μ is carried by E \B̂. Moreover, for such μ, if D̃_{A} := inf{t > 0 : X̃_{t} ∈ A} and D̂_{A}^{+} := inf{t > 0 : X̂_{t}^{+} ∈ A}, then the joint law of (D̃_{A}^{+}, (X̃_{t} : t ≥ 0)) under P̃^{μ} is the same as that of (D̂_{A}^{+}, (X̂_{t}^{+} : t ≥ 0)) under P̂^{μ}. These facts will be used freely in the remainder of this section.

(8.7) Proposition. Suppose ξ = h · m = μÛ ∈ P̂ot(m) and A ∈ E with Ã ⊂ E \ B̂. Then R̂_{A}ξ = μ_{A}Û ∈ P̂ot(m), where

(8.8)  μ_{A} = μP̃_{D_{A}} := P̃^{μ}(X̃_{D_{A}} ∈ ·) = P̂^{μ}(X̂+ ; _{D_{A}^{+}} ∈ ·).

Moreover, μ_{A} is carried by the union of A and the points that are regular for A with respect to X̃. In particular, μ_{A} is carried by Ã.

Proof. Since μ is carried by E \ B̂, the assertions in the last sentence of (8.7) are clear; in particular, μ_{A} is carried by E \ B̂. Since the resolvent kernels Û^{q}(x, ·) and Ũ^{q}(x, ·) are equal for q > 0 and x ∈ E \ B̂ and E is Lusin, it follows that {t : P̂_{t}(x, ·) ≠ P̃_{t}(x, ·)} is at most countable for each x ∈ E \ B̂. Therefore for f ∈ pE,

μ_{A}Û(f) = lim_{n} ∫_{1/n}^{∞} μ_{A}P̂_{t}(f)dt = lim_{n} ∫_{0}^{∞} μ_{A}P̃_{t+1/n}(f)dt.

Using the strong Markov property of X̃ under P̃^{μ} for the second equality,

μ_{A}P̃_{t+1/n}(f) = P̃^{μ}[P̃_{t+1/n}f^{∘}X̃_{D_{A}}] = P̃^{μ}[f^{∘}X̃_{_{D_{A}}+t+1/n}],

and consequently

\begin{aligned}μ_{A}Û(f) & = lim_{n} P̃^{μ} ∫_{_{D_{A}}+1/n}^{∞} f^{∘}X̃_{t}dt \\ & = P̃^{μ} ∫ ∞ ; _{D_{A}} f^{∘}X̃_{t} dt = P̂^{μ} ∫ ∞ ; _{D_{A}^{+}} f^{∘}X̂_{t} dt. \end{aligned}

Since h · m ∈ P̂ot(m), h ∈ S_{p}(m) and ν̂_{t} := μP̂_{t} is the unique loose entrance law with μÛ = ∫_{0}^{∞} ν̂_{t}dt. Thus from (6.5), Q_{m}^{h} = ∫ σ_{t}Q̂_{ν}dt. But μ is carried by E \ B̂, and so (X̂_{t} : t ≥ 0) has the same law under P̂^{μ} and Q̂_{ν} when X̂_{0} := X̂_{0+}^{cr} according to (7.6). This implies that the joint law of (X̂_{t}, D̂_{A}^{+}) is the same under P̂^{μ} and Q̂_{ν}. Therefore μ_{A}Û(f) = Q̂_{ν} ∫ ∞ ; _{D_{A}^{+}} f^{∘}X̂_{t}dt. But Q̂_{ν}(β ≠ 0) = 0 and so Q̂_{ν}-a.e.,

D̂_{A}^{+} = inf{t ≥ 0 : X̂_{t}^{+} ∈ A} = − sup{t ≤ 0; Y_{t−}^{cr} ∈ A} = −S_{A}.

Hence

\begin{aligned}μ_{A}Û(f) & = Q̂_{ν} ∫_{−∞}^{S_{A}} f^{∘}Y_{t}dt = ∫_{−∞}^{∞} Q̂_{ν}[f^{∘}Y_{t}; S_{A} > t]dt \\ & = Q_{m}^{h}[f^{∘}Y_{0}; S_{A} > 0] = R̂_{A}(h · m)(f), \end{aligned}

because S_{A} − t = S_{A}^{∘}σ_{t}. As noted before μ_{A} is carried by Ã ⊂ E \ B̂ and so R̂_{A}(h · m) ∈ P̂ot(m).

Proposition (8.7) describes R̂_{A}ξ when ξ ∈ P̃ot(m). The next result describes R̂_{A}ξ when ξ ∈ Ĥar(m). In its statement, L̃ denotes the energy functional of the Borel right process X̃.

(8.9) Proposition. Let A ∈ E with A ⊂ E \ B̂ and ξ = h · m, where h ∈ S(m).

(i) If m ∈ Con, let ϕ := P^{•}(T_{A}^{−} < ∞). Then ϕ(x) is either 0 or 1 for m-a.e. x, and p_{A}h = ϕ · h.

(ii) If m ∈ Dis and ξ = h · m ∈ Ĥar(m), then

R̂_{A}ξ(f) = L̃(h · m, P̃_{A}Ũf),

where P̃_{A}g := P̃^{•}(g^{∘}X̃_{_{T}(A)}) and T̃(A) := inf{t > 0 : X̃_{t} ∈ A}.

Proof. (i) Fix m ∈ Con. Theorem 2.16 in [G90] asserts that P^{x}[h(X_{t}) ≠ h(X_{0}) for some t > 0] = 0 for m-a.e. x. In particular, P_{t}h(x) = h(x) for all t ≥ 0, for m-a.e. x. But this implies that P_{t}^{h}1 = 1, m-a.e. Consequently, P^{^{•}/h}(ζ = ∞) = 1, m-a.e. Therefore

p_{A}h = h · P^{^{•}/h}(T_{A}^{−} < ∞) = P^{•}(h^{∘}X_{T_{A}^{−}}; T_{A}^{−} < ∞) = hϕ.

Set T := T_{A}^{−} for notational convenience. Since T is an exact terminal time, we have for t > 0 and m-a.e. x,

\begin{aligned}ϕ(x) & = P^{x}(T < ∞) = P^{x}(T ≤ t) + P^{x}(t < T < ∞) \\ & = P^{x}(T ≤ t) + P^{x}(ϕ(X_{t}); t < T). \end{aligned}

Let t → ∞ above and use the fact that ϕ is excessive (so that P^{x}[ϕ(X_{t}) ≠ ϕ(X_{0}) for some t > 0] = 0 for m-a.e. x) to obtain ϕ = ϕ + ϕ(1 − ϕ), which proves (i).

(ii) For each x ∈ E_{Δ}, Theorem 5.8 implies that (X̂_{t} : t > 0) under P̂^{x} and (X̃_{t−} : t > 0) under P̃^{x} have that same law. Of course limits involving X̃ refer to the co-Ray topology of Ẽ. If, as in section 7, X̂_{0} := X̂_{0+}^{cr} and also X̃_{0−} := X̃_{0}, then this assertion extends to t = 0. It was pointed out at the end of section 5 that ξ ∈ Ẽxc. Let Q̃_{ξ} and Ỹ be the Kuznetsov measure and process corresponding to ξ and the semigroup (P̃_{t}) of X̃. Thus (Ỹ_{t}) is right continuous with values in D̃ and left limits Ỹ_{t−} in Ẽ for α̃ < t < β̃. Topological statements about Ỹ refer to the co-Ray topology of Ẽ. Since ξ is carried by E, in fact by E \ B̂, (Y_{−t}^{#}) under Q_{m}^{h} has the same law as (Ỹ_{t−}^{#}) under Q̃_{ξ}. But ξ = h · m ∈ H̃ar(m) and so Y_{−t}^{#} is equivalent to Y_{−t}^{∗} under Q_{m}^{h}, which has the same law as Ỹ_{t−}^{∗} = Ỹ_{t−} under Q̃_{ξ}. Also

S_{A} = sup{t > α : Y_{t−}^{cr} ∈ A} = − inf{t < −α : Y_{(−t)−}^{cr} ∈ A}.

Hence S_{A} under Q_{m}^{h} and −τ̃_{A} under Q̂_{ξ} have the same law, where τ̃_{A} := inf{t : Ỹ_{t} ∈ A} because the processes (Y_{(−t)−}^{cr}, Q_{m}^{h}) and (Ỹ_{t}, Q̃_{ξ}) are identical in law. But for each t, Q_{m}^{h}(Y_{t−}^{cr} ≠ Y_{t}) = 0. Consequently,

(8.10)  \begin{aligned}R̂_{A}(h · m)(f) & = Q_{m}^{h}(f^{∘}Y_{0}; S_{A} > 0) \\ & = Q_{m}^{h}(f^{∘}Y_{0−}^{cr}; S_{A} > 0) \\ & = Q̃_{ξ}(f^{∘}Ỹ_{0}; τ̃_{A} < 0) = R̃_{A}ξ(f), \end{aligned}

where R̃_{A}ξ is the balayage of ξ on A relative to the Kuznetsov process (Ỹ, Q̃_{ξ}); see [G90; (7.3), (7.9)].

Since m ∈ Dis, there exists an increasing sequence (Uf_{n}) of potentials with Uf_{n} ↑ h, m-a.e. Define η_{n} := f_{n}m. Then η_{n} is carried by E \ B̂ ⊂ D̃ and

η_{n}Ũf = η_{n}Ûf = (f_{n}, Ûf) = (Uf_{n}, f) ↑ m(hf) = ξ(f).

Hence ξ ∈ D̃is. Therefore [G90; (4.12)] implies that R̃_{A}ξ(f) = L̃(ξ, P̃_{A}Ũf).

Remark. The proof of (8.9)(ii) does not require m ∈ Dis. It suffices that h · m ∈ H̃ar(m) and that there exists an increasing sequence (η_{n}Û) ⊂ P̂ot(m) with η_{n}Û ↑ h · m.

Propositions (8.7) and (8.9) combine to yield the following description of R̂_{A}ξ when m is dissipative.

(8.11) Theorem. Let A ∈ E with Ã ⊂ E \ B̂, m ∈ Dis, and h ∈ S(m). Let μÛ + η be the Riesz decomposition of ξ := h · m into its co-potential and co-harmonic parts. Then, for f ∈ pE,

(8.12)  R̂_{A}ξ(f) = μP̃_{_{D}(A)}Ûf + L̃(η, P̃_{A}Ũf) = sup{νP̃_{_{D}(A)}Ûf : νŨ ≤ ξ}.

Proof. The first equality in (8.12) is an immediate corollary of (8.7) and (8.9). For the second let u := P̃_{_{D}(A)}Ûf = P̃_{_{D}(A)}Ũf, since Ã ⊂ E\B̂. Now D̃(A) is an exact terminal time for X̃ whose exact regularization is T̃_{A}. It is easy to check that u is strongly supermedian with excessive regularization \overline{u} := P̃_{A}Ũf; that is, P̃_{t}u ↑ \overline{u} as t ↓ 0. The second equality in (8.12) now follows from Theorem (4.7) of [FG03].

We next list some properties of the map ξ ↦ R̂_{A}ξ from Êxc(m) to Êxc(m), for A ∈ E. Recall that if A ∈ E with A ⊂ E \ B̂ then Ã denotes the fine closure of A in D̃ relative to X̃. It was noted earlier that R̂_{A}ξ ≤ ξ and R̂_{A}(ξ + η) = R̂_{A}ξ + R̂_{A}η for ξ, η ∈ Êxc(m).

(8.13) Proposition. Let m ∈ Dis, A ∈ E with Ã ⊂ E \ B̂.

(i) If ξ, η ∈ Êxc(m) with ξ ≤ η, then R̂_{A}ξ ≤ R̂_{A}η.

(ii) If ξ ∈ Ĥar(m), then there is a sequence (μ_{n}Û) ⊂ P̂ot(m) with μ_{n}Û ↑ ξ, and for

any such sequence we have R̂_{A}(μ_{n}Û) ↑ R̂_{A}ξ. (iii) For any m ∈ Exc, ξ ∈ Êxc(m), and A, B ∈ E with A ⊂ B, we have R̂_{A}R̂_{B}ξ =

R̂_{B}R̂_{A}ξ = R̂_{A}ξ ≤ R̂_{B}ξ.

Proof. Assertion (i) is an immediate consequence of the second expression for R̂_{A}ξ in (8.12). For (ii), as in the last paragraph of the proof of (8.9), there exists a sequence (μ_{n}Û) ⊂ P̂ot(m) with μ_{n}Û ↑ ξ. If (μ_{n}Û) is any such sequence, it follows from (i) that R̂_{A}(μ_{n}Û) increases and lim_{n} R̂_{A}(μ_{n}Û) ≤ R̂_{A}ξ. If f ∈ pE, then R̂_{A}(μ_{n}Û)(f) = μ_{n}(u), with u = P̃_{_{D}(A)}Ũf, according to (8.7). Now μ_{n}(u) ↑ a ≤ R̂_{A}ξ(f) = L̃(ξ, \overline{u}) where \overline{u} = P̃_{A}Ũf is the X̃-excessive regularization of u as before. But L̃(ξ, \overline{u}) =↑ lim_{n} μ_{n}(\overline{u}) because μ_{n}Ũ = μ_{n}Û ↑ ξ. However \overline{u} ≤ u, so μ_{n}(u) ≥ μ_{n}(\overline{u}) which forces a ≥ L̃(ξ, \overline{u}), establishing (ii). For (iii), A ⊂ B implies S_{A} ≤ S_{B} and so from the definition (8.2), R̂_{A}ξ ≤ R̂_{B}ξ. Also, k_{S_{A}}^{∘}k_{S_{B}} = k_{S_{B}}^{∘}k_{S_{A}} = k_{S_{A}}. Hence the remaining assertions in (iii) follow from (8.3).

The next result is a direct generalization of Theorem 13.65 in [CW05]. For its statement we need some notation. If μ is a measure on E, let Supp(μ, cr) denote the co-Ray support of μ and for A ∈ E, define M(A) to be the set of measures μ on E such that Supp(μ, cr) is compact (in the co-Ray topology) and contained in A.

(8.14) Theorem. Suppose m ∈ Dis, h · m ∈ Êxc(m), A ∈ E and A ⊂ E \ B̂. Then

(8.15)  R̂_{A}(h · m) = sup{μÛ ∈ P̂ot(m) : μÛ ≤ h · m, μ ∈ M(A)}.

Proof. Suppose μÛ ∈ P̂ot(m), μÛ ≤ h · m and μ ∈ M(A). Let K = Supp (μ, cr) ⊂ A ⊂ E \ B̂. Then from (8.13) and (8.7),

R̂_{A}(h · m) ≥ R̂_{A}(μÛ) ≥ R̂_{K}(μÛ) = μ_{K} Û

where μ_{K} = μP̃_{_{D}(K)}. But K is co-Ray compact and carries μ. Therefore μ_{K} = μ and μÛ = μ_{K} Û ≤ R̂_{A}(h · m). Thus R̂_{A}(h · m) dominates the supremum in (8.15). For the reverse inequality, we shall first show that there exists an increasing sequence (K_{n}) of co-Ray compact subsets of A with R̂_{K_{n}}(h · m) ↑ R̂_{A}(h · m). A theorem of Dellacherie (see [G75; (12.15)]) implies that there is an increasing sequence (K_{n}) of co-Ray compact subsets of A with T̃_{K_{n}} ↓ T̃_{A}, P̃^{ξ}-a.s., where ξ = h · m as usual. It follows that τ̃_{K_{n}} ↓ τ̃_{A}, Q̃_{ξ}-a.e., where the notation is as in the proof of (8.9)(ii). Thus, by (8.10),

\begin{aligned}R̂_{K_{n}}ξ(f) & = Q̃_{ξ}[f^{∘}Ỹ_{0}; τ̃_{K_{n}} < 0] \\ & ↑ Q̃_{ξ}[f^{∘}Ỹ_{0}; τ̃_{A} < 0] = R̂_{A}ξ(f). \end{aligned}

Now let ρÛ + η be the decomposition of ξ into its co-potential part ρÛ and its coharmonic part η. From (8.13) there exists a sequence (μ_{k}Û) ⊂ P̂ot(m) increasing to η with

R̂_{K_{n}}(μ_{k}Û) ↑ R̂_{K_{n}}η. Let λ_{k} := ρ + μ_{k}. Then R̂_{K_{n}}(λ_{k}Û) = μ_{n,k}Û, where μ_{n,k} := λ_{k}P̃_{_{D}(K_{n})}

and supp(μ_{n,k}, cr) ⊂ K_{n} ⊂ A. Hence μ_{n,k} ∈ M(A). Now μ_{n.k}Û = R̂_{K_{n}}(ρÛ) + R̂_{K_{n}}(μ_{k}Û) increases with both n and k, and

lim_{n} lim_{k} μ_{n,k}Û = lim_{n} R̂_{K_{n}}ξ = R̂_{A}ξ.

Therefore μ_{n,n} ∈ M(A), and μ_{n,n}Û ↑ R̂_{A}ξ, completing the proof of (8.14).

Concluding Remarks. In comparing the results in this section to those in section 13.12 of [CW05], one should note that it is assumed there that X and X̂ are right continuous strong Markov process in strong duality although they allow both X and X̂ to have branch points. Moreover most of their deeper results such as 13.59, 13.62 and 13.65 depend on the hypothesis (ŜF) on page 371. This is not made explicit but their proofs use 13.50. Under (ŜF) one may use the original topology in defining the co-Ray compactification Ẽ as remarked at the end of the first paragraph of section 5. But then (15.1) and (15.3) of [G75] imply that the co-Ray topology agrees with the original topology. Therefore those results are corollaries of the results in this section.

It seems that in 13.65 of [CW05] it is necessary to suppose that A ⊂ E \ B̂.

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