ADDITIVE FUNCTIONALS AND EXCURSIONS OF KUZNETSOV PROCESSES

Let B be a continuous additive functional for a standard process ( X t ) t ∈ ℝ + and let ( Y t ) t ∈ ℝ be a stationary Kuznetsov process with the same
semigroup of transition. In this paper, we give the excursion laws of ( X t ) t ∈ ℝ + conditioned on the strict past and future without duality hypothesis. We study excursions of a general regenerative system and of a regenerative system consisting of the closure of the set of times the regular points of B are visited. In both cases, those conditioned excursion laws depend only on two points X g − and X d , where ] g , d [ is an excursion interval of the regenerative set M . We use the ( F D t ) -predictable exit system to bring together the isolated points of M and its perfect part and replace the classical optional exit system. This has been a subject in literature before (e.g., Kaspi (1988)) under the classical duality hypothesis. We define an “additive functional” for ( Y t ) t ∈ ℝ with B , we generalize the laws cited before to ( Y t ) t ∈ ℝ , and we express laws of pairs of excursions.


Introduction
Let X be a standard process, and let M be a closed random and homogeneous subset of R + .Kaspi [8] constructs an additive functional B associated to M and gives, under the classical duality hypothesis, the probability measures allowing the law of excursions to be associated to B with respect to the σ-algebra K = σ(Z t : t ∈ R + ), known to start at x and end at y (Z t = X St where S t = inf{u : B u > t}).The purpose of this paper is to give, without duality, the conditional law P x,y of the excursion straddling an arbitrary random time, given the initial state x and the final state y, as regular probabilities in terms of the (F Dt )-predictable exit measures for M and also for a regenerative system consisting of the closure of the set of times the regular points of an arbitrary continuous additive functional are visited.We also give the conditional laws of pairs of excursions for a Markov process with random birth and death (Y t ) t∈R having the same semigroup as X.In this respect, we define an "additive functional" for (Y t ) t∈R and we extend this result concerning the probability measures P x,y to (Y t ) t∈R .
In Section 2, we introduce our notations, preliminaries, and Maisonneuve's result [12] on the strict past conditioning with respect to the filtration (F Dt ).In Section 3, we construct the probability measures P x,y , which allows the law of the excursion to straddle an arbitrary random time, given the initial state x and the final state y.Section 4 deals with excursions associated to a continuous additive functional B. The measures P x,y which govern these excursion are the same as defined in Section 3 corresponding to the regenerative set M where contiguous intervals are of the form ]S t − ,S t [, t is a time of discontinuity of S. Laws of excursions and of pairs of excursions for (Y t ) t∈R are discussed in Section 5.

Notations and preliminaries
Let (Ω,F,F t ,X t ,θ t ,P x ) be a canonical realization for a Borel standard semigroup (P t ) with lifetime ζ, and let M be a closed random and homogeneous subset of ]0,ζ[ such that R = inf M is F * -measurable, where F * is the universal completion of the σ-algebra F 0 = σ(X t : t ∈ R + ).We assume that the state space E is Lusinian, and we denote by Ᏹ its σalgebra of Borel sets.The cimetry point δ is outside of E. We denote by G 0 the set of the left endpoints of the contiguous intervals of M.
Let ( • P x ) x∈E∪δ be the family of (F D t )-predictable exit measures for the process (X D t ) = (X Dt ) in the sense of Maisonneuve [11], and let µ be a fixed law on E. Then ( • P x ) x∈E∪δ is a universally measurable family of σ-finite measures on (Ω,F * ), under which the process (X t ) is Markov with respect to (P t ).
For all t ∈ R + , let k t be the killing operator at t defined by Let T be a random time on (Ω,F) such that T < D T on {T < ζ} (D t = inf{s ≥ t : s ∈ M} for t ∈ R + with the convention inf ∅ = +∞), and let g = sup{s ≤ T : s ∈ M} and d = inf{s > T : s ∈ M}.Then with the following notations: if υ is a measure on (Ω,F * ), we have the basic Maisonneuve formula [12].
For almost all ω ∈ {g < ∞} we have (if δ is nonabsorbent) for every F * -measurable function f ≥ 0, where P is the probability measure defined by P( f ) = P x ( f )µ(dx).If we assume that δ is absorbent, then this formula is true on {g < ζ} instead of {g < ∞}.
Note that if T is an (F D t ) = (F Dt )-stopping time, we can replace C ω by the condition A ω < R, and if T ∈ G 0 on {X T ∈ E}, the set C ω can be replaced by the condition A ω = 0.

The excursion straddling T
For the conditional law of the excursion e = k R • θ g straddling T, with respect to F D g − and θ d , we assume that δ is absorbent.In this respect, we consider, for (x, y) ∈ E × E, Hacène Boutabia 2033 the measures H x , H x 1 and P x,y on (Ω,F * ) "defined by" Since (Ω,F 0 ) is a U-space, and according to a classical lemma of Doob, the measures P x,y can be chosen measurable for the pair (x, y).
and let the probability measure µ x be defined on (Ω,F * ) by where where which follows from the Markov property at time R with an argument of monotone classes, the definitions of P x,y and H x , and the fact that The following theorem gives the conditional law of the excursion e with respect to F D g − and θ d .
Theorem 3.2.For all ω ∈ {g < ∞}, let the subset of Ω be defined by By formula (2.2) and the definition of µ x with x = X D g − (ω) and A = C ω , the left side of formula (3.7) is equal to which by formula (3.2) is equal to and using formula (2.2) again, we obtain the right side of (3.7).Formula (3.6) is argued in the same manner using formula (3.4).
Remark 3.3.Maisonneuve [12] gives several examples where the set C ω is independent of ω.In these cases Theorem 3.2 implies that the excursion e is conditionally independent of Remark 3.4.Theorem 3.2 contains results of Kaspi [8, Section 5] under duality hypothesis.In fact if M is perfect, then X D g − = X g − and F D g − = F g − .If T is the beginning of the set {t ∈ R + : (X t − ,X t ) ∈ J}, where J ∈ Ᏹ ⊗ Ᏹ, then with the assumption that • P x (X 0 − = x) = 0, the conditions 0 < g(ω) < T(ω) and θ g (ω) ∈ C ω are equivalent to the condition θ g (ω) ∈ {(X 0 − ,X 0 ) / ∈ J; 0 < T < R}, and formula (2.2) becomes According to the same argument used in Theorem 3.2 and the fact that and formula (3.6) becomes (3.12)

Excursions associated to an additive functional
Let (B t ) be a continuous additive functional and let C = {x : P x (R = 0) = 1} be its fine support, where R is the perfect exact terminal time inf{u : B u > 0}.We associate to the right inverse S t = {u : B u > t} of (B t ) the notations Z t = X St , ᏹ t = F St , and θ t = θ St .It is well known that the process Z = (Ω,F,ᏹ t ,Z t ,θ t ,P x ) is strong Markov with semigroup (P t ) (P St ) and takes values on (C,C ∩ Ᏹ * ) (cf.Jacobs [7]).
In this section, we assume that δ is nonabsorbent and we consider the random homogeneous set M = {t + R • θ t : t ∈ R + } and its family of (F Dt )-predictable exit measures ( 0 P x ) x∈E∪{δ} .If S t − = S t , then D S t − = S t .The excursion associated to t is then defined by We denote by (K t ) t∈R+ the filtration, where K t is the intersection of the P πcompletions of the σ-algebra K 0 t + where π is in the set of all the bounded measures on E; (K 0 t ) is the natural filtration of the process (Z t ).For the following lemma we put K 0 − = F 0 by convention.Lemma 4.1.Let T be a (K t )-stopping time such that S T − < S T a.s.Then Proof.According to the fact that S T − is not an isolate point of M, we have Note that S T − = sup r<T {S r : r ∈ Q + }, which implies that where Q + is the set of positive rationals.For all u ≤ t we have X u = Z Bu , then F t ⊂ K Bt , which implies that and The following theorem which gives the conditional law of the excursion e T associated to a (K t )-stopping time T, with respect to the σ-algebra K generated by K t (t ≥ 0), was proved by Kaspi [8] under the duality hypothesis.Theorem 4.2.Let T be a finite (K t )-stopping time.Then (1) S T − is an (F Dt )-stopping time, (2) it is assumed that S T − = S T and Z T − = Z T a.s., then the following formula: holds for every positive and Let (T n ) n∈N be the nondecreasing dyadic approximation of T, then which implies that (2) For every continuous (K t )-adapted process U ≥ 0, and for every positive F *measurable function ϕ, we have by formula (3.5) with S T − instead of T and the fact that Formula (4.5) follows from the fact that K is generated by K T − and θ T .

Excursions of Kuznetsov processes
Let W be the set of applications w : R → E ∪ {δ} which satisfies the following properties: there exists an open interval of R on which w is E-valued right-continuous with left limits and out of which w equals δ.We denote by (Y t ) t∈R the coordinate process on W. Let (Ᏻ 0 t ) t∈R be the natural filtration of (Y t ) t∈R and let Ᏻ 0 = ᐂ t∈R Ᏻ 0 t .Then the birth and the death times of (Y t ) t∈R are, respectively, (5.1) We define the families of operators on W by Let η be an excessive measure with respect to (P t ) and let Q be the Kuznetsov measure on W that corresponds to (η,(P t )) (cf.[9,10]).We denote by Ᏻ t and Ᏻ the Q-completions of Ᏻ 0 t and Ᏻ 0 , and we assume that the semigroup (P t ) satisfies "les hypothèses droites de Meyer."It follows by [13] that the process Y = (W,Ᏻ,Ᏻ t ,(Y t ) t∈R ,τ t ,α,β,Q) is stationary (i.e., σ t (Q) = Q) and strong Markov with semigroup (P t ).
For the generalization of Theorem 4.2, we consider the additive functionals B and S given in the previous section.We also denote by B the random measure on W, carried by ]α,β[ such that (5.3) We assume that the characteristic measure υ B Q of B is purely excessive for the semigroup (P t ) (i.e., P t f (x)υ B (dx) → 0 as t → ∞ if υ B ( f ) < ∞).It was shown in [9] that Q a.e.B]α,t] < ∞ for all t > α.
Let (V t ) t∈R be the nondecreasing process defined on W by and let (U t ) t∈R be the right-continuous inverse of (V t ) t∈R , that is, We also denote by M the closed random subset of ]α,β[ defined by M = α<t<β {t + R • τ t } which verifies the following property of homogeneity (cf.[4]): (5.6) , and Ᏼ 0 = ᐂ t∈R Ᏼ 0 t .We denote by Ᏼ t (resp., Ᏼ) the Q-completion of Ᏼ 0 t + (resp., Ᏼ 0 ).Note that for all the following formulas, the σ-finiteness of Q is guaranteed by the argument used in [1].It is not hard to show that (Φ t ) has the same properties as (Z t ) and that the following result holds.Proposition 5.2.(1) The process (U t ) is right-continuous, has left limits, and satisfies U t = U β for all t ≥ β Q a.e.
If U t = U t − , let E t be the excursion associated to B and defined by (5.7) According to the previous proposition, the process (V t ) t∈R has got the same role as B for the process (Y t ) t∈R .We say that (V t ) t∈R is an "additive functional" for (Y t ) t∈R .We have the extension of Theorem 4.2 on W.
Theorem 5.3.(1) The process Φ = (W,Φ t ,Ᏻ,Ᏻ t ,τ t ,Q) is strong Markov in the sense that for all (Ᏻ t )-stopping time T and s > 0, Q f Φ T+s | Ᏻ T = P s f ,Φ T on α < U T < β (5.8) for every positive and F-measurable function f .
Let Z be a positive F D g − -measurable random variable carried by {g < ζ; d − g < ∞}, and let ϕ be a positive F 0 -measurable function.We have to prove that .6) It follows that if T is an (F Dt )-stopping time such that T ∈ G 0 on {X T ∈ E}, then formulas (3.5) and (3.6) hold without conditioning by U ω d in the right sides.2034Additivefunctionals and excursions of Kuznetsov processesProof.