ON THE EXISTENCE OF RECURRENT EXTENSIONS OF SELF-SIMILAR MARKOV PROCESSES. P.J. Fitzsimmons University of California San Diego.

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1 ON THE EXISTENCE OF ECUENT EXTENSIONS OF SELF-SIMILA MAKOV POCESSES P.J. Fitzsimmons University of California San Diego March 29, 26 Abstract. Let X = (X t) t be a self-similar Markov process with values in,, such that the state is a trap. We present a necessary and sufficient condition for the existence of a self-similar recurrent extension of X that leaves continuously. This condition is expressed in terms of the Lévy process associated with X by the Lamperti transformation. 1. Introduction In his pioneering study L72 of the structure of self-similar Markov processes with state space,, Lamperti posed the problem of determining those self-similar Markov processes that agree with a given self-similar Markov process up to the time the latter process first hits. Our goal in this paper is to present a necessary and sufficient condition for the existence of a recurrent extension that, in addition, leaves continuously. Our work was inspired by that of Vuolle-Apiala VA94 and ivero 5. To state our results precisely, we introduce some notation and recall some of the basic theory of self-similar Markov processes. A Borel right process X = ((X t ) t, (P x ) x ) with values in, ) is self-similar provided there exists H > such that, for each c > and x, the law of the rescaled process c H X ct, t, is P x/ch when X has law P x. The number H is the order of X, and when there is a need to emphasize H, we shall describe X as being H-self-similar. By the discussion in section 2 of L72, we can (and do) assume that X is a Hunt process; thus in addition to being a right-continuous strong Markov process, the sample paths of X are quasi-left-continuous. One of several zero-one laws developed by Lamperti states that if (1.1) T := inf{t > : X t = }, then either P x T < = for all x > or P x T < = 1 for all x >. Our interest is in the latter situation, which we assume to be the case throughout the rest of the paper. For definiteness, we take X to be realized as the coordinate process X t (ω) = ω(t) on the sample space Ω of all right-continuous left-limited paths from, to itself. We assume that is a trap 1991 Mathematics Subject Classification. Primary 6G18, secondary 6G51, 6J45, 6J55. Key words and phrases. self-similar, semi-stable, Lamperti transformation, recurrent extension, Cramér condition, excursion. 1 Typeset by AMS-TEX

2 2 P.J. FITZSIMMONS for X, so that each of the laws P x governing X is carried by {ω Ω : ω(t) =, t T (ω)}. The natural filtration on Ω is (F t ), and F := t F t. We write P t f(x) = P t (x, f) := P x f(x t ) for the transition semigroup of X, and U q := e qt P t dt, q >, for the associated resolvent operators. Define, for c >, Φ c : Ω Ω by Φ c ω(t) := c H ω(ct). The H-self-similarity of X means that (1.2) Φ c P x (B) := P x Φ 1 c B = P x/ch B, B F, x, c >. Observe that (1.3) T Φ c = c 1 T, identically on Ω. (1.4) Definition. A Borel right Markov process X = (X t,p x ) with state space, is a recurrent extension of X provided (i) the stopped process ((X t T ) t, P x ) has the same law as (X t,p x ), for each x, and (ii) is not a trap for X. Typically, X will be realized as the coordinate process on Ω, in which case X t (ω) = ω(t). What distinguishes X from X is the collection of laws (P x ) x. For emphasis or clarity we may at times write T instead of T, etc. In view of (1.3), if a recurrent extension X is self-similar, then its order must be the same as that of X. Let X be a recurrent extension of X, with resolvent U q, q >. Writing T for the hitting time of by X, we have ψ q (x) := P x exp( qt ) = P x exp( qt ), and by the strong Markov property of X at time T, (1.5) U q f(x) = U q f(x) + ψ q (x)u q f(), x, q >. Excursion theory I72, Ma75, B92, FG6 leads to an expression for U q f() in terms of a certain entrance law for X. Let M denote the closure of the zero set {t : X t = }, and let G denote the set of strictly positive left endpoints of the maximal intervals in the complement of M. The excursions of X from are indexed by the elements of G: The excursion e s associated with s G is the Ω-valued path defined by { e s Xs+t, t < T θ s, t :=, t T θ s, where θ s is the shift operator on Ω. Let L = (L t ) t denote X-local time at, normalized so that P e t dl t = 1. Then there is a σ-finite measure n on (Ω, F ) such that, for predictable Z and F -measurable F, (1.6) P Z s F(e s ) = P Z t dl t nf. s G Formula (1.6) determines n uniquely, and under n the coordinate process (X t ) t> is a strong Markov process with transition semigroup (P t ) and one-dimensional distributions (1.7) n t (B) := nx t B, t < T, t >, B B,.

3 ECUENT EXTENSIONS OF SELF-SIMILA MAKOV POCESSES 3 One can show that there exists l such that t 1 {}(X s )ds = l L t for all t, P -a.s.; it follows from this and (1.6) (applied to Z s = e qs and F = T e qt f(x t )dt, first with a general f and then with f = 1) that (1.8) U q f() = lf() + nq (f) ql + qn q (1), f bpb,, where n q := e qt n t dt. (Here bpb, denotes the class of bounded real-valued Borel functions on, ). The denominator in (1.8) is equal to T ql + n qe qt dt = ql + n1 e qt, which is finite for all q >, as follows from (1.6) with F = 1 exp( qt ) and Z s = e qs. The family (n t ) t> is an entrance law: n t P s = n t+s for all t, s >. The measure n is uniquely determined by (n t ) and (P t ). If the recurrent extension X is self-similar, then (i) l = VA94, p. 551 and (ii) either nx = = or nx > = VA94, Thm (1.9) Definition. A recurrent extension X is said to leave continuously provided nx > =. It follows easily from (1.6) that X leaves continuously if and only if P X s = for all s G = 1. In this paper we focus on recurrent extensions that leave continuously, referring the interested reader to VA94 and 5 for discussion of extensions for which nx = =. We shall produce an H-self-similar recurrent extension of X by constructing a suitable excursion measure n as above. Motivated by the preceding discussion, we make the following definitions. ecall that (P t ) t is the transition semigroup for X. (1.1) Definitions. (a) A measure n on (Ω, F ) is an excursion measure provided (i) the measure n t defined on B, by formula (1.7) is σ-finite for each t >, (ii) (X t ) t> under n is Markovian with transition operators (P t ) and one-dimensional distributions (n t ), and (iii) n puts no mass on the zero path : t. (b) We say that an excursion measure n is self-similar if (1.11) Φ c n = c γ n, c >, for some γ >. Equivalently, (1.12) n ct (c H B) = c γ n t (B), c >, t >, B B,, (c) An excursion measure n is admissible provided n1 exp( T ) <. As noted above, the excursion measure associated with a recurrent extension of X is necessarily admissible. Itô I72 showed that the excursion measure n determined by a recurrent extension X (as in (1.6)) satisfies a set of six necessary conditions. These conditions are not quite sufficient for

4 4 P.J. FITZSIMMONS the existence of a recurrent extension, but Salisbury S86a, S86b discovered that if two of the conditions are strengthened, then the resulting set of conditions is sufficient (and still necessary). These results hold for processes with very general state spaces. The special case of, -valued Feller processes was treated by Blumenthal in B83. Vuolle-Apiala VA94 verified that the conditions imposed by Blumenthal are satisfied in the setting of self-similar Markov processes on,. Thus, if n is an admissible excursion measure, then X admits recurrent extensions X l (one for each l ) such that (1.8) holds. The process X has the additional property that U q (x, 1 {} ) = for all x, a condition that is necessary for self-similar recurrent extensions, as was noted above. By VA94, (1.5) (see also 5, Lem. 2), this extension is self-similar if and only if n is self-similar. Our hypotheses will be stated in terms of the Lévy process associated with X by the Lamperti transformation. For this consider the continuous additive functional A defined by (1.13) A t := t and its right continuous inverse τ defined by X 1/H s ds, t, (1.14) τ(u) := inf{t > : A t > u}, u, in which we follow the usual convention that inf = +. According to L72, Thm. 4.1, the, -valued process (1.15) Z u := log X τ(u), u, is a Lévy process (i.e., a process with stationary independent increments). Moreover, by L72, Lem we have either (1.16) P x X T >, T < = 1, x >, or (1.17) P x X T =, T < = 1, x >. If (1.16) holds then the random variable ζ := A T is exponentially distributed (with rate δ >, say) and there is a real-valued Lévy process Z independent of ζ such that Z is Z killed at time ζ: { Zu, u < ζ; (1.18) Z u =, u ζ. If (1.17) holds then ζ = A T = +, P x -a.s. for all x >, and Z is a real-valued Lévy process that drifts to. Let Q z denote the law of Z under the initial condition Z = z. The process Z = (Z t,q z ) is referred to as the Lévy process underlying X. We shall write (Q t ) for the transition semigroup of Z. Because A and τ are strictly increasing on their respective domains of finiteness, the Lamperti transformation can be inverted as follows. Let (Z t,q z ) be a, -valued Lévy process that drifts to in case ζ := inf{t : Z t = } = +, a.s. (The state is a trap, and serves as the cemetery state for Z.) Then (1.19) τ(u) := u exp(z s /H)ds

5 ECUENT EXTENSIONS OF SELF-SIMILA MAKOV POCESSES 5 is finite for all u ; see BY5, Thm.1. Define the inverse of τ: (1.2) A(t) = A t := inf{u : τ(u) > t}, t. For each x >, the process defined by (1.21) exp(z A(t) ), t, has, under Q log x, the same law as X under P x. Here is the main result of the paper. Define I := τ( ) = exp(z v /H)dv. (1.22) Theorem. (a) There is an excursion measure n satisfying (1.11) such that nx > = if and only if κ := γ/h > satisfies Cramér s condition: (1.23) Q exp(κz t ) = 1, for some (or all) t >. (b) The H-self-similar Markov process X admits a self-similar recurrent extension that leaves continuously if and only if (1.23) holds for some κ, 1/H and (1.24) Q I κh 1 <. There is at most one self-similar recurrent extension that leaves continuously. (1.25) emarks. (a) If the excursion measure promised by part (a) of Theorem (1.22) is to correspond to a self-similar recurrent extension X, then κh must lie in, 1, because the inverse local time of X at is a stable subordinator of index κh. (b) Let C, denote the class of continuous real-valued functions f on, such that lim x f(x) = lim x + f(x) =. Vuolle-Apiala VA94 introduced the following condition (VA): There exists k > such that the limit (VA.1) exists in,, and the limit P x 1 exp( T ) lim x + x k U q f(x) (VA.2) lim x + x k exists for all q > and all f C,, and is strictly positive for at least one non-negative f C,. Vuolle-Apiala showed that (VA) implies the existence (and uniqueness) of a selfsimilar recurrent extension of X that leaves continuously. (c) The meaning of the parameter k in (VA) was clarified by ivero 5, who introduced the following condition (), expressed in terms of the underlying Lévy process Z: (.1) The law of Z 1 is not supported by a lattice rz; there exists θ > such that (.2) Q exp(θz t ) = 1, for some (or all) t > and (.3) Q Z + 1 exp(θz 1) <. Suppose that () holds. ivero showed that if θh 1, then condition (VA) must fail, while if < θh < 1 then (VA) holds for k = θ. Conversely, ivero showed that if (VA) holds then < kh < 1 and (.2) and (.3) hold with θ = k. Thus, modulo the technical condition (.1), (VA) and () are equivalent, for k = θ, 1/H. Moreover, under (VA), k = θ = κ (as in (1.23)). (d) The essentially equivalent conditions (VA) and () imply our conditions (1.23) (with < κh < 1) and (1.24). It is not clear how broad the gap is between these conditions.

6 6 P.J. FITZSIMMONS 2. Proof of Theorem (1.22). (a) Let us first suppose that X admits an excursion measure n such that nx > = and such that the scaling property (2.1) Φ c n = c γ n, c >, holds for some γ >. The mean occupation measure m defined on the Borel subsets of, by T (2.2) m(b) := n 1 B (X s )ds, B B,, is an X-excessive measure; see FG6, (4.5). As a consequence of (2.1), m takes the form (2.3) m(dx) = Cx 1+(1 γ)/h dx, for some constant < C < ; see 5, Lem. 3. Briefly, (2.1) implies that m scales as follows: m(b H B) = b 1 γ m(b); this in turn implies (2.3). By the theory of time changes for Markov processes, as found for example in FG88, the measure ν := dx x 1 γ/h dx is therefore excessive for the time-changed process t X τ(t). Define κ := γ/h. Then ξ : dz e κz dz (the image of ν under the mapping x log x) is excessive for the Lévy process Z. Let (Y t, Q) denote the Kuznetsov measure associated with Z and ξ. In more detail, let W be the space of paths w :, + that are -valued and right-continuous with left limits on an open interval α(w), β(w) and that hold the value outside that interval. (Thus serves as cemetery state for Y.) The coordinate maps Y t (w) := w(t), t, generate G := σ{y t : t }, and Q is the unique measure on (W, G) not charging the dead path : t, and such that (2.4) QY t1 dx 1, Y t2 dx 2,..., Y tn dx n = ξ(dx 1 )Q t2 t 1 (x 1, dx 2 ) Q tn t n 1 (x n 1, dx n ), for all real t 1 < t 2 < < t n and x 1, x 2,..., x n. Thus Y = (Y t ) under Q is a stationary Markov process with random times of birth and death (namely, α and β), with one-dimensional distributions (while alive) all equal to ξ, and with transition probabilities those of the Lévy process Z. For the construction and various properties of such processes the reader is referred to Mi79, F87b, G9. We now use a time-reversal argument to show that ξ is in fact invariant for Z. To this end define a semigroup ( Q κ t ) by the formula (2.5) Qκ t f(x) := Q f(x Z t )e κzt, and observe that ( Q κ t ) is the semigroup dual to (Q t ) with respect to ξ. That is, f Q t g dξ = g Q κ t f dξ, t >, f, g pb ; cf. the computation in (2.15) below. Thus (2.6) QY t1 dx 1, Y t2 dx 2,..., Y tn 1 dx n 1, Y tn dx n = ξ(dx n ) Q κ t n t n 1 (x n, dx n 1 ) Q κ t 2 t 1 (x 2, dx 1 ),

7 ECUENT EXTENSIONS OF SELF-SIMILA MAKOV POCESSES 7 The moment generating function Q exp(λz t ), λ, is necessarily of the form (2.7) Q exp(λz t ) = exp(tψ(λ)), where ψ :, +. By Hölder s inequality, log ψ is convex (strictly convex on the interior of the interval where it is finite). Now either Z drifts to (in which case ψ() = and Q Z 1 = ψ (+), ) or jumps to (in which case ψ() < ). Clearly 1 Q κ t 1(x) = exp(ψ(κ)t), so ψ(κ). If ψ(κ) < then Q κẑκ t = = 1 exp(ψ(κ)t) >. If ψ(κ) = then Q κẑκ t = Q Z t e κzt = ψ (κ ), by convexity. Therefore Ẑκ either jumps to at a finite time (if ψ(κ) < ) or drifts to (if ψ(κ) = ). In either case, the exponential integral Î := is finite, Q z κ-a.s., for all z. Thus if we define (2.8) ρ(u) := u then (2.9) Qρ(u) =, α < u < β = exp(ẑκ s /H)ds exp(y s /H)ds, u, Q z κ Î = ξ(dz) =, by Mi79, (4.7). In particular, lim ρ(u) =, Q-a.s. u α It follows that the inverse process C defined by C(t) := inf{u : ρ(u) > t}, t, is strictly increasing and continuous on, ρ( ). ecalling from section 1 the discussion of the (inverse) Lamperti transformation, we see that the measure m is proportional to the image under the mapping z exp(z) of the evuz measure of the CAF τ relative to the Z-excessive measure ξ. By a result of Kaspi K88, (2.3), (2.8), the formula (2.1) η t (f) := Qf(exp(Z C(t) )), < C(t) 1, f pb,, t >, defines an entrance law for (P t ) and (2.11) m = η t dt. But by (2.2) and Tonelli s theorem we also have m = n t dt, so by the uniqueness of such a representation G9, (5.25), η t = n t for all t >. Let Ω + be the space of right-continuous left-limited paths from, to,, and let n + be the image of n under the mapping Ω ω ω, Ω +. Let F + be the σ-field on Ω + generated by the coordinate maps X + t, t >. Theorem (2.1) of K88 now tells us that if Π : W Ω + denotes the (inverse Lamperti) transformation w (t exp(y C(t,w) (w)), t > ) then (2.12) n + A, t < T = QΠ 1 (A), < C(t) 1, A F +.

8 8 P.J. FITZSIMMONS In particular Q lim sup Y u >, < C(t) 1 u α = n + lim sup X s + >, t < T + s = nx >, t < T =, for all t >, from which it follows that lim u α Y u =, Q-a.s. By time reversal (as in (2.9)), this means that the dual process Ẑκ does not jump to ( Q z -a.s. for ξ-a.e. z ). That is, ψ(κ) = ; equivalently, Q e κzt = 1, t >, which is (1.23). Conversely, suppose that (1.23) holds for some κ >. Then ξ(dz) = e κz dz is an invariant measure for Z: ξq t (f) = e κz Q z f(z t ) dz = e κz Q f(z + Z t ) dz (2.13) = Q e κz f(z + Z t )dz = Q e κ(zt y) f(y)dz = e κy f(y)q e κzt dy = e κy f(y), dy = ξ(f). Let (Y, Q) be the Kuznetsov process for Z and ξ as before. Because ξ is invariant, Qα > = ; see G9, (6.7). Making use of (2.5) and the discussion following (2.6) we see that Q κẑκ t = tψ (κ ),, so that Ẑκ drifts to. As before, (2.6) holds; consequently the time-reversal argument used before to show the finiteness of the integral in (2.8) now permits us to deduce that Q is carried by {α =, lim t α Y t = }. Define m(dx) = x 1 κ+1/h dx. The evuz measure of the continuous additive functional τ(u) := u exp(z v/h)dv, relative to ξ, is the measure µ(dz) := e z/h ξ(dz) on. The image of µ under the mapping z exp(z) is the measure m; another application of Kaspi s theorem shows that m is purely excessive for X, so there is a uniquely determined entrance law (η t ) t> such that m = η t dt. As before let Ω + be the space of right-continuous left-limited paths from, to,. Let n + be the measure on Ω + under which coordinate process (X + t ) t> is Markovian with one-dimensional distributions (η t ) t> and transition semigroup (P t ); see GG87 for the construction of such measures. Then by K88, Thm. 2.1, writing Π : W Ω + for the map w (t exp(y C(t,w) (w)), t > ), we have (2.14) n + B, t < T = QΠ 1 (B), < C(t) 1, B F +. In particular, the choice B = B := {ω Ω + : lim t ω(t) = } in (2.14) shows that that n + is carried by B because Q is carried by {w W : lim t w(t) =, α(w) = }. Identifying Ω with B and F with F + B, we see that n + induces an excursion measure n on (Ω, F ) as in Definition (1.1). Now write Ψ x : w (w(t) + x) t for the spatial translation operator on W; a check of finite dimensional distributions shows that Ψ x Q = e κx Q.

9 ECUENT EXTENSIONS OF SELF-SIMILA MAKOV POCESSES 9 This, in combination with (2.14), implies that n is self-similar, with similarity index γ = κh. This concludes the proof of part (a) of Theorem (1.22). (b) For any self-similar excursion measure n (with γ = κh) we have, from the proof of Lemma 3 in 5, that (2.15) n1 exp( T ) = CH Γ(1 κh) Q I κh 1, where the constant C is as in (2.3). Thus, a self-similar excursion measure n is admissible if and only if (1.24) holds. Finally, turning to the uniqueness assertion, let X 1 and X 2 be self-similar recurrent extensions of X that leave continuously. In view of (1.5) and (1.8), the resolvent (and so the distribution) of X j is uniquely determined by the associated entrance law (n j t ), j = 1, 2. Because of a uniqueness theorem G9, (5.25) cited earlier, each entrance law is in turn uniquely determined by its integral m j = n j t dt. But as noted at the beginning of this section, mj has the form m j (dx) = C j x 1+(1 γ)/h dx, where γ = κh. (Both X 1 and X 2 satisfy (1.23), by the part of Theorem (1.22) already proved.) It follows that n 1 t = Cn 2 t for all t > (C = C 1 /C 2 ), so X 1 and X 2 have the same resolvent. (2.16) emark. If n is a self-similar excursion measure (with similarity index γ) then (1.3) implies that there is a constant C, such that (2.17) nt > t = Ct γ, t >. In particular, if n is the excursion measure guaranteed by part (a) of Theorem (1.22), then (2.15) and (2.17) combine to show that C = CHQ I γ 1, and so condition (1.24) holds if and only if nt > 1 <. 3. An Application. Suppose that our self-similar Markov process X satisfies (1.16). In this section we shall apply part (a) of Theorem (1.22) to obtain the following improvement of Theorem 6(i) of Chaumont and ivero C6. (3.1) Theorem. Suppose there exists κ > such that (3.2) Q exp( κz t ) = 1, for some (or all) t >. Then h : x x κ is a purely excessive function for X, and the h-transform process X h with laws (3.3) P x h F, t < T = x κ P x FX κ t, t >, F bf t, x >, is X conditioned to converge to : (3.4) P x hx T =, T < = 1, x >. Proof. Let us start with the Lévy process Ẑ := Z, the dual of Z with respect to Lebesgue measure on, and apply the inverse Lamperti transformation to obtain a self-similar Markov

10 1 P.J. FITZSIMMONS process X = (( X t ) t, ( P x ) x ) on, with as a trap. The process X is in weak duality with X with respect to the measure η(dx) := x 1+1/H dx; see VG86. By hypothesis, X satisfies the condition (1.23) of part (a) of Theorem (1.22). In particular, P x T < = 1 for all x >. (Otherwise, the Lévy process Ẑ has infinite lifetime and limsup t + Ẑt = +, a.s., hence the moment generating function ψ of Ẑ (defined by analogy with (2.7)) vanishes at as well as at κ. Since ψ in convex on, κ, this yields ψ (+),, implying that Ẑ drifts to, a contradiction.) By Theorem (1.22)(a), applied to X, there is an X-excursion measure n under which the coordinate process on (Ω, F ) is a strong Markov process with transition semigroup ( P t ) (that of X) and entrance law ( n t ) t>, say. Moreover, (3.5) m(b) := n t (B)dt = C x 1 κ+1/h dx, B B,, B for some constant < C <. By Nagasawa s theorem, as found for example in the section 4 of F87a (see also D85 and the appendix of F99), the process X defined by { X(T t), if t < T, (3.6) Xt :=, otherwise, is Markovian under n, with transition semigroup (3.7) P h t f(x) := x κ P x f(x t )X κ t ; t < T i.e., the h-transform of (P t ) corresponding to h(x) = x κ. In other words, the image ñ of n under the mapping ω X(ω) is an X h -excursion measure. Because ñt = =, we have, for ǫ >, = ñǫ < T = = ñ P Xǫ h T = ; T > ǫ, so P x h T = =, first for Lebesgue a.e. x >, then for all x > because this probability does not depend on x apply (1.2) and (1.3) to X h. Similarly, so if ǫ >, = ñ ñx = n lim sup t e T Xt > {T > ǫ} θǫ 1 {lim sup X t > } t T = ñ = nx =, P Xǫ h lim sup t T X t > ; T > ǫ. It follows that P x h lim sup t T X t > =, first for m-a.e. x >, and then for all x > by the reasoning used above.

11 ECUENT EXTENSIONS OF SELF-SIMILA MAKOV POCESSES 11 eferences BY5 Bertoin, J. and Yor, M., Exponential functionals of Lévy processes, Probab. Surv. 2 (25), B83 Blumenthal,.M., On construction of Markov processes, Z. Wahrsch. verw. Gebiete 63 (1983), B92 Blumenthal,.M., Excursions of Markov Processes, Birkhäuser, Boston, C6 Chaumont, L. and ivero, V., On some transformations between positive self-similar Markov processes, preprint, 26. D85 Dynkin, E.B., An application of flows to time shift and time reversal in stochastic processes, Trans. Amer. Math. Soc. 287 (1985), F87b Fitzsimmons, P.J., Homogeneous random measures and a weak order for excessive measures of a Markov process, Trans. Amer. Math. Sci. 33 (1987), F87a Fitzsimmons, P. J., On the excursions of Markov processes in classical duality, Probab. Theory elated Fields 75 (1987), F99 Fitzsimmons, P. J., Markov processes with equal capacities, J. Theoret. Probab. 12 (1999), FG88 Fitzsimmons, P.J. and Getoor,.K., evuz measures and time changes, Math. Z. 199 (1988), FG6 Fitzsimmons, P.J. and Getoor,.K., Excursion theory revisited, To appear in Illinois J. Math. (Doob volume), 26. GG87 Getoor,. K. and Glover, J., Constructing Markov processes with random times of birth and death, Seminar on Stochastic Processes, 1986, Birkhäuser, Boston, 1987, pp G9 Getoor,.K., Excessive Measures, Birkhäuser, Boston, 199. I72 Itô, K., Poisson point processes attached to Markov processes, Proc. Sixth Berkeley Symp. Math. Stat. Prob., vol. 3, 1971, pp K88 Kaspi, H., andom time changes for processes with random birth and death, Ann. Probab. 16 (1988), L72 Lamperti, J., Semi-stable Markov processes. I, Z. Wahrsch. verw. Gebiete 22 (1972), Ma75 Maisonneuve, B., Exit Systems, Ann. Probab. 3 (1975), Mi79 Mitro, J.B., Dual Markov processes: construction of a useful auxiliary process, Z. Wahrsch. verw. Gebiete 47 (1979), ivero, V., ecurrent extensions of self-similar Markov processes and Cramér s condition, Bernoulli 11 (25), S86a Salisbury, T., On the Itô excursion process, Probab. Theory elated Fields 73 (1986), S86b Salisbury, T., Construction of right processes from excursions, Probab. Theory elated Fields 73 (1986), VA94 Vuolle-Apiala, J., Itô excursion theory for self-similar Markov processes., Ann. Probab. 22 (1994), VG86 Vuolle-Apiala, J. and Graversen, S. E., Duality theory for self-similar processes, Ann. Inst. H. Poincaré Probab. Statist. 22 (1986),