ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 51-65 A decomposition for Markov processes at an independent exponential time Received 29 September 2015, accepted 12 October 2015, published online 22 April 2016 The path of Markov process X run up to an independent exponential random time Sg can be decomposed into the part prior to the last exit time from a point before Sg, and the remainder up to Sg. In this paper the laws of the two segments are identified under suitable assumptions using excursion theory. Keywords: Markov processes, excursions, last exit decomposition, diffusions, Brownian motion. Math. Subj. Class.: 60J25, 60J55, 60J60, 60J65, 60G51. 1 Introduction Considering a Markov process X up to an independent exponential time Sg with rate 6 > 0 has been used effectively to compute functionals of X. The computations are often based on decompositions of the path of X up to Sg into fragments before and after the last exit time from a set before time Sg. The known results described below are more general in the sense that the path is decomposed at the last exit from a set before either fixed times or random times belonging to a suitable family. Choosing an independent exponential time in some cases leads to simpler descriptions of the laws of the two fragments involved. They are often conditionally independent given suitable conditioning variables and their laws are related to laws of known processes. Williams [27] uses a decomposition of Brownian motion with drift run up to an independent Sg to prove result of Ray [23] on the distribution of local times in the space *This work is supported in part by the Slovenian Research Agency (research program P1-0285). E-mail address: mihael.perman@fmf.uni-lj.si (Mihael Perman) Mihael Perman * Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia and Department of Mathematics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 52 Ars Math. Contemp. 12 (2017) 37-50 variable. In their investigations of Ray-Knight theorems for Brownian motion B at fixed times Biane and Yor [4] considered the pair of processes (Bt: 0 < gSe) and (BSg -t: 0 < t < Sg — gSg) where Sg is an exponential random variable independent of B and gt — sup{s < t: Bs =0} is the last exit time from 0 before time t. Under P0 the two processes are shown to be independent and their conditional laws given the local time L(Sg) at zero of Brownian motion and BSg respectively are identified. This decomposition has been exploited by Jeanblanc, Pitman and Yor [13] to derive Feynman-Kac formulae for Brownian motion. Salminen, Vallois and Yor [26] extend the decomposition for Brownian motion to linear diffusions on [0, to) with 0 a recurrent point and use them to study the excursion of the diffusion straddling an independent exponential time. For general Markov processes Pittenger and Shih [22] investigated the dependence of the fragments of the path of a cadlag strong Markov process X before coterminal time Lt < t and the fragment on the interval between Lt and t. Last exit times LfF before time t from a closed set F are coterminal times. It is shown that given a suitably defined a-algebra FLt, the conditional law of the process (XLt+s: 0 < s < t - Lt) only depends on Lt and Xlf or XLt- and is an inhomogenous strong Markov process. Getoor and Sharpe [7] give related results. General and elegant treatments of last exit decompositions are given in Maisonneuve [18] and Pitman [21]. Kallenberg [14] proves that for Levy processes the fragments considered by Pittenger and Shih are conditionally independent given XL^ - and LF where the last exit time from a set F is an instance of a backward time. Under suitable conditions the laws of the two fragments are described. Another example of considering a decomposition of a Markov process at last exit time from 0 before and independent exponential time Sg is the proof of fluctuation equalities for Levy processes given by Greenwood and Pitman [9]. If X is a Levy process then it is known that the process reflected at the supremum defined by Yt — sups 0 which implies the Chapman-Kolmogorov equations for all s, t > 0 and x, y G E. M. Perman: A decomposition for Markov processes at an independent exponential time 53 To formulate the results the existence of a dual strong Markov process X on (E, E) relative to the measure £ will be assumed. This means that £ is an invariant measure for both X and X and Px(Xt e dy) = pt(x, y)£(dy) and Px(Xt e dy) = p(y, x)£(dy) (2.1) for all t > 0, x, y e E with pt(x, y) = pt(y, x). See [6] and the references therein for details. Denote by re(x, y) = / e_etp (x, y) dt and re (x,y)= / e_etpt(x,y) dt ./0 ./0 the resolvent densities of X and X respectively. For the sake of simplicity it will be assumed that X and X have infinite lifetimes Z under Px and Py for all x, y e E respectively. The assumptions on X imply that it is possible to define bridge laws PX,y (■)= Px(-|Xt = y) (2.2) for t > 0 and for x, y e E. By Proposition 1 in Fitzsimmons, Pitman, Yor [6] for any x, y e E and t > 0 with pt(x, y) > 0 there is a unique law PX,y on (Q, Ft) such that for any Fs-measurable functional F for 0 < s < t Et,y(F) ■ pt(x,y) = Ex (F ■ Pt_s(Xs,y)) , (2.3) where EX ,y and Ex are expectations with respect to measures PX,y and Px respectively. The laws Px,y provide a regular version of the family of conditional distributions P( |Xt = y). Furthermore by Corollary 1 in Fitzsimmons, Pitman, Yor [6] the law of the reversed bridge (X(t_s)_ : 0 < s < t) under Px,y has the law of the bridge of the dual process Py,x. The subject of this paper is the law of the process X started at a and run to an independent exponential time Se with rate 0 and conditioned on |XSe = b}. Conditionally on {Se = t, Xt = b} the law of the process will be the law of the bridge P^ b and the laws P,t b will serve as the regular version of the family of conditional distributions. Assume that a is a recurrent point of the process X. Let Ta = inf {t > 0: Xt_ = a or Xt = a}. Since a is assumed to be recurrent the assumptions imply that Pb(Ta < to) = 1 for all b e E. For t > 0 define the last exit time from a before time t as gt = sup{s < t: Xt_ = a or Xt = a} . Let (LJ1: t > 0) be the local time for the process X at a. We will assume that such a right continuous nondecreasing additive functional exists and only increases on the set M = {t > 0: Xt_ = a or Xt = a}. All the results will be valid for any choice of normalization of the local time. Let for s > 0 ts = inf{t > 0: > s} (2.4) be the right continuous inverse of the local time. From the strong Markov property of X it follows that (ts : s > 0) is a subordinator. Since we are assuming recurrence the local time at a will be unbounded and hence ts is well defined for all s. For simplicity we will assume that the set M has Lebesgue measure 0 almost surely. This means that the subordinator 54 Ars Math. Contemp. 12 (2017) 37-50 (ts : s > 0) has no drift. From properties of subordinators, see Bertoin [2], Ch. 3, it follows that Ea (e-0Tu) = e-^(e)u . (2.5) The notation X(t), La(t) and g(t) will be used for Xt, La and gt whenever necessary. The theorem to be proved is stated as follows. Theorem 2.1. Assume that a is a recurrent point for the process X and pt(a, b) > 0 for all t > 0. Let X0 = a and assume (La : t > 0) is the local time of X at a. If Sg is an exponential random variable with parameter 0 independent of X then, under the measure Pa: (i) The random variables La (Sg) and X(Sg) are independent with distributions Pa(La(Sg) € du) = ^(0) e-'^g)udu and Pa(X(Sg) € dy) = 0rg(a,y)dy (2.6) where ^(0) is the Laplace exponent defined in Equation 2.5. (ii) The processes (Xt: 0 < t < gse) and (XgSg +u :0 < u < Sg - gse) are independent. (iii) For bounded measurable Junctionals F and G Ea \F(Xs: 0 < s < tu) e-gTu 1 Ea [F(Xs :0 < s < gsg)| Lse = u] = a[ ( °E <_g<, u)-(2.7) Ea [e gTu ] Ea[G(X{se-s)- : 0 < s < Sg - gse)|Xs9 = b] = (2.8) Eb G(Xs: 0 < s < fa) e ...-gTa a) Eb[e-gTa ] where Eb refers to expectation under the law Pb of the dual process, and Ta = inf {t : Xt = a or Xt- = a} is the hitting time of a for X. The novelty lies in the fact that known special cases are covered by the more general Markov setup. Excursion arguments used are standard. 3 Excursion arguments Let n be a point process on an abstract space (S, S) with mean measure A. If n is a Poisson process then by Campbell's Theorem, see Kingman [16], p. 28, for any measurable f > 0 E (exp(—J f(x)n(dx)^ =exp J (1 - e-f (x)) A(dx)^ . (3.1) Conversely, if Equation 3.1 holds for any measurable f > 0, then n is a Poisson process with mean measure A. M. Perman: A decomposition for Markov processes at an independent exponential time 55 Assume that n is a Poisson process with mean measure A, and let h: S ^ [0, to) be a measurable function such that f (1 - e-h(x))A(dx) < to . J S (3.2) The random variable Sh defined by Sh = / h(x)n(dx) S is almost surely finite and non-negative with E (exp(-Sh)) = exp ^- (1 - e-h(x))A(dx)^ . (3.3) Define a new probability measure Q by dQ exp(-Sh ) dP E [exp(-Sh)] " (3.4) The following lemma is known in the literature, see Proposition 2.1 in James [12] and the discussion therein, or Proposition 2.4 in Bertoin [3]. Lemma 3.1. Under the measure Q, n is a Poisson process with mean measure e-h(x) • A(dx). Proof. It suffices to check that Equation 3.1 holds. Denote c = 1/E(exp(-Eh)) and let = fS f (x)n(dx) for a measurable funtion f > 0. One has Eq [exp(-Sf )] = cEP cEP exp (-Sh) • exp ^- J f (x)n(dx) ex^ -J (f (x) + h(x)) n(dx) ex^ -J (l - e-f (x)-h(x^ A(dx)^ = c exp (1 - e-f(x)) e-h(x) + (1 - e-h(x))l A(dx) ex^ -J (1 - e-f (x)) e-h(x) A(dx)^) =c S □ Let X be a cadlag strong Markov process. The set M = {t > 0: Xt- = a or Xt = a} is closed under the assumptions. Since we are assuming recurrence of X the complement of M is a countable union of bounded open intervals. The segments of the path of X on these open intervals are called the excursions of X away from the point a. By definition the open intervals coincide with the complement of the range of the subordinator (ts : s > 0) defined in Equation 2.4. Let Us be the space of cadlag paths 56 Ars Math. Contemp. 12 (2017) 37-50 w: [0, to) ^ E such that w(0) = a and there is a t > 0 such that for 0 < s < t we have w(s) = a, w(s-) = a, and w(t) = a or w(t-) = a and w(s) = S for s > t where S is the coffin state added to E. Let U be the a-algebra generated by the coordinate maps in Us. Define the point process (es: s > 0) of excursions of X in the sense of Ito as S if ts - rs- =0 (35) es(u) = XTs_+„ for u < ts - ts- and S else. ( . ' The process e is a Poisson process in the sense of Ito governed by the measure A x n where A is a multiple of the Lebesgue measure on [0, to) and n is the Ito excursion law. We can change the normalization of the local time, if needed, in order to ensure that A is the standard Lebesgue measure. See Rogers and Williams [25], Ch. 8 or Revuz and Yor [24], Ch. 8 for background on excursion theory. The connection between excursion theory and the law of a Markov process run up to an independent exponential time Sg is established through marking excursions. Let n be a Poisson process on an abstract space (S, S). If conditionally on n each point x e n is assigned a mark with probability p(x) independently of all the other points in n then the resulting marked and unmarked processes are both Poisson and are independent. If A is the mean measure of n the marked process will have mean measure p ■ A and the unmarked process (1 - p) ■ A. See Kingman [16], Ch. 5 for definitions and proofs. The excursion of the Markov process X straddling the independent exponential random time Sg can be interpreted as the first marked excursion of X where conditionally on e marks are assigned to an excursion e with probability 1 - e-gR(e) where R(e) stands for the duration of the excursion i.e. the length of the open interval of Mc containing Sg. Intuitively we can think that Sg is the first point in a Poisson process N on (0, to) with rate 0 and independent of X. Excursions straddling a point of N are considered marked and other excursions are considered unmarked. By independence properties of Poisson processes conditionally on e the marks are assigned independently and an excursion of length R(e) contains a point of the Poisson process with probability 1 -eR(e) which follows from the Poisson distribution of points contained in the excursion interval. See Sec. 49 in Rogers and Williams [25] for definitions and proofs. The following theorem is stated in slightly more general terms allowing the Poisson process (es: s > 0) to be killed at a rate q. This would correspond to excursions of X that have infinte length. The conclusions of the theorem are well known, see Sec. 49.4 in Rogers and Williams [25]. Theorem 3.2. Let (es : s > 0) be a possibly killed excursion process of X from a recurrent point a in the sense of Ito, and let Sg be an exponential random variable with parameter 0 independent of X. Denote by (La : t > 0) the local time process of X at a and (es: s > 0) the associated excursion process governed by A x n where A is the Lebesgue measure. (i) The local time La(Sg) during the excursion straddling Sg is an exponential random variable with parameter i (1 - e-gR(e)) n(de; R< to) + q Ju where q is the rate of arrival of excursions with infinite lifetime. Moreover, it is independent of the excursion e* = e( LaSg) which may have infinite lifetime. M. Perman: A decomposition for Markov processes at an independent exponential time 57 (ii) Given La (gSg) = u the process of excursions (es : 0 < s < u) is a Poisson process in the sense oflto which is governed by the measure n given by n(de) = e-6R{e) n(de; R(e) < to) where n islto's excursion law and R(e) denotes the length of the excursion. Moreover, e* is conditionally independent of (es : 0 < s < u) given {La(So) = u}, and is independent of La(So). Proof. The first marked excursion in (es: s > 0) will arrive at an exponential time. The processes of finite length excursions and those of infinte length are independent so the rates of arrivals add. The rate of arrivals of marked finite length excursions is by definition equal to i (1 - e-0R(e)) n(de; R< to) . Ju The two processes of marked and unmarked excursions are independent. This means that conditionally on La (So) = u the process (es : 0 < s < u) is a Poisson process on (0, u) x Us. The first marked excursion is picked according to the normalized law (1 - e-0R) • n irrespective of the local time La( So). This and the independence of marked and unmarked excursions conclude the proof. □ 4 Proofs Recall that under the assumptions on X and X and if pt(a, b) > 0 there is a measure Pa b corresponding to the bridge of X starting at a and conditioned to be b at time t. The family of Pt b is a family of regular conditional laws of X given Xt = b. If So is an exponential random variable of rate 0 then under Pa the family Pa b is a regular conditional law of X given {So = t,Xt = b}. The assumptions made on X and X also imply that X and X have no jumps at fixed times. See (3.18) in Getoor and Sharpe [8] for a proof. Let Tb = inf{t > 0: Xt = b or Xt- = b}. Assume further that Pb(Tb > 0) = 0 which in conjunction with right continuity and strong Markov property implies that XTb = b. Lemma 4.1. Assume that Pa(Tb < to) = 1. Assume that pt(a, b) > 0 for all t > 0. Then esn re(M) Pa(Tb e ds\X(Se) = b)= e- Pa(Tb e ds) re(a, b) ' where re (x, y) is the resolvent density. Proof. By assumptions on b and X we have Pa(Tb < So|X(So) = b) = 1. For fixed 0 < s 0 for y G B. Let 58 Ars Math. Contemp. 12 (2017) 37-50 F =\(Tb e ds). We have Pa(Tb e ds,Tb < Se,Xse e B) = = i 6e-et dt i pt(a,y)P*ay(Tb e ds) £(dy) Js J B /CO r Oe-et dt j Pa (Tb e ds) pt-s (b, y) £(dy) = e-esPa(Tb e ds) f ady) i de-eupu(b,y) du J B J 0 = e-esPa(Tb e ds) dt re (b,y)£(dy) JB f re (b,y) Ib re (a,y) e-esPa(Tb e ds) i ■ dre (a,y) £(dy) Jb re(a,y) It follows that esn rm re(b,b) Pa (Tb e ds\X (Se) = b)= e-esPa(Tb e du) re (a,b) ' □ Remark 4.2. Integrating the equation with respect to s over (0, to) in Lemma 4.1 gives the well known formula Ea(e^) = ^ . (4.2) rg(b, b) See e.g. Rogers and Williams [25], (50.7) on p. 293. Let us consider the process (Xt: 0 < t < Tb) given {XSe = b}. The following lemma gives the conditional distribution of this process given X(Sg) = b. Lemma 4.3. Assume that Pa(Tb < to) = 1. Assume thatpt(a, b) > 0 for all t > 0. Let F be a non-negative measurable functional of the process (Xt: 0 < t < Tb). Then Ea(F\X(Se) = b) = Ea (e-eTb ■ F) ■ ^^ , (4.3) re(a, b) where re (x, y) is the resolvent density. Proof. As in Lemma 4.1 we compute for an open neighbourhood B e E of b such that M. Perman: A decomposition for Markov processes at an independent exponential time 59 rg (a, y) > 0 and a bounded measurable functional F Ea [F •1(Tb < Sg, Tb e ds, Xse e B)] = = f 6e-gt dt f pt(a,y)E (F • 1(Tb e ds)) dy Js J B = f 9e-gt dt f Ea (F • 1(Tb e ds)) pt-s(b,y) dy Js J B = e-gsEa (F • 1(Tb e ds)) f dy f 6e-gupu(b,y) du J B J 0 = e-gsEa (F • 1(Tb e ds)) 9 f rg(b,y) dy JB = e-gsEa (F • 1(Tb e ds)) f • 9rg(a,y) dy B rg(a, y) This in conjunction with the distribution of Tb from Lemma 4.1 completes the proof. □ The conclusions of Lemma 4.3 apply equally to the dual process X. Moreover under Pa the conditional law of the process (X(Sg-t)- : 0 < t < Sg) given XSg = b is equal to the law of X started at b run to an independent exponential time and conditioned to be a at the end. This implies that under the assumptions on a for any bounded functional G G^X(se-t)- : 0 < t < Sg - gs0^j XSg = b = Eb , (4.4) G (Xt: 0 < t < Ta) e-gf* 1 • V / J rg(b, a) because the last exit time gSg from a is the first hitting time of a for the reversed process. This is in accordance with Theorem 7.6 in Getoor and Sharpe [8] that excursions straddling a fixed time reversed and conditioned on the length are the excursions of the dual process. See also formula (3.12) in Ikeda, Nagasawa, Sato [11] who give the law of the process reversed from the lifetime of a killed Markov process. The case treated here considers killing at a constant rate. We are now in position to give the proof of Theorem 2.1. Proof. The second assertion in (i) is the definition of the resolvent density. Let (es: s > 0) be the excursion process of X from a. Marked excursions arrive at an exponential rate so we know that La(Sg) will be exponential. Since excursions are marked by an independent Poisson process, the event {La(Sg) > u} is equal to the event that there is no mark in the interval [0, tu], and has conditional probability e-gTu. Integration gives Pa(La(Sg) >u)= Ea (e-gTu) = e-u^(g), which by differentiation gives the density. To prove (ii) note that by Theorem 3.2 conditionally on {La(Sg) = u} the process of excursions (es: 0 < s < u) is independent of the excursion e* straddling Sg. Because marks to excursions are assigned by an independent Poisson process conditionally on R(e*) = r the mark is distributed at the distance U from the left endpoint with density 9e-gu/(1 - e-gr) Ea 60 Ars Math. Contemp. 12 (2017) 37-50 on [0, r] independently of the process of unmarked excursions and of the local time La (Sg). So the excursion e* together with the position of Sg within the duration of e* are independent of the process of unmarked excursions and of La(Sg). This proves the independence of the two processes in (ii). For the first assertion in (iii) note that conditionally on La(Sg) = u the excursions of (Xt: 0 < t < gSe) from the point a are a Poisson process with excursion law e-gR ■ n by Theorem 3.2 (i). On the other hand, if we let (es: 0 < s < u) be the Poisson process of excursions of X from a and choose h(e) = 6 R(e) in Lemma 3.1, under the new measure the process is still Poisson but with the mean measure e-Sh ■ n. But under the assumption that the set M has Lebesgue measure 0 we have = 6 ■ tu. The proof of the second formula in (iii) follows from Lemma 4.3 applied to the reversed process. □ Note the connection with Lemma 4.1 in Kallenberg [14] which states that for Levy processes with continuous densities Pa (F(Xs: 0 < s < Tu)|Tu = t) = Pa,a (F(Xs: 0 < s < t)|Lt = u) (4.5) where Pt refers to the law of the bridge of length t. Noting that Lemma 3.1 gives Pa (gs0 e dt|L(Sg) = u) = ^Effuf)dt) . (4.6) Ea (e gTu) Equations 4.5 and 4.6 imply part (ii) in Theorem 2.1. 5 Examples 5.1 Linear diffusions Let X be a regular diffusion on an interval I C R with speed measure m. It is well known that X has a jointly continuous density p(t, x, y) with respect to m: Px(Xt e A)= i p(t,x,y) m(dy) . (5.1) J A The density is symmetric in x and y which implies that for diffusions the dual process is the diffusion itself. Assume that the X has a recurrent point a and that the point a is not an atom of the speed measure m. This implies that the inverse local time at a has no drift. With such assumptions the conclusions of the Theorem 2.1 hold with X = X. Moreover, it is known that Eb (e-eT*) = and Ea (e-gT") = e-r» . (5.2) rg(a, a) See Rogers and Williams [25], Sec. 50. As the first example one can take X to be Brownian motion and a = 0. All the assumptions are satisfied. It follows that the two processes (Bt: 0 < t < gSe) and (BSe-t: 0 < t < Sg — gSg) are independent. For the first process we get s, , s E0 (F(Bt: 0 < t < TU)e-gTu) Eo (F(Bt: 0 < t < gse)|L(gs9) = l) = 0 1 ( ' ~~) U)-. (5.3) Eo (e gTu) M. Perman: A decomposition for Markov processes at an independent exponential time 61 It is well known that Eo (e-gT) = and Eb (e-gT) = exp (—|b|V28) . where T0 = inf{t > 0: Bt = 0}. The first assertion follows from (i) in Theorem 2.1 and the fact that Lt = |Bt|, see e.g. Revuz and Yor [24], p. 289. The hitting time distribution is an elementary consequence of the reflection principle for Brownian motion. The law of the second process, given BSg = a, is described by s, Eb (F(Bs: 0 < s < T0)e-gTo) E(G(Bse-t: 0 < t < Sg — gse )|Bse = b) = bl ( SE° (<-g<} 0)-1 . (5.4) which yields the result first obtained by Biane and Yor in [4]. See Leuridan [17] for an alternative elementary proof and Yen and Yor [28], Ch 9. for an alternative proof. Since the bridge laws for Brownian motion with drift B(m) = Bt + pt are exactly the same for all drifts the conditional law of (B(m) : 0 < t < Sg) given {B%? = b} does not depend on p. This means that Equations 5.3 and 5.4 hold for Brownian motion with drift with B instead of B(m) . The only change is that the resolvent density changes to that of Brownian motion with drift rg(a, b) 1 eM(b-a)-|b-a|^2g+M2 ^28 + p 2 The skew Brownian motion X(a) with parameter a e (0,1) is constructed by independently flipping the excursions of |Bt| up with probability a and down with probability 1 — a. A pair of dual processes with respect to Lebesgue measure are the processes X (a) and X(1-a). Both processes behave like Brownian motion away from 0 and the distribution of their local time at a fixed point is equal to the distribution of the local time of Brownian motion X(1/2). From the known transition densities of X(a), see p. 82 in Revuz and Yor [24], it follows rg(0, b) = (2a1(b > 0) + 2(1 — a)1(b < 0)) e-|b|72g . The skew Brownian motion satisfies all the assumptions made on the Markov process X. Equation 2.7 holds with the same E(e-gTu) as in the case of Brownian motion. Equation 2.8 holds with X replaced by standard Brownian motion started at 0. Let X be a Bessel process of dimension S e (0,2). Denote v = 5/2 — 1 e ( — 1,0). It is well known, see Revuz and Yor [24], that 0 is a recurrent point for X, satisfying all the assumptions and that the time X spends at 0 has Lebesgue measure 0. The results of Theorem 2.1 apply. Bessel processes are dual to themselves under the speed measure with density £(dx) = — ^^ dx on [0, to). Let Iv(z) and Kv(z) be the modified Bessel functions with index v. With respect to £ the transition density of X for a, b > 0 is given by 1 fab\ pt(a,b) = we 2t habj. Using formula 15.55 in Oberhettinger [19] for 0 < a < b we get that 2 rg(b, a) = ^Iv(aV2d)Kv(bV2d) . av bv 62 Ars Math. Contemp. 12 (2017) 37-50 By Equation 4.2 ) = ( b y ' \ a) Kv (bV26) in accordance with Theorem 3.1 in Kent [15]. By Pa(T0 < to) = 1 and the continuity of paths we have Ta t T0 as a I 0. Letting a ^ 0 and taking into account that Kv (z) ~ (z/2)v for z ^ 0 we get Eb (e-ST) = (bV2^)V/2 Kv (bv^fl) . By Pitman, Barlow and Yor [1] there is a bicontinuous family of local times L of the process X such that pt -i f (Xs)ds = - / bs-1Lbt db J 0 2 J 0 for bounded measurable functions f. With this choice the inverse local time is a stable subordinator of index —v with Laplace transform - ( -st ) ( 2!+ve-vr(i + v) \ eo (e - )=—u —r—)—L). 5.2 Levy processes For a Levy process X the Lebesgue measure is invariant and the dual process is —X. If the process has continuous densities for t > 0, is recurrent and spends Lebesgue measure 0 at points the conclusions of Theorem 2.1 can be applied. An example is provided by symmetric stable processes of index a e (1, 2). These processes are recurrent and by scaling property the inverse local time is a subordinator of index 1 — 1/a. See Bertoin [2], Ch. 8. The independence of (Xt: 0 < t < gSe) and (Xgs +u: 0 < u < Ss —gSe) and scaling imply that given g1 the process (Xt: 0 < t < g1) is conditionally independent of (Xfl1+„: 0 < u < 1 — gi). This means that the two processes (Vgi : 0 < t < 0 and fXv+u(1-gi) :0 < u < ^ (5.5) Vvgi < < J V vr—gi < < J are independent. Scaling also implies that the inverse local time tu is a stable subordinator of index 1 — 1/a with E(e-dTu) = e-u^(s) = e-cSl 1/a for some constant c depending on the normalization of the local time. From Equation 2.7 we can compute E (e-(A+0K) E (e-A3S* lL(S) = u) = ¿(e-ST„) . (5.6) Using the form of ^(6) and unconditioning using Equation 2.6 gives ( 6 \1-1/a E (e-X'") = (AT?) . It follows that gSe ~ r(1 — 1/a, 6) and by independence Ss — gSe ~ r(1/a, 6). Using scaling again this gives the arc-sine law g1 ~ Beta(1 — 1/a, 1/a). This result is due to Chaumont [5]. See also Bertoin [2], p. 230. M. Perman: A decomposition for Markov processes at an independent exponential time 63 For another application let Y be a Levy process. Assume P(Y0 = 0) = 1 and define Yt = sups u} be the right continuous inverse of L. Denote by (eu : u > 0) the excursion process attached to the process X. It is in general not possible to reconstruct Y from the exursion process of the reflected process X. As noted by Greenwood and Pitman in their remark on p. 899 in [10], however, the process of excursions can be extended into a two dimensional Poisson point process such that X can be reconstructed. The idea is to add to the excursion at time u > 0 the jump of the ladder height process Hu = YTu. Denote Ju = Hu — Hu-. The resulting point process ((eu, Ju) : u > 0) is a Poisson point process in the sense of Ito in the space U x (0, to). Let Sg be an exponential random variable with rate 0 independent of Y. If the assumptions of Theorem 3.2 are met the following conclusions can be made: (i) The pairs of random variables (gSg, XSg ) and (Sg — gSg, YSg — YSg ) are independent. This follows from Theorem 2.1. (ii). The random pair (gg, Xg) is infinitely divisible. To prove this statement first recall a standard result about Levy processes: if Z is a d-dimensional Levy process and Sg is an independent exponential random variable, then the random variable (Sg, ZSg ) is infinitely divisible. See Bertoin, [2] p. 162. By Equation 2.8 applied to the reflected process X we find that the law of (gSg ,XSg ) given {Lg = u} is just like the sum of the points of the process ((Ru, Ju) : u > 0) where Ru is the excursion length at local time u and Ju is the jump of the ladder height process H. This last two-dimensional process is a map of the extended excursion process ((eu, Ju) : u > 0) and as such a Poisson point process on (0, to)2. Sums of Poisson processes are infinitely divisible so it follows that (gSg ,XSg ) given {LSg = u} is infinitely divisible. But LSg is exponentially distributed and infinite divisibility follows. The infinite divisibility of the pair (Sg — gSg, YPSg — YSg ) follows by duality arguments. See Lemma 9 in Bertoin [2], p. 164. The assertions about infinite divisibility and independence are true in general without additional assumptions on the reflected process X. See Greenwood and Pitman [9] for details. References [1] M. Barlow, J. Pitman and M. Yor, Une extension multidimensionnelle de la loi de l'arc sinus, in: Séminaire de Probabilités, XXIII, Springer, Berlin, volume 1372 of Lecture Notes in Math., pp. 294-314, 1989, doi:10.1007/BFb0083980, http://dx.doi.org/10.10 07/ BFb0083980. [2] J. Bertoin, Levy processes, volume 121 of Cambridge Tracts in Mathematics, Cambridge University Press, 1st edition, 1996. [3] J. Bertoin, Random fragmentation and coagulation processes, volume 102 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2006, doi:10.1017/ CBO9780511617768, http://dx.doi.org/10.1017/CBO97805116177 68. [4] P. Biane and M. Yor, Sur la loi des temps locaux browniens pris en un temps exponentiel, in: Seminaire de Probabilites, XXII, Springer, Berlin, volume 1321 of Lecture Notes in Math., pp. 454-466, 1988, doi:10.1007/BFb0084151, http://dx.doi.org/10.10 07/ BFb0084151. [5] L. Chaumont, Excursion normalisee, meandre et pont pour les processus de Levy stables, Bull. Sci. Math. 121 (1997), 377-404. 64 Ars Math. Contemp. 12 (2017) 37-50 [6] P. Fitzsimmons, J. Pitman and M. Yor, Markovian bridges: construction, palm interpretation, and splicing, in: Seminar on Stochastic Processes, 1992, Birkhauser Boston, Inc., Boston, MA, volume 33 of Progress in Probability, pp. 101-134, 1993, doi:10.1007/978-1-4612-0339-1, http://dx.doi.org/10.1007/978-1-4612-033 9-1. [7] R. K. Getoor and M. J. Sharpe, Last exit decompositions and distributions, Indiana Univ. Math. J. 23 (1973/74), 377-404. [8] R. K. Getoor and M. J. Sharpe, Excursions of dual processes, Adv. in Math. 45 (1982), 259-309, doi:10.1016/S0001-8708(82)80006-6, http://dx.doi.org/10. 1016/S0001-8708(82)80006-6. [9] P. Greenwood and J. Pitman, Fluctuation identities for Levy processes and splitting at the maximum, Adv. in Appl. Probab. 12 (1980), 893-902, doi:10.2307/1426747, http://dx.doi. org/10.2307/1426747. [10] P. Greenwood and J. Pitman, Fluctuation identities for random walk by path decomposition at the maximum, Adv. in Appl. Probab. 12 (1980), 291-293, doi:10.2307/1426564, http: //doi.org/10.2 30 7/142 6564. [11] N. Ikeda, M. Nagasawa and K. Sato, A time reversion of Markov processes with killing, Kodai Math. Sem. Rep. 16 (1964), 88-97. [12] L. F. James, Bayesian Poisson process partition calculus with an application to Bayesian Levy moving averages, Ann. Statist. 33 (2005), 1771-1799, doi:10.1214/009053605000000336, http://dx.doi.org/10.1214/00905360500000033 6. [13] M. Jeanblanc, J. Pitman and M. Yor, The Feynman-Kac formula and decomposition of Brown-ian paths, Mat. Appl. Comput. 16 (1997), 27-52. [14] O. Kallenberg, Splitting at backward times in regenerative sets, Ann. Probab. 9 (1981), 781799, http://www.jstor.org/stable/2243738. [15] J. Kent, Some probabilistic properties of Bessel functions, Ann. Probab. 6 (1978), 760-770. [16] J. F. C. Kingman, Poisson processes, volume 3 of Oxford Studies in Probability, The Clarendon Press, Oxford University Press, New York, 1993, oxford Science Publications. [17] C. Leuridan, Une demonstration elementaire d'une identite de Biane et Yor, in: Séminaire de Probabilités, XXX, Springer, Berlin, volume 1626 of Lecture Notes in Math., pp. 255-260, 1996, doi:10.1007/BFb0094653, http://dx.doi.org/10.10 07/BFb0 0 94 653. [18] B. Maisonneuve, Exit systems, Ann. Probab. 3 (1975), 399-411. [19] F. Oberhettinger and L. Badii, Tables of Laplace transforms, Springer-Verlag, New York-Heidelberg, 1973. [20] E. A. Pecerskil and B. A. Rogozin, The combined distributions of the random variables connected with the fluctuations of a process with independent increments, Teor. Verojatnost. i Primenen. 14 (1969), 431-444. [21] J. W. Pitman, Levy systems and path decompositions, in: Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981), Springer, volume 1 of Progr. Prob. Statist., 1981 pp. 79-110. [22] A. O. Pittenger and C. T. Shih, Coterminal families and the strong Markov property, Bull. Amer. Math. Soc 78 (1972), 439-443. [23] D. Ray, Sojourn times of diffusion processes, Illinois J. Math. 7 (1963), 615-630. [24] D. Revuz and M. Yor, Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2nd edition, 1994. M. Perman: A decomposition for Markov processes at an independent exponential time 65 [25] L. C. G. Rogers and D. Williams, Diffusions, Markov processes, and martingales. Vol. 2: Ito calculus, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1987. [26] P. Salminen, P. Vallois and M. Yor, On the excursion theory for linear diffusions, Jpn. J. Math. 2 (2007), 97-127, doi:10.1007/s11537-007-0662-y, http://dx.doi.org/10.10 07/ s11537-007-0662-y. [27] D. Williams, Path decomposition and continuity of local time for one-dimensional diffusions. I, Proc. London Math. Soc. (3) 28 (1974), 738-768. [28] J.-Y. Yen and M. Yor, Local times and excursion theory for Brownian motion, volume 2088 of Lecture Notes in Mathematics, Springer, Cham, 2013, doi:10.1007/978-3-319-01270-4, a tale of Wiener and It6 measures, http://dx.doi.org/10.1007/978-3-319-01270-4.