Table Of Content

On some transformations between positive self(cid:21)similar Markov pro esses ∗ † 6 Loï CHAUMONT Ví tor RIVERO 0 0 2 This version: 2nd February 2008 n a J 1 1 Abstra t ] Apathde ompositionatthein(cid:28)mumforpositiveself-similarMarkovpro esses(pssMp) R is obtained. Next, several aspe ts of the onditioning to hit 0 of a pssMp are studied. P X, X↓ Asso iated to a given a pssMp that never hits 0, we onstru t a pssMp that hits 0 . X h in a (cid:28)nite time. The latter an be viewed as onditioned to hit 0 in a (cid:28)nite time and t X. a we prove that this onditioning is determined by the pre-minimum part of Finally, we m provide a method for onditioning a pssMp that hits 0 by a jump to do it ontinuously. [ 1 Key words: Self-similarMarkov pro esses, Lévy pro esses, weak onvergen e, de omposition at v h 3 the minimum, onditioning, -transforms. 4 2 MSC: 60 G 18 (60 G 17). 1 0 6 0 1 Introdu tion / h t [0,∞[ a This work on erns positive self(cid:21)similarMarkov pro esses (pssMp), that is -valued strong m α > 0 Markov pro esses that have the s aling property: there exists an su h that for any : 0 < c < ∞ v , (d) i {(cX ,t ≥ 0), } = {(X ,t ≥ 0), }, x > 0. X tc−1/α IPx t IPcx r a This lass of pro esses has been introdu ed by Lamperti [22℄ and sin e then studied by several authors, see e.g.[4, 6, 7,10,11,25,26℄. We willmake systemati use of a result due toLamperti that establishes that any pssMp is the exponential of a Lévy pro ess time hanged, this will be re alled at Se tion 2. Some of the motivations of this work are some path de ompositions and onditionings that an be dedu ed from [9, 10℄ and that we will re all below, in the parti ular ase where the positive self(cid:21)similar Markov pro ess is a stable Lévy pro ess onditioned to stay positive. ∗ LaboratoiredeProbabilitésetModèlesAléatoires,UniversitéPierreetMarieCurie,4,Pla eJussieu-75252 Par†is Cedex 05, Fran e. CIMATA.C.CalleValen ianas/n,C.P.36240,Guanajuato,Gto. Mexi o;andUniversitéParisX,Nanterre, MODAL'X, 200, Avenue de la République, 92001, Nanterre Cedex, Fran e; Email: rivero imat.mx 1 X, Let IP be a law on the the spa e of àdlàg paths under whi h the anoni al pro ess is an α 0 < α ≤ 2 -stable Lévy pro ess, , i.e. a pro ess with independent and stationary in rements 1e/α (X, ), that is -self(cid:21)similar. Asso iated to this pro ess we an onstru t a pssMp, say IP that (X, ) an be viewed as IP onditioned tostay positive. The onstru tion an be performedeither (X, ) h (X, ) via the Tanaka(cid:21)Doney [15℄ path transform of IP or as an -transform of the law of IP e as in [9, 13℄ or also, in the spe trally one sided ase, via Bertoin's transformation [2℄. e e (X, ) (X, ↓), Another interesting pro ess related to IP is IP· whi h was introdu ed in [9℄, an (X, ) be viewed as IP onditioned to hit 0 ontinuously and is onstru ted via an h-transform (X, ) ]−∞e ,0]. of IP killed at its (cid:28)rst hitting time of e (X, ), Usineg(Xth,e re↓)sults o(fXM, ill)ar [24℄, in [9℄ it has been proved the following results f(oXr, ).IP relating IP and IP started at 0, with the pre and post minimum parts of IP IX = inf{X ,s > 0} m = sup{t > 0 : X ∧ X = IX}. s t− t Fa t 1. Let and Under IP, the pre- X {X ,0 ≤ t < m}, X {X ,t > 0} t m+t minimum part of , i.e. and the post minimum part of , i.e. IX. x > 0, , x aIrXe= oyn,di0ti<onyal≤lyxin,dependent given the valu(eXo+f y, F↓or a)ny under IP on(Xdit+ioyn,ally o)n, x−y 0+ the law of the former is IP and that of the later is IP w (X, ) 0 −→ x → 0+. 0+ · x 0+ where IP isthelimitinglawof IP asthestartingpointtendsto , IP IP as Furthermore, it an be veri(cid:28)ed using the previous result, and it is intuitively lear, that X, x under IP the law of the pre-minimum (respe tively, post-minimum) of onditionally on the {IX < ǫ}, ǫ → 0, ↓, , x 0+ event onverges as to the law IP respe tively IP in the sense that, lim (F ∩{t < m},G◦θ |IX < ǫ) = ↓(F ∩{t < T }) (G), F ∈ G ,G ∈ G , x m x 0 0+ t ∞ Fa t 2. ǫ→0+IP IP IP {G ,t ≥ 0} X. t where is the natural (cid:28)ltration generated by Our (cid:28)rst purpose is to extend Fa ts 1 & 2, to a larger lass of positive self-similar Markov pro esses. That is the ontent of se tions 3 & 4, respe tively. ↓ 0 (X, ) Here is another interpretation of the law IP . Let IP be the law of the pro ess IP killed ]−∞,0]. (X, 0) at its (cid:28)rst hitting time of The pro ess IP still has the strong Markov property (X, ), e and inherits the s aling property from IP so it is a pssMp and it hits 0 in a (cid:28)nite time. (X, ) (Y, 0) 0 Moreover, whenever IP has negative jumps, the pro ess IP hits for the (cid:28)rst time e with a negative jump: e 0(T < ∞,X > 0) = 1, ∀x > 0, IPx 0 T0− T = inf{t > 0 : X = 0}. ↓ h 0 0 t where It has been proved in [9℄, that IP is an transform of IP x 7→ xα(1−ρ)−1, x > 0, ρ (X, ), via the ex essive fun tion where is the positivity parameter of IP ρ = (X ≥ 0). (X, ↓) 1 IP Furthermore, IP hits 0 ontinuously and in a (cid:28)nite time, i.e.: e ↓(T < ∞,X = 0) = 1, ∀x > 0, e IPx 0 T0− ↓ 0 and Proposition 3 in [9℄ des ribes a relationship between IP and IP that allows us to refer to ↓ (X, 0) 0 IP asthelawof IP o{nIdYi0ti=on0e}dtohit ontinuously. Th{eIYla0t<terǫ} o,nǫd>it0io.ningisperformed by approximating the set by the sequen e of sets In Se tion 5, we obtainan analogousresult fora larger lass of self(cid:21)similarMarkov pro esses. Namelythose asso iated to a Lévy pro ess killedatan independent exponential timeand whi h 2 satisfy a Cramér's type ondition. Furthermore, an alternative method for onditioning a self- similarMarkov pro ess that hits 0 by a jump, to hit 0 ontinuously, is provided by making tend to 0 the height of the jump by whi h the pro ess hits the state 0. The approa h used to aboard these problems is based on Lamperti's representation between real valued Lévy pro esses and pssMp whi h we re all in the following se tion. 2 Some preliminaries on pssMp D [0,∞) ∪ ∆ ∆ Let be the spa e of àdlàωg∈paDths de(cid:28)ned onω = ∆, with valute≥s iinnfI{Rt : ω ,=w∆he}re:= ζ(isω)a t t emetery point. Ea h path fis:sRu →h tRhat R∪∆ , fofr(a∆n)y= 0. D . As usual we extend the fun tions to by The spa e is endowed σ X with the Skohorod topo(lFog)y and its Borel -(cid:28)eld. We will denote by Xthe anoni al proP ess t of the oordinates and will be tDhe natural (cid:28)ltration generateξd, by . Moreover, let be a referen e probability measure on under whi h the pro ess, is a Lévy pro ess; we will (D ,t ≥ 0), ξ. t denote by the omplete (cid:28)ltration generated by α > 0 ( ,x > 0) α (ξ,P) x Fix and let IP be the laws of an -pssMp asso iated to via the Lamperti representation. Formally, de(cid:28)ne t A = exp{(1/α)ξ }ds, t ≥ 0, t s Z0 τ(t) and let be its inverse, τ(t) = inf{s > 0 : A > t}, s inf{∅} = ∞. x > 0, x with the usual onvention, For we denote by IP the law of the pro ess xexp{ξ }, t > 0, τ(tx−1/α) ∆ τ(tx−1/α) = ∞. withthe onventionthattheabovequantityis if TheLampertirepresentation ( ,x > 0) α x ensures that the laws IP are those of a pssMp with index of self-similarity . (ξ,P) Besides, re all that any Lévy pro ess with lifetimehas the same law as a Lévy pro ess q ≥ 0. T = inf{t > 0 : X = 0} 0 t within(cid:28)nitelifetimethathasbexe1n/αkAilledatarPate Itfollowsthat x ζ has the same law under IP as under with ζ A = exp{(1/α)ξ }ds. ζ s Z0 q > 0, A q = 0, ζ So, if then the random variable is a.s. (cid:28)nite; while in the ase we have A ζ two possibilities, either is (cid:28)nite a.s. or in(cid:28)nite a.s.; the former happens if and only if lim ξ = −∞, limsup ξ = ∞, t→∞ t t→∞ t a.s. and the latter if and only if a.s. Lamperti proved that any pssMp an be onstru ted this way and obtained the following lassi(cid:28) ation of pssMp's: q > 0, (LC1) if and only if (T < ∞, X > 0, X = 0, ∀s ≥ 0) = 1, x > 0. IPx 0 T0− T0+s for all (2.1) 3 q = 0 lim ξ = −∞ t→∞ t (LC2) and a.s. if and only if (T < ∞, X = 0, X = 0, ∀s ≥ 0) = 1, x > 0, IPx 0 T0− T0+s for all (2.2) q = 0 limsup ξ = ∞ t→∞ t (LC3) and a.s. if and only if (T = ∞) = 1, x > 0. x 0 IP for all (2.3) α = 1 Observe that without loss of generality we an and we will suppose that in Lamperti's α > 0 onstru tion of pssMp be ause all our results may trivially be extended to any by Xα α onsidering whi h is a pssMp with index of self-similarity . In this work we will be mostly interested by those pssMp that belong to the lass LC3; nevertheless, in Se tion 5 we will prove that some elements of the lass LC1 an be transformed into elements of the lass LC2. 3 Path de omposition at the minimum (ξ,P) We suppose throughout this se tion that is a Lévy pro ess with in(cid:28)nite lifetime whi h +∞ lim ξ = +∞ t→+∞ t drifts to , tha(tξ,iPs) , a.sI.ξ =Weinfstartξ by reρ a=llinsugpa{tW: ξill∧iamξ 's=tyIpξe}path t≥0 t t t− de omposition of at its minimum. Let and . We de(cid:28)ne the post minimum pro ess as ξ (d=ef) (ξ −Iξ, t ≥ 0). → ρ+t The following result is due to Millar [24℄, proposition 3.1 and Theorem 3.2. ((ξ , t < m),P) ( ξ,P) t → Theorem 1. The pre-minimum pro ess and the post-minimum pro ess are independent. Moreover, the three following exhaustive ases hold: (i) 0 (−∞,0) (0,∞) P is regular for both and and -a.s., there is no jump at the minimum, (ii) 0 (−∞,0) (0,∞) Iξ = ξ < ξ P ρ− ρ is regular for but not for and , -a.s. (iii) 0 (0,∞) (−∞,0) Iξ = ξ < ξ P ρ ρ− is regular for but not for and , -a.s. P (ξ , t < m) ξ Iξ t → In any ase under , the pro ess and are also onditionally independent given ξ → and the pro ess is strongly Markovian. A tually, Millar's result is mu h more general and asserts that for any Markov pro ess, whi h admits a minimum, the pre-minimum pro ess and the post minimum pro ess are onditionally independent given both the value at the minimum and the subsequent jump and the post- minimum pro ess is strongly Markovian. When this Markov pro ess is a pssMp that belongs to the lass (LC3), we may omplete Millar's result as in the following proposition. First of X +∞ ξ, all, observe that derives towards as well as and so the following are well de(cid:28)ned IX = inf X m = sup{t : X ∧X = IX} t≥0 t t t− and . 4 x > 0, , (X ,t < m) (X , t ≥ 0) x t t+m Proposition 1. For any under IP the pro esses and are IX onditionally independent given , and with the representation given by Lamperti's transformation (Se tion 2), we have ρ ((X ,0 ≤ t < m), ) = xexpξ , 0 ≤ t < x expξ ds ,P , t x τ(t/x) s IP (3.1) (cid:18)(cid:18) Z0 (cid:19) (cid:19) ((X , t ≥ 0), ) = IX exp ξ , t ≥ 0 ,P , t+m IPx →→τ(t/IX) (3.2) (cid:18)(cid:18) (cid:19) (cid:19) s τ(t) = inf s : exp ξ du > t t ≥ 0 where → 0 →u , for . n (cid:16) (cid:17) o R (X, ) x Proof. The expression of the pre-minimum part of IP follows dire tly from Lamperti's τ transformation (Se tion 2). Note that in parti ular, sin e is a ontinuous and stri tly in reas- ing fun tion, one has: ρ A = expξ ds, τ(A ) = ρ, xA = m IX = xexpIξ. ρ s ρ ρ and (3.3) Z0 (X, ) x To express the post-minimum part of IP , (cid:28)rst note that X = xexpξ , t ≥ 0. m+t τ(Aρ+t/x) Then we an write the time hange as follows: s τ(A +t/x) = inf{s > 0 : expξ du > A +t/x} ρ u ρ Z0 s−ρ = inf{s > ρ : expξ du > t/x} u+ρ Z0 s = inf{s > 0 : exp ξ du > (t/x)exp(−Iξ)}+ρ → u Z0 = τ((t/x)exp(−Iξ))+ρ = τ(t/IX)+ρ, → → so that ξτ(Aρ+t/x) = ξ→τ(t/IX)+ρ = →ξ→τ(t/IX) +Iξ (X, ) x and the expression (3.2) for the post-minimum part of IP follows. (X ,t < m) (ξ ,t < ρ) t t From (3.1), we see that is a measurable fun tional of and from (3.2), (X , t ≥ 0) IX ξ IX = xexpIξ m+t → is a fun tional of and . Sin e , the onditional independen e follows from Theorem 1. X ξ When has no positive jumps (or equivalently when has no positive jumps), it makes sense y ≥ x to de(cid:28)ne the last passage time at level as follows σ = sup{t : X = y}. y t X Then the post-minimum pro ess of be omes more expli it as the following result shows; its proof is an easy onsequen e of Proposition 1. 5 y ≤ x IX = y (X , t ≥ 0) t+m Proposition 2. Let . Conditionally on , the post-minimum pro ess (X , t ≥ 0) has the same law as σy+t , and ((X , t ≥ 0), ) (=d) yexp ξ , t ≥ 0 ,P . σy+t IPx →→τ(t/y) (3.4) (cid:18)(cid:18) (cid:19) (cid:19) (X, ) As we have just seen, the post-m(ξi,nPim)um pro ess of IP an be( ξo,mPp)letely des ribed → using the underlying Lévy pro ess onditioned to stay positive . Nevertheless, the des ription of the pre-minimum obtained in (3.1) is not so expli it. So our next purpose is to make some ontributions to the understanding of the pre-minimum pro ess of a positive self-similar Markov pro ess. (X, ) Let u(sξ,stPa)rt by the ase where the pro ess IP (or equivalently the underlying Lévy pro ess ) has no negative jumps, be ause in this ase we an provide a more pre ise des ription of the pre-m(inξi,mPu)m, −pIrξo, ess using known results for Lévy pro essγes.>R0e all that γthe overall minimum of followPs a.n exponential law of parameter for some whi h is determined in terms of the law (See Bertoin [3(℄,ξ,CPh)apter VII.) Furthermore, it has been proved by Berto(inξ,[P1℄↓)t,hat the pre-minimum part of −ehas thee same law as a real valu(ξe,dPL↓é)vy pro ess, say killed at its (cid:28)rst hitting time oγf. with a r.v.(ξi,nPde)pendent of and that follows an exponential law of parameter (The pro ess an be (ξ,pr) −∞. viewed as onditioned to drift to ) The translation of Bertoin's results for positive ↓, self-similar Markov pro ess leads to the following Proposition. Denote by IP(ξ,tPhe↓).law of the pro ess obtained by applying Lamperti's transformation to the Lévy pro ess (X, ), (ξ,P), Proposition 3. If IP equivalently has no negative jumps, then there exists a real γ > 0 x > 0 su h that for any (IX ≤ ǫ) = (ǫ/x)γ ∧1, ǫ ≥ 0, x IP ((X ,0 ≤ t < m), ) ((X ,0 ≤ t < T(Z)), ↓), t x t x and the law of IP is the same as that of IP where Z (X, ↓) (−log(Z/x), ↓) is a random variable independent of IPx and su h that IPx follows an γ > 0. exponential law of parameter Proof. This follows from Proposition 1 and Theorem 2 in [1℄, des ribed above. So to rea h our end, we will next provide a des ription of the pre-minimum of a real val- ∞, ued Lévy pro ess that drifts to whi h generalizes Bertoin's result and is analogous to the des ription of the pre-minimum of a Lévy pro ess onditioned to stay positive that has been obtained in [9℄ and [17℄. V(dx),x ≥ 0 Let be the renewal measure of the downward ladder height pro ess, sPee, e.g. [3℄ or [13℄ for ba kground. In the remaining of this Se tion we will assume that under b 0 ]−∞,0[ is regular for ξ +∞  (H) derives towards  V(dx) the measure is absolutely ontinuous w.r.t Lebesgue's measure.   b 6 (ξ,P) In order to onstru t the Lévy pro ess whi h des ribes the pre-minimum parPt]−o∞f ,0[ we will nee(ξd,Pth)e following Lemma whi h is remin]−is ∞en,t0o[.f Theorem 1 in [28℄. Let be the law of killed at its (cid:28)rst hitting time of (H) V(dx) ϕ : R → LRe+m, ma 1. Under the assumptions the r(eξn,Pew]0a,∞l m[)easur0e< ϕ(x)h<as∞a density, sxa∈y R+. whi h is ex essive for the semigroup of and for a.e. b (ξ,P) (−ξ,P) Proof. It is known that the pro esses and (ξa,rePi]−n∞w,0e[a)k dua(l−ityξ,wP.]r−.t∞.,0L[)ebesgue's measure, so by Hunt's swit hing identity we have that and are also in weak duality w.r.t. Lebesgue measure, see e.g. [20℄. On the other hand, it is known that V(dx) S −ξ = {sup ξ − ξ ,t ≥ 0}, ξ s≤t s t the measure is an invariant measure for the pro ess, V(dx) re(cid:29)eS t−edξa,t itsbsupremum, see e.g. [3℄ Chapte0r,VI exer (i−seξ,5P. ]S−o∞t,0h[)e.measure, isex essive for killed at its (cid:28)rst hitting time of so for Thus the (cid:28)rst assertion of b Lemma 1 is a dire t onsequen e of Theorem in Chapter XII paragraph 71 in [14℄. To prove the se ond assertion we re all that V[0,x] = kP(− inf ξ ≤ x), x ≥ 0, s 0≤s<∞ k ∈]0,∞[ b ϕ < ∞ with a onstant, see [3℄ Proposition VI.17. So a.e. and by the regularity for ]−∞,0[ inf ξ ]−∞,0[, 0 < ϕ 0≤s<∞ s of 0, the support of the law of is thus a.e. Pց, h P]−∞,0[, ϕ. Pց Let be the -transform of the law, via the ex essive fun tion That is, {0 < ζ} is the unique measure whi h(Pisց ,atr≥rie0d),by and under whi h the anoni al pro ess is t Markovian with semi-group 1 E]−∞,0[(f(ξ )ϕ(ξ )) x ∈ {z ∈ R : 0 < ϕ(z) < ∞}, Pցf(x) = ϕ(x) x t t if t (0 x ∈/ {z ∈ R : 0 < ϕ(z) < ∞}. if Λ = {z ∈ R : 0 < ϕ(z) < ∞}. Pց {ξ ∈ Λ,ξ ∈ t t− Let Furthermore, the measure is arried by Λ,t ∈]0,ζ[}, G T t and for any -stopping time ϕ(ξ ) Pց1 = T 1 P]−∞,0[, G . x {T<ζ} ϕ(x) {T<ζ} x on T (ξ,P) Pց In the ase where the semigroup of is absolutely ontinuous, (ξ,Ph)as been introdu ed in [9℄ where it is proved that this m(ξe,aPsu)re an be viewed asϕthe law of onditioned to hit 0 ontinuously. In the ase where reeps downward an be made expli it: ϕ(x) = cP(ξ = −x) > 0, x > 0, T]−∞,−x[ 0 < c < ∞, with a onstant, see [3℄ Theorem VI.19, and then we have the right onditioning: Pց = P]−∞,0[( · |ξ = 0). x x T]−∞,0[ Pց (ξ,P) 0. So in the sequel we will refer to as the law of onditioned to hit 7 ξ (H) ϕ Lemma 2. Let be a real valued Lévy pro ess that satis(cid:28)es the hypotheses and be the V density of the renewal measure as in Lemma 1. Then for any bounded measurable fun tional F , 1 E(F (ξ −ξ ,0 ≤ s <b ρ)) = daϕ(a)Eց(F (ξ ,0 ≤ s < ζ)). s ρ− a s V]0,∞[ Z]0,∞[ P Iξ = a Iξn+paarti ularPuցnde.r onditionally on b , the pre-minimum pro ess has the same law as −a under Observe that Bertoin's [1℄ Theorem 2 an be dedu ed from this Lemma sin e in the ase ξ ϕ, ϕ(x) = γe−γx,x > 0, where has no negative jumps is given by and so we have that E(F (ξ ,0 ≤ s < ρ)) = daγe−γaEց(F (ξ −a,0 ≤ s < ζ)) s a s Z]0,∞[ = daγe−γaE]−∞,0[ F (ξ −a,0 ≤ s < ζ)| T < ∞ a s ]−∞,0[ Z]0,∞[ (cid:0) (cid:1) = γ daE F ξ ,0 ≤ s < T ,T < ∞ s ]−∞,−a[ ]−∞,−a[ Z]0,∞[ (cid:0) (cid:0) (cid:1) (cid:1) = daγe−γaE F ξ ,0 ≤ s < T e−γξT]−∞,−a[, T < ∞ s ]−∞,−a[ ]−∞,−a[ Z]0,∞[ (cid:16) (cid:17) = E↓ F ξ ,0 ≤ s < T(cid:0) , (cid:1) s ]−∞,−e[ P↓ e (cid:0) (cid:0) (cid:1)(cid:1) where and are as explained just before Proposition 3. Proof. To prove the laimed identity, we will start by al ulating for any ontinuous and F, bounded fun tional Ee/λ(F (ξ −ξ ,0 ≤ s < ρ)), s ρ Ee/λ (ξ,P) e/λ e where is th(ξe,Pla)w. of killed at time , w{iLth,u a≥n0e}xponential random variable u independent of To do that we will denote by the lo al time at 0 of the {ξ −I ,t ≥ 0}, g ξ −I t, t t t strong Markov pro ess by the last hitting time of 0 by before time g = sup{s ≤ t : ξ −I = 0}, N ξ −I t s s and by the ex ursion measure of away from 0. Indeed, using Maisonneuve's exit formula of ex ursion theory it is justi(cid:28)ed that ∞ Ee/λ(F (ξ −I ,0 ≤ s < ρ)) = dtλe−λtE(F (ξ −I ,0 ≤ s < g )) s ρ s gt t Z0 ∞ t = dtλe−λtE dL F (ξ −I ,0 ≤ s < u)N(t−u < ζ) u s u Z0 (cid:18)Z0 (cid:19) ∞ = E dL e−λuF (ξ −I ,0 ≤ s < u) N(1−e−λζ). u s u (cid:18)Z0 (cid:19) λ Next, making tend to 0, the left hand term in the previous equality tends to E(F (ξ −ξ ,0 ≤ s < ρ)), s ρ− while the right hand term tends to ∞ E dL F (ξ −I ,0 ≤ s < u) N(ζ = ∞). u s u (cid:18)Z0 (cid:19) 8 Finally, a straightforward extension of Lemma 3 in [10℄ to our weaker hypothesis allows us to ensure that ∞ E dL F (ξ −I ,0 ≤ s < u) = daϕ(a)Eց(F (ξ ,0 ≤ s < ζ)), u s u a s (cid:18)Z0 (cid:19) Z]0,∞[ V]0,∞[= (N(ζ = ∞))−1. whi h on ludes the proof given that b a ∈ R We next introdu e the law of a Lévϕy pro ess onditio(nξe,dPt)o hit by above a given level . aOw∈inRg to the fa t thϕat t:hRe fu→n Rti+on is ex essϕive(xfo)r= ϕ(x −kilale)d,xat∈0R, w, e have that for any a a the fun tion de(cid:28)ned by is ex essive for the ξ ]−∞,a[. semigroup of killed at its (cid:28)rst hitting time of Indeed, E (ϕ (ξ ),t < T ) = E (ϕ(ξ ),t < T ) ≤ ϕ(x−a) = ϕ (x), x > a, x a t ]−∞,a[ x−a t ]−∞,0[ a lim E (ϕ (ξ ),t < T ) = ϕ (x). t→0+ x a t ]−∞,a[ a aPnցdaanalohgously it is veri(cid:28)ed that ξ ]−∞,Wa[e wilϕl denote by a Gthe -transformTof the law of killed at it (cid:28)rst hPi]t−t∞in,ga[time of via , i.e.: for any t-stopping time , with an obvious notation for x , ϕ (ξ ) Pցa1 = a T 1 P]−∞,a[, G . x {T<ζ} ϕ (x) {T<ζ} x on T a ξ The followingaelementary Lemma will enable us to refer to this meaPsuցr0e as the lawPoցf . onditioned to hit ontinuously and by above. Of ourse the measure x is simply x (ξ,P) (H) Lae∈mRm, a 3. Letx > a be a realξv+aluaed LévyPpցro ess that satis(cid:28)es the hyξpothesesPցa.. For x−a x and any the law of under is the same as that of under As a x > a onsequen e, for a.e. Pցa(ξ = x; ζ < ∞; ξ > a t < ζ; ξ = a) = 1. x 0 t for all ζ− G t Prootf.>To0.prove the (cid:28)rst assertion it su(cid:30) es t(oξ,vPe)rify that both lawts>ar0e equal over for any Indeed, the spatial homogeneity of implies that for and any bounded F measurable fun tional 1 Pցa F(ξ ,0 ≤ s < t)1 = P (F(ξ ,0 ≤ s < t)1 ϕ (ξ )) x s {t<ζ} ϕ (x) x s {t<T]−∞,a[} a t a (cid:0) (cid:1) 1 = P (F(ξ +a,0 ≤ s < t)1 ϕ(ξ )) ϕ(x−a) x−a s {t<T]−∞,0[} t = Pց (F(ξ +a,0 ≤ s < t)1 ). x−a s {t<ζ} 0 Now, the se ond assertion is an easy onsequen e of Lemma 2 and the hypothesis that is ]−∞,0[. regular for A rewording of Lemma 2 using Lemma 3 reads: (ξ,P) (H) Theorem 2. Let be a real valued Lévy pro ess that satis(cid:28)es the hypotheses . The F following identity holds for any bounded measurable fun tional , 1 E (F (ξ ,0 ≤ s < ρ)) = daϕ (x)Eցa(F (ξ ,0 ≤ s < ζ)). x s V]0,∞[ a x s (3.5) Z]−∞,x[ b 9 We have now all the elements to state the main result of this se tion whose proof follows easily from Lemma 3 & Theorem 2. (ξ,P) (H) T(Xh,eor)em 3. Let be a real valued Lévy pro ess th(ξa,tPs)atis(cid:28)es the hypotheses and IP be the self-similar Markov pro ess asso iated to via Lamperti's representation. F Then for any bounded measurable fun tional , 1 (F (X ,0 ≤ s < m)) = ν (dv) ցvx(F(X ,0 ≤ s < ζ)) x s 1 x s IE IP Z0 1 = ν (dv) ց1(F(vxX ,0 ≤ s < vxζ)), 1 IP1/v s/vx Z0 ν ]0,1[ 1 where is a measure over with density ν (dv) 1 = (V]0,∞[)−1v−1ϕ(−lnv), 0 < v < 1, dv ցvx b (ξ, Pցlog(vx)). and IPx is thelawofthepro essobtainedbyapplyingLamperti'srepresentationto log(x) 4 The asymptoti behavior of the pre- and post-minimum as the minimum tends to 0. (H) Throughout this se tion we will leave aside the assumptions . We only assume that the ξ ∞ underlying Lévy pro ess drifts to , it is not a subordinator and it is non latti e. Some an illary hypothesis will be stated below. 4.1 Post-minimum Iξ ]−∞,0] Under these hypotheses, it is known that the support of the law of is . From (3), IX [0,y] y > 0 y the support of is then under IP , for any . Proposition 1 shows that a regular (X , t ≥ 0) IX = x m+t y version of the law of the post-minimum pro ess under IP given , for x ∈]0,y] x ξ , t ≥ 0 ,P is given by the law of the pro ess →→τ(t/x) . In parti ular, this law (cid:18)(cid:18) (cid:19) (cid:19) y x → does not depend on . Let us denote it by IP . A straight onsequen e of this representation ( x) ]0,∞[ → is that the family IP is weakly ontinuous on . In Theorem 4 below, we show that if 0 < E(ξ ) < ∞, x x 1 → 0+ moreover then IP onverges weakly as tends to 0 towards the law IP . x → 0+ x This measure is the weak limit of IP as , whose existen e is ensured by Theorem 2 in [11℄. x > 0, (X, x) → Re all that Millar's results implies that for any the pro ess IP is strongly [x,∞[ Markov with values in . 0 < E(ξ ) < ∞. x D x → 0+ 1 → Theorem 4. Assume that The laws IP onverge weakly in as to . x > 0, 0+ the law IP As a onsequen e, for any (·◦θ |IX < ǫ) −−w→ (·). x m 0+ IP ǫ→0 IP 10