On the equivalence of some eternal additive coalescents

www.imstat.org/aihp 2008, Vol. 44, No. 6, 1020–1037

DOI: 10.1214/07-AIHP154

© Association des Publications de l’Institut Henri Poincaré, 2008

On the equivalence of some eternal additive coalescents ^*

Anne-Laure Basdevant

Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 175 rue du Chevaleret, 75013 Paris, France.

E-mail: Anne-Laure.Basdevant@ens.fr

Received 17 November 2006; revised 18 May 2007; accepted 1 June 2007

Abstract. In this paper, we study additive coalescents. Using their representation as fragmentation processes, we prove that the law of a large class of eternal additive coalescents is absolutely continuous with respect to the law of the standard additive coalescent on any bounded time interval.

Résumé. Nous étudions dans ce papier les coalescents additifs. En utilisant leur représentation en tant que processus de fragmen- tation, nous prouvons que certains coalescents additifs éternels ont une loi absolument continue par rapport à la loi du coalescent additif standard sur n’importe quel intervalle de temps borné inférieurement.

MSC:60J25; 60G09

Keywords:Additive coalescent; Fragmentation process

1. Introduction

This paper deals with additive coalescent processes, a class of Markov processes first introduced by Evans and Pitman [12]. In the simple situation of a system initially composed of a finite number k of clusters with masses m1, m2, . . . , mk, the dynamic is such that each pair of clusters (mi, mj) merges into a unique cluster with mass m_i+m_j at ratem_i+m_j, independently of the other pairs. In the sequel, we always assume that the initial total mass is equal to 1 (i.e.m₁+ · · · +m_k=1). This assumption induces no loss of generality since we can then deduce the law of any additive coalescent process through a linear time-change. Hence, an additive coalescent takes values on the compact set

S^↓=

x=(x_i)_i≥1, x1≥x2≥ · · · ≥0,

i≥1

x_i≤1

,

endowed with the topology of uniform convergence.

Evans and Pitman [12] proved that one can define an additive coalescent on the whole real line for a system starting at timet= −∞with an infinite number of infinitesimally small clusters. Such a process is called aneternal additive coalescent process. More precisely, if we denote by(Cⁿ(t ), t≥0)the additive coalescent starting from the config- uration(1/n,1/n, . . . ,1/n), they proved that the sequence of processes (Cⁿ(t+¹₂lnn), t≥ −¹₂lnn)converges in distribution on the space of càdlàg paths with values in the setS^↓ toward some process(C^∞(t ), t∈R), called the standard additive coalescent. It should be noted that this process is defined for all timet∈R. Remarkably, the stan- dard additive coalescent becomes, through time-reversal, a fragmentation process. Namely, the process(F (t ), t≥0)

*Sur l’équivalence de certains coalescents additifs éternels

defined byF (t )=C^∞(−lnt )is a self-similar fragmentation process with index of self similarityα=1/2, without erosion and with dislocation measureνgiven by

ν(x1∈dy)=

2πy³(1−y)³₋1/2

dy fory∈ 1

2,1

, ν(x3>0)=0.

We refer to [8,9] for the definition of erosion, dislocation measure, and index of self similarity of a fragmentation process and a proof of the property mentioned above. Let us just recall that, in a fragmentation process, distinct fragments evolve independently of each other.

Aldous and Pitman [1] constructed such a fragmentation processF by cutting the skeleton of the Brownian contin- uum random tree according to a Poisson point process. In another paper [2], they gave a generalization of this result:

consider for eachn∈Na decreasing sequencern,1≥ · · · ≥r_n,n≥0 with sum 1, setσ_n²= ⁿ_i=1r_n,i² and suppose that

nlim→∞σn=0 and lim

n→∞

r_n,i

σ_n =θi for alli∈N.

Assume further that either _iθ_i²<1 or _iθ_i = ∞. Then, according to [2], ifMⁿ=(Mⁿ(t ), t ≥0)denotes the additive coalescent process starting withnclusters with massesrn,1≥ · · · ≥rn,n, then(M⁽ⁿ⁾(t−lnσn), t≥lnσn)has a limit distribution asn→ ∞, which can be obtained by cutting a specific inhomogeneous random tree with a point Poisson process. Furthermore, any extreme eternal additive coalescent may be obtained this way up to a deterministic time translation (i.e. any eternal additive coalescent can be obtained as a mixture of such additive coalescents).

Bertoin [5] gave another construction of the limit of the process(M⁽ⁿ⁾(t−lnσn), t≥lnσ_n)in the following way:

letbθbe the bridge with exchangeable increments defined, fors∈ [0,1], by bθ(s)=σ bs+

∞ i=1

θi(1_{s≥V_i}−s),

where(b_s, s∈ [0,1])is a standard Brownian bridge,(V_i)_i≥1 is an i.i.d. sequence of uniform random variables on [0,1]independent ofbandσ=1− _iθ_i². Letε_θ=(ε_θ(s), s∈ [0,1])be the excursion obtained fromb_θby Vervaat’s transform, i.e.εθ(s)=bθ(s+mmod 1)−bθ(m),wheremis the point of[0,1]wherebθ reaches its minimum. For allt≥0, consider

ε^{(t )}_θ (s)=t s−ε_θ(s), S^{(t )}_θ (s)= sup

0≤u≤s

ε^{(t )}_θ (u),

and defineF^θ(t )as the sequence of the lengths of constancy intervals of the process(S_θ^{(t )}(s),0≤s≤1). Then, the limit of the process(M⁽ⁿ⁾(t−lnσ_n), t≥lnσn)has the same law as(F^θ(e^−t), t∈R).

Miermont [15] studied the same process in the special case whereεθ is the normalized excursion below the supre- mum of a spectrally negative Lévy process. More precisely, let(X_t, t≥0)be a Lévy process with no positive jump, with unbounded variation and with positive, finite mean. LetX(t )=sup_0≤s≤tXt and letεX=(εX(s), s∈ [0,1])de- note the normalized excursion with duration 1 of the reflected processX−X. We now define the processesε_X^{(t )}(s), S_X^{(t )}(s)andF^X(t )in the same way as forbθ. Then, the process(F^X(e^−t), t∈R)is a mixture of some eternal ad- ditive coalescents (see [15] for more details). Furthermore, (F^X(t ), t≥0)is a fragmentation process in the sense that distinct fragments evolve independently of each other (but is not necessarily homogeneous in time). It is quite remarkable that the Lévy property ofXensures the branching property ofF^X. We emphasize that there exist other eternal additive coalescents for which this property fails. Notice that when the Lévy processXis the standard Brown- ian motionB, the process(F^B(e^−t), t∈R)is the standard additive coalescent and(F^B(t ), t≥0)is a self-similar and time-homogeneous fragmentation process.

In this paper, we study the relationship between the laws of F^X andF^B. From now on, we denote by F the canonical process on the space of functionsR→S^↓and(Ft, t≥0)the natural filtration, i.e.Ft=σ (F (s),0≤s≤t ).

We also defineP^(X)(resp.P^(B)) on the space of functionsR→S^↓such thatF underP^(X)(resp.P^(B)) has the law of F^X(resp.F^B).

We prove that, for certain Lévy processes(X_t, t≥0), the lawP^(X)_|Ft is absolutely continuous with respect toP^(B)_|Ft and we explicitly compute its density.

Theorem 1.1. Let(Γ (t ), t≥0)be a subordinator without drift.Assume thatE(Γ₁) <∞and choose anyc≥E(Γ₁).

We defineXt =Bt−Γt+ct,whereB denotes a Brownian motion independent ofΓ.Let(pt(u), u∈R)and(qt(u), u∈R)stand for the respective densities ofBt andXt.In particularpt(u)=^√₂¹_πtexp(−^u_2t²).LetS1be the space of positive sequences with sum1.We consider the functiong:R+×S1→Rdefined by

⎧⎨

⎩

g(0,1)=^q_p¹₁⁽⁰⁾₍₀₎ with1^def=(1,0,0, . . .), g(t,x)=e^{t c}_∞

i=1

q_xi(−t xi)

p_xi(−t x_i) withx=(x_i)_i≥1. (1)

Then,for allt≥0,the functiong(t,·)is bounded onS1and has the following properties:

1. UnderP^(B),g(t, F (t ))is anFt-martingale.

2. For everyt≥0,the law ofP^(X)_|Ft is absolutely continuous with respect toP^(B)_|Ft with densityg(t, F (t ))/g(0,1).

Let us notice that the first part of the theorem is a direct consequence of the second part. In terms of coalescent processes, this theorem yields the following corollary.

Corollary 1.2. Let C^X be a mixture of additive coalescents associated with the Lévy processX. Since we have (C^X(t ), t∈R)^law= (F^X(e^−t), t∈R),the law of(C^X(s), s≥t )is absolutely continuous with respect to the law of the standard additive coalescent started at timet∈R,with density given byg(e^−t, C(t ))/g(0,1),C being the canonical process on the space of càdlàg functionsR→S^↓.

Remark 1.3. It is known that,ifΓ is a subordinator,the density(qt(u), u∈R)of the Lévy processXt=Bt−Γt+ct is well defined fort >0.Moreover,the function(t, u) →q_t(u)isC^∞onR^∗₊×R(see Chapter5of[18]).

Let us notice thatg(t,·)is a multiplicative function, i.e. it can be written as a product of functions, each of them depending only on the size of a single fragment. In the sequel we use the notation

g(t, x)=e^{t cx}qx(−t x)

px(−t x) forx∈(0,1]andt≥0, so thatg(t,x)=

ig(t, x_i). As a consequence of the branching property ofF^Band the multiplicative form ofg, the processF^Xalso has the branching property. Indeed, for every multiplicative bounded continuous functionf:S^↓ → R₊, for allt> t >0 andx∈S^↓, sinceg(t, F (t ))is anFt-martingale underP^(B), we have

E^(X) f

F t

|F (t )=x

= 1

g(t,x)E^(B) g

t, F t

f F

t

|F (t )=x .

Using the branching property ofF underP^(B)and the multiplicative form ofg(t,·), we get E^(X)

f F

t

|F (t )=x

= 1 g(t,x)

i

E^(B) g

t, F t

f F

t

|F (t )=(x_i,0, . . .) .

And we deduce that E^(X)

f F

t

|F (t )=x

= 1 g(t,x)

i

g(t, xi)E^(X) f

F t

|F (t )=(xi,0, . . .)

=

i

E^(X) f

F t

|F (t )=(x_i,0, . . .) .

LetM_x(resp.M_xi) be the random measure on (0,1) defined byM_x= _iδ_si where the (random) sequence(s_i)_i≥1 has the law ofF (t)conditioned onF (t )=x(resp.F (t )=(x_i,0, . . .)). Then, for every bounded continuous function k:R→R, we have

E^(X) exp

−k, Mx

= ∞ i=1

E^(X) exp

−k, Mx_i ,

which shows thatM_xhas the same law as _iM_xi where the random measures(M_xi)_i≥1are independent. Hence the processF has the branching property underP^(X). Let us note that other multiplicative martingales have already been studied in the case of branching random walks [10,11,14,16].

This paper is divided in two sections. The first section is devoted to the proof of Theorem 1.1. In the second section, we give an integro-differential equation solved bygusing the fact thatg(t, F (t ))is anFt-martingale underP^(B). 2. Proof of Theorem 1.1

The assumptions and notations of Theorem 1.1 are implicitly enforced throughout this section.

2.1. Absolute continuity

In order to prove Theorem 1.1, we first prove the absolute continuity of the law ofF (t )underP^(X)with respect to the law ofF (t )underP^(B)for any fixed timet >0 and for any finite number of fragments. We start with:

Definition 2.1. Let x=(x₁, x₂, . . .) be a sequence of positive numbers with sum1. We call the random variable

˜

x=(x_j1, x_j2, . . .)a size-biased rearrangement ofxif we have:

∀i∈N, P(j₁=i)=x_i, and,by induction,

∀i∈N\ {i1, . . . , i_k}, P(j_k+1=i|j1=i1, . . . , j_k=i_k)= xi

1− ^k_l=1x_il.

Of course, when the sequencexis itself random, the same construction can be done for each realization ofxand thus we obtain another random variablex˜ called a size-biased rearrangement ofx. In particular, for anyt ≥0, we have ^∞_i=1F_i(t )=1P^(X)-a.s. (this follows from the construction from an excursion ofXsince the Lévy process has unbounded variation, cf. [15], Section 3.2). Thus, in what follows, considering a possibly enlarged probability space, we shall denote byF (t )˜ =(F˜₁(t ),F˜₂(t ), . . .)a size-biased rearrangement ofF (t )(well definedP^(X)almost surely).

The following lemma gives the distribution of thenfirst fragments ofF (t )underP^(X), chosen with a size-biased pick.

Lemma 2.2. For alln∈N,for allx1, . . . , xn∈R+such thatS= ⁿ_i=1xi<1,we have P^(X)F˜1(t )∈dx1, . . . ,F˜_n(t )∈dxn

= tⁿ

q1(0)q1−S(St ) n i=1

qx_i(−t xi)

1− ⁱ_k=1x_k dx1· · ·dxn.

Proof. On the one hand, Miermont [15] gave a description of the law ofF (t )underP^(X): letT^{(t )}be a subordinator with Lévy measurez⁻¹q_z(−t z)1_z>0dz. Then,F (t )has the law of the sequence of the jumps of T^{(t )}before timet conditioned onT_t^{(t )}=1.

On the other hand, consider a subordinatorT on the time interval[0, u]conditioned byT_u=y and pick a jump of T by size-biased sampling. Its distribution has density

zuh(z)f_u(y−z) yf_u(y) dz,

wherehis the density of the Lévy measure ofT andf_uis the density ofT_u(see Theorem 2.1 of [17]). In the present case, we have

u=t, y=1, h(z)=z⁻¹q_z(−t z), f_u(z)=u

zq_z(u−zt ) (cf. Lemma 9 of [15]).

Hence, we get

P^(X)F˜1(t )∈dz

=t q_z(−t z)q_1−z(zt ) (1−z)q₁(0) dz.

This proves the lemma forn=1. We prove the general case by induction. Assume that the result holds forn−1. We have

P^(X)F˜1(t )∈dx1, . . . ,F˜_n(t )∈dxn

=P^(X)F˜1(t )∈dx1, . . . ,F˜n−1(t )∈dxn−1

P^(X)F˜n(t )∈dxn| ˜F1(t )∈dx1, . . . ,F˜n−1(t )∈dxn−1

.

Furthermore, Perman, Pitman and Yor [17] proved that thenth size-biased picked jumpΔnof a subordinator before timeu, conditioned byT_u =y andΔ1=x1, . . . , Δ_n−1=x_n−1, has the law of a size-biased picked jump of the subordinatorT before timeuconditioned byT_u=y−x₁− · · · −x_n−1.Hence we get

P^(X)F˜₁(t )∈dx1, . . . ,F˜_n(t )∈dxn

=

tⁿ⁻¹

q1(0)q_1−Sn−1(S_n−1t )

n−1 i=1

qx_i(−t xi) 1−Si

t qx_n(−t xn)q1−S_n(Snt )

(1−Sn)q1−S_n−1(Sn−1t )dx1· · ·dxn,

whereS_i= ⁱ_k=1x_k. This completes the proof of the lemma.

Of course, the lemma also holds forP^(B)(takeΓ =c=0), thus we obtain:

Corollary 2.3. For alln∈N,for allx1, . . . , xn∈R₊such thatS= ⁿ_i=1xi<1,we have P^(X)(F˜₁(t )∈dx1, . . . ,F˜_n(t )∈dxn)

P^(B)(F˜₁(t )∈dx1, . . . ,F˜_n(t )∈dxn)=g_n(t, x₁, . . . , x_n) g(0,1) , with

gn(t, x1, . . . , xn)=q_1−S(St ) p_1−S(St )

n i=1

q_xi(−t x_i) p_xi(−t x_i).

In order to establish that the law ofF (t )underP^(X)is absolutely continuous with respect to the law ofF (t )under P^(B)with densityg(t,·)/g(0,1), it remains to check that the functiong_nconverges, asntends to infinity, togP^(B)-a.s.

and inL¹(P^(B)). To this aim, we first prove two lemmas:

Lemma 2.4. We have_p^qy(−ty)

y(−ty)<1for ally >0sufficiently small.As a consequence,if(xi)i≥1is a sequence of positive numbers withlimi→∞x_i=0,then the product_n

i=1

q_xi(−t x_i)

p_xi(−t xi) converges asntends to infinity.

Proof. Recall thatXhas the formXt=Bt−Γt+t c, thus we have

∀s >0, ∀u∈R, qs(u)=E

ps(u+Γs−cs) .

Replacingps(u)by its expression √¹

2πsexp(−^u_2s²), we get q_s(u)

p_s(u)=exp

cu−c²s 2

E

exp

−Γ_s² 2s −Γ_s

u s −c

, (2)

i.e., for ally >0, for allt≥0, q_y(−ty)

p_y(−ty)=exp

−y

ct+c² 2

E

exp

−Γ_y²

2y +Γy(t+c)

.

In view of the inequality(γ−α)(γ−β)≥ −(^β−₂^α)², we have

−Γ_y²

2y +Γ_y(t+c)≤y(t+c)² 2 so we deduce that

q_y(−ty)

p_y(−ty)≤e^t2y/2.

Fixc∈(0, c)and letf (y)=P(Γ_y≤cy). Since Γ_t is a subordinator without drift, we have limy→0f (y)=1 (indeed,Γ_y=o(y)a.s.,cf.[4]). On the event{Γ_y≤cy}, we have

exp

−y

ct+c² 2

exp

−Γ_y²

2y +Γ_y(t+c)

≤exp

−y 1

2

c−c2

+t

c−c

≤exp(−εy), withε=¹₂(c−c)².This gives us the upper bound

qy(−ty)

p_y(−ty)≤e^−εyf (y)+

1−f (y) e^yt2/2. Sincef (y)→1 asy→0, we obtain

e^−εyf (y)+

1−f (y)

e^yt2/2=1−εy+o(y).

Thus, we have ^q_p^y(−ty)

y(−ty)<1 forysmall enough, and so the product converges for every sequence(xi)i≥0which tends

to 0.

Lemma 2.5. We have

s→lim1⁻

q_1−s(st ) p_1−s(st )=e^{t c}.

Proof. In view of equation (2), we have:

q_1−s(st ) p_1−s(st )=exp

t sc−c² 2 (1−s)

E

exp

− Γ₁²_−s

2(1−s)−Γ1−s

t s 1−s −c

.

Forsclose enough to 1, we have ₁^{t s}_−s −c≥0, hence exp

− Γ₁²_−s

2(1−s)−Γ_1−s t s

1−s−c

≤1.

SinceΓ_t is a subordinator without drift, we have limu→0Γu

u =0 a.s., and so

s→lim1exp

− Γ₁²_−s

2(1−s)−Γ1−s

t s 1−s−c

=1 a.s.

Using the dominated convergence theorem, we get lim

s→1⁻

q_1−s(st ) p_1−s(st )=e^{t c}.

We can now prove the absolute continuity of the law ofF (t )underP^(X)with respect to that ofF (t )underP^(B). SinceS_n= ⁿ_i=1F_i(t )convergesP^(B)-a.s. to 1, Lemmas 2.4 and 2.5 imply thatG_n=g_n(t,F˜₁(t ), . . . ,F˜_n(t ))con- verges toG=g(t, F (t ))P^(B)-a.s.

It suffices to check thatG_nis uniformly bounded, which implies theL¹convergence. Recall that there existsε >0 such that:

∀x∈(0, ε), qx(−t x) px(−t x)≤1.

Besides, the density function (t, u)→qt(u) is continuous on R^∗₊×R, so that the function x→ ^q_p^{x(−t x)}

x(−t x) is also continuous on[ε,1], therefore, bounded on this interval by some constantA. Since there are at most ¹_ε fragments of F (t )larger thanε, we deduce the upper bound:

∞ i=1

qF_i(t )(−t Fi(t ))

p_{Fi(t )}(−t F_i(t ))≤A^1/ε.

Likewise, the functionS→ ^q_p¹_1−S^{−S(St )}_{(St )} is continuous on[0,1)and has a finite limit at 1⁻, so it is bounded by some constantD >0 on[0,1]. Therefore, we get

Gn≤A^1/εD, P^(B)-a.s.

In consequence,G_nconverges toGP^(B)-a.s. and inL¹(P^(B)). Furthermore, by construction,G_nis a martingale under P^(B)for the filtrationGn

def=σ (F˜1(t ), . . . ,F˜n(t )). Hence, for alln∈N, E^(B)

G| ˜F1(t ), . . . ,F˜_n(t )

=G_n.

This implies, for every bounded continuous functionf:S1→R, that E^(X)

f F (t )

= 1

g(0,1)E^(B) f

F (t ) g

t, F (t ) .

We have proved that, for any fixed timet ≥0, the law of F (t )underP^(X) is absolutely continuous with respect to that ofF (t )underP^(B)with densityg(t, F (t ))/g(0,1). On the other hand, Miermont [15] proved that the process (F (e^−t), t∈R)is an eternal additive coalescent underP^(X)as well as underP^(B)(but with different entrance laws).

Therefore, the semi-groups are the same, which implies the absolute continuity ofP^(X)_|Ft with respect toP^(B)_|Ft with densityg(t, F (t ))/g(0,1).

2.2. Sufficient condition for equivalence

In this section, we give a sufficient condition for the equivalence of the measuresP^(B)_|F

t andP^(X)_|F

t, i.e. a sufficient condition for the strict positivity ofg(t, F (t )),P^(B)-a.s.

Proposition 2.6. Letφbe the Laplace exponent of the subordinatorΓ,i.e.

∀s≥0, ∀q≥0, E

exp(−qΓ_s)

=exp

−sφ (q) .

Assume that there existsδ >0such that

xlim→∞φ (x)x^δ−1=0, (3)

then the functiong(t, F (t ))defined in Theorem1.1is strictly positiveP^(B)-a.s.

Let us note that condition (3) is quite weak. Indeed, if is the Lévy measure of the subordinator andI (x)= _x

0 (t )dt where(t ) denotes((t,∞)), it is well known thatφ (x) behaves likexI (1/x) asx tends to infinity (see [4], Section III). Thus, condition (3) is equivalent toI (x)=o(x^δ)asx tends to 0 (recall that we always have I (x)=o(1)).

Proof of Proposition 2.6. Lett >0 and letP^(B)t denote the law ofF (t )underP^(B). We must check that_∞

i=1

q_xi(−t xi) p_xi(−t xi)

is strictly positiveP^(B)t (dx)-almost surely. Using (2), we have:

qy(−ty) p_y(−ty)=exp

−y

ct+c² 2

E

exp

−Γ_y²

2y +Γ_y(t+c)

.

Since we have ^∞_i=1x_i=1,P^(B)t -a.s., we get ∞

i=1

qx_i(−t xi) px_i(−t xi)≥exp

−ct+c² 2

_∞

i=1

E

exp

−Γ_x²

i

2xi +cΓ_xi

.

Let us now obtain a lower bound forE[exp(−^Γ_2y^y2 +cΓ_y)]. Sincec≥E(Γ1), we have E

exp

−Γ_y² 2y +cΓy

≥E

exp Γ_y

y

E(Γy)−Γ_y 2

.

Set A=E(Γ₁)and letK >0. Notice that the inequalityE(Γ_y)−^Γ₂^y ≥ −Ky may be rewritten Γ_y≤2(A+K)y.

Markov’s inequality yields P

Γ_y≥2(A+K)y

≤ A

2(A+K). Therefore, we get

E

exp

−Γ_y² 2y +cΓy

≥E

exp Γ_y

y

E(Γy)−Γ_y 2

1_{Γ_y≤2(A+K)y}

≥E

exp(−KΓ_y)1_{{Γy≤2(A+K)y}}

≥E

exp(−KΓ_y)

−E

exp(−KΓ_y)1_{{Γy>2(A+K)y}}

≥exp

−φ (K)y

− A

2(A+K).

This inequality holds for allK >0. Hence, choosingK=y^−1/2−εwithε >0, we get E

exp

−Γ_y² 2y +cΓ_y

≥exp

−φ

y^−1/2−ε y

−Ay^1/2+ε.

Furthermore, the product_∞

i=1E[exp(−^Γ_2x^xi2_i +cΓ_xi)]is strictly positive whenever ∞

i=1

1−E

exp

−Γ_x²

i

2xi +cΓ_xi

converges. Hence, we obtain the following sufficient condition:

∃ε >0 such that ∞ i=1

1−exp

−φ

x_i^−1/2−ε x_i

+Ax_i^1/2+ε

<∞, P^(B)t -a.s.

Recall that the distribution of the Brownian fragmentation at timet is equal to the distribution of the jumps of a stable subordinatorT with index 1/2 before timetconditioned onTt=1 (see [1]). In particular, for allε >0

∞ i=1

x_i^1/2+ε<∞, P^(B)t -a.s. (see Formula (9) of [1]).

Thus, the measuresP^(B)_|Ft andP^(X)_|Ft are equivalent whenever there exist two strictly positive numbersε, εsuch that, for xsmall enough,

φ

x^−1/2−ε

x≤x^1/2+ε.

One can easily check that this condition is equivalent to (3).

In Theorem 1.1, we assumed thatX_tis of the formB_t+Γ_t−ct, withc≥E(Γ₁)and whereΓ_tis subordinator. It is natural to ask whether the theorem holds for a wider class of Lévy processes. Let us first notice that the processXmust fulfill the conditions of Miermont’s paper [15] recalled in the Introduction, i.e.Xhas no positive jumps, unbounded variation and finite positive mean. A simple example would be to chooseXt =σ²Bt +Γt−ct, withσ >0,σ =1.

However, it is clear that Theorem 1.1 fails in this case. Indeed, if we chooseXt =2Bt, using Proposition 3 of [15], we get

F^X(2t ), t≥0law

=

F^B(t ), t≥0 .

But, it is well known that we have

nlim→∞n²F_n(t )=t 2

π, P^(B)-a.s. (see [7]).

Hence, the lawsP^(B)t andP^(B)_2t are mutually singular.

3. An integro-differential equation

Let L be the infinitesimal generator of the Brownian fragmentation. Assuming that g belongs to its domain and using the fact thatg(t, F (t ))is aP^(B)-martingale, we would have∂_tg+Lg=0. In this section, we first compute the infinitesimal generator of a fragmentation process, which enables us to deduce an integro-differential equation solved by the functiong.

3.1. The infinitesimal generator of a fragmentation process

We start by recalling an unpublished result obtained by Bertoin and Rouault [9].

LetDdenote the space of functionsf:[0,1] →(0,1]of classC¹withf (0)=1. Givenf ∈Dandx∈S^↓, we set f(x)=^∞

i=1

f (x_i).

Forα∈R+andνmeasure onS^↓such that

S^↓(1−x1)ν(dx) <∞, we define the operator Lαf(x)=f(x)

∞ i=1

x_i^α

ν(dy) f(xiy)

f (x_i)−1

forf ∈Dandx∈S^↓.

Proposition 3.1. Let(Y (t ), t≥0)be a self-similar fragmentation with index of self-similarityα≥0,with dislocation measureνand without erosion.Then,for every functionf∈D,the process

f Y (t )

− _t

0

Lαf Y (s)

ds is a martingale.

We shall use the following lemma.

Lemma 3.2. Forf∈D,y∈S^↓, r∈ [0,1],we have f(ry)

f (r)−1

≤2Cfe^Cfr(1−y1),

withCf = _f^f2∞.

Notice that, sincef isC¹on[0,1]and strictly positive,Cf is always finite.

Proof of Lemma 3.2. We first write lnf (ry₁)−lnf (r)≤

f f

∞(1−y₁)r≤C_f(1−y₁)r, from which we deduce

f(ry)

f (r)−1≤ f (ry1)

f (r) −1≤e^Cf(1−y1)r−1≤C_fe^Cf(1−y1)r.

Besides, we have ln 1

f (xi)≤ 1

f (xi)−1≤C_fx_i, which impliesf(x)≥f (x1)exp

−C_f ∞ i=2

x_i

.

Hence, we get f(ry)

f (r)≥ f (ry1) f (r) exp

−C_f(1−y1)r

≥exp

−2Cf(1−y1)r ,

and we deduce 1−f(ry)

f (r)≤2Cf(1−y1)r.

Proof of Proposition 3.1. Assume first thatf ≡1 in a neighbourhood of 0. In consequence,f(x)depends only on a finite number of terms of the sequencex. Thus, we can write

f(x)=b(x1, x1+x2, . . . , x1+ · · · +xn),

wherebis a function from[0,1]ⁿto[0,1]of classC¹. Furthermore, Berestycki [3] proved that, ifYis a fragmentation process, then, for anyk∈N,(Y₁(t )+ · · · +Y_k(t ), t≥0)is a pure jump process. Considering only the jumps of Y larger thanε and letting ε tend to 0 with the help of a dominated convergence theorem, it is easily checked that (f(Y (t )), t≥0)is also a pure jump process with finite variation. Therefore, ifT denotes the set of times where some dislocation occurs (it is a countable set), we have

f Y (t )

−f Y (0)

=

s∈[0,t]∩T

f Y (s)

−f

Y (s−) .

Let, fors∈T,k_s denote the index of the fragment which has been broken at times, and letΔ_s be the element ofS^↓ according to whichYk_s(s−)has been broken. With these notations, we can write

s∈[0,t]∩T

f Y (s)

−f Y (s−)

=

s∈[0,t]∩T

f

Y (s−)_∞

i=1

1_ks=if(Yi(s−)Δs) f (Yi(s−)) −1

.

Recall that a dislocation of a fragment of massroccurs with rateν_r(dy)=r^αν(dy), thus the predictable compensator of this quantity is (see Section I.3 of [13] or [7])

t 0

dsf Y (s−)

S^↓ν(dy) ∞ i=1

Y_i^α(s−)

f(Yi(s−)y) f (Yi(s−)) −1

= _t

0

Lαf Y (s)

ds.

This completes the proof whenf is constant on some neighbourhood of 0. We now turn our attention to the general case. Given a functionf∈D, we consider a sequence(f_n)_n∈Nof functions ofDsuch that:

• fnconverges uniformly tof.

• f_n≡1 on[0,1/n].

• sup_nf_n∞<∞.

• For alln∈Nand for allx∈ [0,1],fn(x)≥f (x).

Such a sequence clearly exists sincef isC¹andf (0)=1. Moreover, with these hypotheses, we have, for allx∈ [0,1], lnfn(x)→lnf (x)and|lnfn(x)| ≤ |lnf (x)|. Hence, using the dominated convergence theorem, we getfn(x)→f(x) for allx∈S^↓. This implies, for alli∈Nandy∈S^↓,

fn(x_iy) f_n(x_i) −1

n−→→∞

f(xiy) f (x_i)−1

.

Thanks to the previous lemma, these quantities are uniformly bounded byCx_i(1−y1), thusLαfn(x)converges to Lαf(x)and we have|Lαfn(x)| ≤C

S^↓(1−y₁)ν(dy). We conclude thatf(X(t ))−_t

0Lαf(X(s))dsis also a martin-

gale.

The next lemma is a generalization of Proposition 3.1.

Lemma 3.3. Letf:R+× [0,1] →(0,1]be aC¹function equal to1onR+× {0}such that

• i|∂tf (t, xi)|<∞,for everyx,uniformly fort on any compact interval,

• ^∂2_{∂t ∂x}^{f (t,x)}exists and is bounded on any compact interval.

Forx∈S^↓,definef(t,x)=

if (t, x_i).Let(Y (t ), t≥0)be a self-similar fragmentation with index of self-similarity α≥0,with dislocation measureνand without erosion.Then,the process

f t, Y (t )

− _t

0

Lαf s, Y (s)

+∂tf s, Y (s)

ds is a martingale.

Proof. Let us first notice that∂_tf(t,x) exists since _i|∂_tf (t, x_i)|<∞ uniformly fort on any compact interval.

Indeed, given T >0, the function f (t, x) is strictly positive and continuous on [0, T] × [0,1] so we can find a constantηsuch that

∀(t, x)∈ [0, T] × [0,1], f (t, x) > η.

Thus ^∞_i=1|^∂t_{f (t,x}^{f (t,xi)}

i) |converges uniformly on[0, T]. Using the dominated convergence theorem, we get

∂_tf(t,x)=f(t,x) ∞ i=1

∂_tf (t, x_i) f (t, xi) .

Furthermore, for anyx,y∈S^↓, we have lnf(t,x)−lnf(t,y)≤sup

x∈ [0,1],∂_xf (t, x) f (t, x)

x−y1

and

i

∂_tf (t, x_i) f (t, x_i) −

i

∂_tf (t, y_i) f (t, y_i)

≤sup

x∈ [0,1], ∂x

∂_tf (t, x) f (t, x)

x−y1,

wherex−y1= _i|x_i−y_i|denotes the¹norm. Using the fact thatf has bounded derivatives and that^∂2_{∂t ∂x}^{f (t,x)} is also bounded, we deduce that the functionx→∂_tf(t,x)is continuous onS^↓(for either the¹norm or the^∞norm which are equivalent onS^↓). Let us now fixt₂> t₁≥0. We have

E f

t1, Y (t2)

−f

t1, Y (t1)

|Ft₁

=E t₂

t1

Lαf

t2, Y (s) dsFt₁

and E

f

t2, Y (t2)

−f

t1, Y (t2)

|Ft₁

=E t₂

t1

∂tf

s, Y (t2) dsFt₁

.

Therefore, for any partitiont₁=s₀< s₁<· · ·< s_n=t₂, we get E

f

t2, Y (t2)

−f

t1, Y (t1)

|Ft1

=E t₂

t1

Lαf

s, Y (s) +∂_tf

s, Y s

dsFt1

, (4)

wheres=skands=sk−1ifs∈ ]sk−1, sk].Using Lemma 3.2, we can get an upper bound for|Lαf(s, Y (s))|. More- over, the process(Y (s), s≥0)is continuous almost everywhere. Therefore, in view of the dominated convergence theorem and letting max|s_k−s_k−1|tend to 0, we deduce from (4) that

E f

t2, Y (t2)

−f

t1, Y (t1)

|Ft1

=E _t2

t1

Lαf s, Y (s)

+∂_tf s, Y (s)

dsFt1

,

which concludes the proof of the lemma.