Feller Property and Infinitesimal Generator of the Exploration Process

DOI 10.1007/s10959-007-0082-1

Feller Property and Infinitesimal Generator of the Exploration Process

Romain Abraham·Jean-François Delmas

Received: 13 December 2005 / Revised: 17 May 2006 / Published online: 26 April 2007

© Springer Science+Business Media, LLC 2007

Abstract We consider the exploration process associated to the continuous random tree (CRT) built using a Lévy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infini- tesimal generator on exponential functionals and give the corresponding martingale.

Keywords Exploration process·Lévy snake·Feller property·Measure valued process·Infinitesimal generator

1 Introduction

The coding of a Lévy continuous random tree (CRT) by its exploration process can be found in Le Gall and Le Jan [7]. They associate to a Lévy process with no negative jumps that does not drift to+∞,X=(X_t, t≥0), a critical or sub-critical continuous state branching process (CSBP) and a Lévy CRT which keeps track of the genealogy of the CSBP. The exploration process (ρ_t, t ≥0) takes values in the set of finite measures onR+. Informally, for an individualt≥0, its generation is recorded by its heightHt =sup Suppρt, where Suppμstands for the closed support of the measure

The research of the second author was partially supported by NSERC Discovery Grants of the Probability group at Univ. of British Columbia.

R. Abraham (

MAPMO, Université d’Orléans, B.P. 6759, 45067 Orleans cedex 2, France e-mail: romain.abraham@univ-orleans.fr

J.-F. Delmas

ENPC-CERMICS, 6-8 av. Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vallee, France e-mail: delmas@cermics.enpc.fr

μonR. Andρ_t(dv)records, for the individual labeledt, the “number” of brothers of its ancestor at heightvwhose labels are larger thant.

For instance, the height process(Ht, t≥0)of Aldous’ CRT [2] is a normalized Brownian excursion, and the exploration process(ρt, t≥0)is the Lebesgue measure, that isρ_t is the Lebesgue measure on[0, Ht].

The height process, which is enough to code a CRT, is not Markov whereas the exploration process is Markov. The exploration process has been first introduced in order to study super-Brownian motion with general branching mechanism (see [4]

and [6]). The strong Markov property of the exploration process is a fundamental property and has been proved in [7]. In [1], the exploration process is used in or- der to construct a fragmentation process associated to a Lévy process using Markov properties and martingale problems.

The goal of this paper is to give some martingales related to the exploration process. We also prove that the exploration process satisfies the Feller property (and hence has càd-làg paths and satisfies the strong Markov property). And we compute the generator of the exploration process for exponential functionals. At this point, let us mention that we will always work with the topology of weak convergence for finite measures. But, as the set of finite measures onR₊ is not locally compact for this topology, the Feller property does not make sense on that space. So, we must first extend the definition of the transition function of the exploration process to the set of finite measures onE=R₊∪ {+∞}endowed with a metric that makes it compact.

We give the general organization of the paper. In Sect.2we recall the construc- tion and definition of the exploration process and some of its properties (which can be found in [4]) that will be used in the sequel. We then state the Feller property, Theorem 2.1. We also give its infinitesimal generator for exponential functionals, Corollary2.3, as well as the related martingales in Corollaries2.5and2.6. We also recall in Remark2.8the relation given in [1] between the infinitesimal generator of the exploration process associated to the Lévy process with Laplace exponentψand the infinitesimal generator of the exploration process associated to the Lévy process with Laplace exponentψ^{(θ )}=ψ (θ+ ·)−ψ (θ ),θ≥0. Section3is devoted to the proof of the Feller property. We eventually compute the infinitesimal generator of the exploration process for exponential functionals in Sect.4.

2 Definitions and Main Results

2.1 Notations and Exploration Process

Let S be a metric space, and B(S) its Borel σ-field. We denote by B₊(S) (resp.

Bb(S)) the set of real-valued non-negative (resp. bounded) measurable functions de- fined on S. Let Mf(S) be the set of finite measures defined on S endowed with the topology of weak convergence. We write Mf forMf(R+). Forμ∈Mf(S), f ∈B₊(S)we writeμ, ffor

Sf (x)μ(dx).

We consider aR-valued Lévy processX=(X_t, t≥0)with no negative jumps, no Brownian part and starting from 0. Its law is characterized by its Laplace transform:

forλ≥0

E e^−λXt

=e^{t ψ (λ)},

and we suppose that its Laplace exponent,ψ, is given by ψ (λ)=α0λ+

(0,+∞)

π(d )

e^−λ −1+λ ,

withα₀≥0 and the Lévy measureπ is a positiveσ-finite measure on(0,+∞)such that

(0,+∞)( ∧ ²)π(d ) <∞and

(0,1) π(d )= ∞. The first assumption (with the conditionα₀≥0) implies the processXdoes not drift to+∞, while the second impliesXis a.s. of infinite variation.

Forμ∈Mf, we define

H^μ=sup Supp(μ)∈ [0,∞], (1)

where Supp(μ)is the closed support ofμ, with the convention sup∅ =0.

We recall the definition and properties of the exploration process which are given in [6,7] and [4]. The results of this section are mainly extracted from [4].

LetI =(I_t, t≥0)be the infimum process ofX:I_t=inf0≤s≤tX_s. We will also consider for every 0≤s≤t the infimum ofXover[s, t]:

I_t^s= inf

s≤r≤tXr.

There exists a sequence(ε_n, n∈N^∗)of positive real numbers decreasing to 0 s.t.

H˜_t= lim

k→∞

1 εk

t 0

1_{X_s<I_t^s+ε_k}ds exists and is finite a.s. for allt≥0.

The point 0 is regular for the Markov processX−I, and−I is the local time ofX−I at 0 (see [3, Chap. VII]). LetNbe the associated excursion measure of the processX−Iout of 0, andσ=inf{t >0;X_t−I_t=0}be the length of the excursion ofX−I underN. Recall that underN,X₀=I₀=0.

From Sect. 1.2 in [4], there exists aMf-valued process,ρ⁰=(ρ_t⁰, t≥0), called the exploration process, such that:

• For eacht≥0, a.s.H_t⁰= ˜H_t, whereH_s⁰=H^ρ0s.

• For everyf ∈B+(R+),

ρ_t⁰, f

=

[0,t]d_sI_t^sf H_s⁰ , or equivalently

ρ_t⁰(dr)=

0<s≤t

Xs−<I_t^s

I_t^s−Xs− δ_H0

s(dr). (2)

• Almost surely, for everyt >0, we haveρ_t⁰,1 =X_t−I_t.

2.2 The Feller Property

In Proposition 1.2.3 [4], Duquesne and Le Gall proved that the exploration process is a strong Markov process with càd-làg paths inMf equipped with the topology of weak convergence (in fact they prove the exploration process is a.s. càd-làg with the variation distance on finite measures). Here, we improve this result by proving that the exploration process fulfills the Feller property.

We setE=R₊∪ {+∞}equipped with a distance that makesEcompact. The set Mf(E)is locally compact.

In the definition of the exploration process, asXstarts from 0, we haveρ0=0 a.s.

To get the Markov property ofρ, we must define the processρ started at any initial measureμ∈Mf(E).

Letμ∈Mf(E). Fora∈ [0,μ,1], we define the erased measurekaμby k_aμ([0, r])=μ([0, r])∧(μ,1 −a), forr∈E.

If a >μ,1, we set k_aμ=0. In other words, the measurek_aμ is the measureμ erased by a massabackward fromH^μ, with the natural extension of (1) toMf(E).

Forν, μ∈Mf(E), we define the concatenation[μ, ν] ∈Mf(E)of the two mea- sures, using the conventionx+ ∞ = +∞forx∈E, by

[μ, ν], f = μ, f + ν, f (H^μ+ ·), f ∈B₊(E).

Eventually, we set for everyμ∈Mf(E)and everyt >0, ρ_t=

k_−Itμ, ρ_t⁰ .

We say thatρ=(ρ_t, t≥0)is the exploration process started atρ0=μ, and writePμ

for its law. We setH_t=H^ρt. The processρis an homogeneous Markov process (this is a direct consequence of Proposition 1.2.3 in [4]).

Theorem 2.1 The exploration process, ρ, defined on Mf(E) equipped with the topology of weak convergence is Feller.

The proof of this theorem is given in Sect.3. Sinceρ has the Feller property we recover it enjoys the strong Markov property and has a.s. càd-làg paths. Notice that ifμ∈Mf, asρ⁰ isMf valued, we get by construction thatρ isMf valued. We recover that the exploration process inMf enjoys the strong Markov property and has a.s. càd-làg paths. We can consider the infinitesimal generator ofρ.

2.3 The Infinitesimal Generator

Letf be a bounded non-negative function defined onR+of classC¹with bounded first derivative such thatf (∞)=lim_t→∞f (t )exists.

We considerF, K∈B(Mf(E))defined by: forμ∈Mf(E), F (μ)=e^−μ,f, and K(μ)=F (μ)

ψ (f (H^μ))−f(H^μ)1_{H^μ<∞}

. (3)

Notice thatF is a continuous function, whereas K belongs only toB(Mf(E)) a priori.

We denote byPt the transition function of the exploration processρat timet and we consider its resolvent: forλ >0,Uλ=_∞

0 e^−λtPt. Theorem 2.2 Letλ >0. We have

U_λ(λF −K)(μ)=e^−μ,f− f (0)

ψ⁻¹(λ)e^{−ψ−1(λ)μ,1}.

The proof of this theorem is given in Sect.4. There exists a local time atμ=0 which has to be considered when computing the infinitesimal generator. This trans- lates into the conditionf (0)=0 for the domain of the infinitesimal generator, as stated in the next corollary.

Corollary 2.3 Assume thatf (0)=0,λ >0. We haveU_λ(λF−K)=F. In particu- lar(F, K)belongs to the extended domain of the infinitesimal generator ofρ.

Remark 2.4 Notice the function K is not continuous. However, it can be proved directly that ifμ=0 with compact support then

λlim→∞λ(λUλF−F )(μ)=K(μ) and ifμ=0

λlim→∞λ

λU_λF −F + f (0) ψ⁻¹(λ)

(0)=ψ (f (0))−f(0).

Standard results on Markov process imply the following corollary, see e.g. [5, Chap. 4].

Corollary 2.5 Assumef (0)=0,λ≥0. The process(M_t, t≥0)defined by M_t=e^{−λt−ρt,f}+

_t

0

e^{−λs−ρs,f}

λ−ψ (f (H_s))+f(H_s)1_{Hs<∞}

ds, whereH_s=H^ρs, is a martingale w.r.t. the filtration generated byρ.

We deduce from the optional stopping theorem and the monotone class theorem the next result. (Notice we don’t assumef (0)=0.)

Corollary 2.6 The process(Mt, t≥0)defined by M_t =e^{−ρt∧σ,f}−

_t∧σ

0

e^−ρs,f

ψ (f (H_s))−f(H_s)1_{Hs<∞}

ds is a martingale w.r.t. the filtration generated byρ.

Remark 2.7 Notice from Definition 1.2.1 in [4] thatH_t <∞ Pμ-a.s. for allt >0 ifμ∈Mf. Hence the indicator 1_{Hs<∞}can be removed in Corollaries2.5and2.6 underPμforμ∈Mf.

Remark 2.8 From [1] (see Theorem 3.8 and Proposition 3.10 in Sect. 3.4), there is a relationship between the infinitesimal generator ofρ and the infinitesimal generator of the exploration process,ρ^{(θ )}, associated to a Lévy process with exponentψ^{(θ )}, where

ψ^{(θ )}(λ)=ψ (λ+θ )−ψ (θ ), λ≥0.

More precisely, letF, K∈B(Mf)bounded such that we haveEμ[σ

0 |K(ρs)|ds]<

∞for anyμ∈Mf andM_t=F (ρ_t∧σ)−_t∧σ

0 K(ρ_s), fort≥0, defines a martingale underPμ. Then, we have for allμ∈Mf,Pμ-a.s.

_σ^{(θ )}

0

du

(0,∞)

1−e^−θ π(d )F

ρ_u^{(θ )}, δ0 −F

ρ_u^{(θ )} <∞, (4)

whereσ^{(θ )}=inf{t >0;ρ_t^{(θ )}=0}, and the process(N_t, t≥0)is a martingale, where fort≥0,

Nt=F ρ^{(θ )}

t∧σ^{(θ )} − _t∧σ^{(θ )}

0

du

K ρ_u^{(θ )} +

(0,∞)

1−e^−θ π(d ) F

ρ_u^{(θ )}, δ0 −F ρ^{(θ )}_u

.

Now, if we takeF andKdefined by (3), and assume thatf≥εfor some constant ε >0. Then, we have

Eμ

_σ

0

|K(ρ_s)|ds

≤Eμ

_σ

0

e^−ερs,1ds

=1−e^−εμ,1 ψ (ε) <∞. Notice thatF ([μ, δ₀])−F (μ)≤0 and

(0,∞)

1−e^−θ π(d )

F ([μ, δ₀])−F (μ)

= −F (μ)

(0,∞)

1−e^−θ 1−e^{− f (Hμ)} π(d )

= −F (μ)

ψ (f (H^μ))−ψ^{(θ )}(f (H^μ)) .

Condition (4) is also satisfied. We get for the martingale(N_t, t≥0)given above that N_t=F

ρ^{(θ )}

t∧σ^{(θ )}

− _t∧σ^{(θ )}

0

du

K ρ_u^{(θ )} +

(0,∞)

1−e^−θ π(d ) F

ρ_u^{(θ )}, δ0 −F ρ_u^{(θ )}

=F ρ^{(θ )}

t∧σ^{(θ )} − _t∧σ^{(θ )}

0

duF (ρs) ψ^{(θ )}

f H^{ρ(θ )s}

−f H^{ρs(θ )} .

And thanks to Remark 2.7, we recover Corollary 2.6for the processρ^{(θ )} for any initial measure inMf.

3 Proof of Theorem2.1

In the sequelx₊=max(x,0)denotes the positive part ofx∈R.

3.1 A Distance which Induces the Topology of Weak Convergence

LetGbe an increasing non-negative bounded function of classC¹onR₊. For con- venience, we write G(∞)for the limit of G(t )as t goes to infinity, and we shall assume that G(0)=0 and G(∞)≤1. The setE=R₊∪ {∞} endowed with the distanced(x, y)= |G(x)−G(y)|is compact. Then the set of finite measures onE, Mf(E), equipped with the topology of weak convergence is locally compact.

Forr∈R₊,μ∈Mf(E), we set H_r^μ=H^krμ. Notice the functionr→H_r^μ is non-increasing, right continuous, satisfies H^μ_μ,1=0. Forr ∈ [0,μ,1], we have Hr^μ=g((μ,1 −r)⁻), where gis the right continuous inverse of the cumulative distribution function ofμ. In particular, if forμ, ν∈Mf(E)we haveμ,1 = ν,1 andH_r^μ=H_r^ν for allr≥0, thenμ=ν.

Lemma 3.1 For allr, v∈R₊,μ∈Mf(E), we have

v < H_r^μ⇐⇒μ((v,+∞]) > r. (5) For anyh∈Bb(E), we have

_μ,1

0

h

H_r^μ dr= μ, h (6)

and ifh(0)=0 this can be also written as _∞

0

h

H_r^μ dr= μ, h. (7)

Proof Notice thatH_r^μ=sup{x∈E;μ([x,+∞]) > r}. This implies (5). Lethbe a non-negative non-decreasing bounded function defined onR₊of classC¹such that

h(0)=0 and

R+|h(v)|μ([v,+∞]) dvis finite. We have _∞

0

h

H_r^μ dr=

R²₊h(v)1_{v<H^μ

r}drdv

=

R²₊h(v)1_{r<μ((v,+∞])}drdv

= _∞

0

h(v)μ((v,+∞]) dv

=

R+×E

h(v)1_{u>v}dvμ(du)

= μ, h,

where we used (5) for the second equality, andh(0)=0 for the first and last equality.

By linearity and monotone class theorem, we get (7) for anyh∈Bb(E), such that h(0)=0. Then (6) is a direct consequence of (7) asH_r^μ=0 forr >μ,1.

In particular, (7) holds for h=G. Thus we have_∞

0 G(Hr^μ)dr ≤ μ,1<∞ for all μ∈Mf(E). We introduce the following function defined onMf(E)²: for μ, ν∈Mf(E)

D(μ, ν)= |μ,1 − ν,1| + _∞

0

d

H_r^μ, H_r^ν dr

= |μ,1 − ν,1| +

_{max(μ,1,ν,1)}

0

d

H_r^μ, H_r^ν dr.

SinceGis bounded by 1, we have

μ,1 ≤D(0, μ)≤2μ,1. (8)

From what precedes Lemma3.1, we get thatD(μ, ν)=0 impliesμ=ν. It then is easy to check thatDis a distance onMf(E).

Remark 3.2 We deduce from (8) that a sequence (μ_n, n≥1) is unbounded in (Mf(E), D)if and only if the sequence(μn,1, n≥1)is unbounded.

Lemma 3.3 The topology induced by D onMf(E)corresponds to the topology induced by the weak convergence.

Notice that(Mf(E), D)is a locally compact Polish space.

Proof We first consider a sequence(μ_n, n≥1)of elements ofMf(E)which con- verges for the distanceDtoμ. We shall prove that this sequence converges weakly toμ. Letf ∈Bb(E)such that|f (x)−f (y)| ≤d(x, y)for allx, y∈E. We setf^∗

forf−f (0). Thanks to (7), we have

|μn, f − μ, f| =

μn, f (0) − μ, f (0) + _∞

0

f^∗

H_r^μn −f^∗

H_r^μ dr

≤ |f (0)||μn,1 − μ,1| + _∞

0

f^∗

H_r^μn −f^∗ H_r^μ dr

≤ |f (0)||μ_n,1 − μ,1| + _∞

0

d

H_r^μn, H_r^μ dr

≤(|f (0)| +1)D(μn, μ).

As this holds for any Lipschitz function, we get that(μn, n≥1)converges weakly toμ.

Let(μ_n, n≥1)be a sequence of elements of Mf(E)which converges weakly to μ. In particular, the sequence (μn,1, n≥1)converges to μ,1. Thus, there exists a finite constantAsuch thatμ,1 ≤Aandμn,1 ≤Afor alln≥1. Then, we have

_∞

0

d

H_r^μn, H_r^μ dr= _∞

0

drG

H_r^μn −G H_r^μ

= _∞

0

dr _∞

0

dv G(v)[1_{r<μn((v,∞])}−1_{r<μ((v,∞])}]

≤

R²₊drdvG(v)|1_{r<μn((v,∞])}−1_{r<μ((v,∞])}|

=

R+

dvG(v)|μ_n((v,∞])−μ((v,∞])|,

where we used (5) for the second equality. The weak convergence of (μ_n, n≥1) towards μ implies that dv-a.e., limn→∞μ_n((v,∞]) = μ((v,∞]). As G(v)|μ_n((v,∞])−μ((v,∞])| ≤2AG(v)and 2AGis integrable, we deduce from dominated convergence Theorem that limn→∞_∞

0d(Hr^μn, Hr^μ) dr=0. This and the convergence of(μ_n,1, n≥1)toμ,1imply that limn→∞D(μ_n, μ)=0.

We introduce the distance D in order to get good continuity properties for the concatenation and the erasing functionska.

Lemma 3.4 Leta≥0. We haveD(kaμ, kaν)≤D(μ, ν).

Proof NoticeH_r^kaρ=H_r^ρ_+a. We have

D(k_aμ, k_aν)= |(μ,1 −a)₊−(ν,1 −a)₊| + _∞

0

d

H_r^μ_+a, H_r^ν_+a dr

= |(μ,1 −a)₊−(ν,1 −a)₊| + _∞

a

d

H_r^μ, H_r^ν dr

≤D(μ, ν),

where we used for the last inequality the fact that the function x →(x−a)₊ is

Lipschitz with constant equal to one.

With the conventionx+ ∞ = ∞for allx∈E, we have H_r^[kaμ,ρ]=

H_a^μ+H_r^ρ ifr≤ ρ,1, H_a^μ_+r−ρ,1 ifr≥ ρ,1.

Let μ∈Mf(E), andAμ⊂R₊ be the set of continuity points of the function r→Hr^μ. Notice thatR₊\Aμis at most countable.

Lemma 3.5 Let (μ_n, n≥1)be a sequence ofMf(E)which converges weakly to μ∈Mf(E). For alla∈Aμ,ρ∈Mf(E), we have that([kaμn, ρ], n≥1)converges weakly to[k_aμ, ρ]:

nlim→∞D([kaμn, ρ],[kaμ, ρ])=0.

Proof Let (μ_n, n≥ 1) be a sequence of elements of Mf(E) which converges to μ∈ Mf(E) for the distance D (i.e. which converges weakly). For all a ≥ 0, we have [k_aμ_n, ρ],1 = ρ,1 +(μ_n,1 −a)₊ and limn→∞[k_aμ_n, ρ],1 = [k_aμ, ρ],1. There exists a finite constantAsuch thatμ,1 ≤Aandμ_n,1 ≤A for all n≥1. Furthermore the sequence of functions(G(H_·^μn), n≥1) converges inL¹([0, A], dr)toG(H_·^μ). SinceG(H_·^μn)andG(H_·^μ)are non-increasing, we de- duce that limn→∞G(Ha^μn)=G(Ha^μ)for any continuity pointa of G(H_·^μ). Since G is increasing and continuous, we deduce that for anya∈Aμ∩ [0, A], we have lim_n→∞H_a^μn=H_a^μ. The results is also trivially true fora > A.

Leta∈Aμ. Forr≤ ρ,1, we have H_r^[kaμn,ρ]=H_a^μn+H_r^ρ−−−→

n→∞ H_a^μ+H_r^ρ=H_r^[kaμ,ρ], and forr≥ ρ,1,dr-a.e.

H_r^[kaμn,ρ]=H_a+r−ρ,1^μn −−−→_n→∞ H_a+r−ρ,1^μ =H_r^[kaμ,ρ].

Notice that forr≥A+ ρ,1, we haveH_r^[kaμn,ρ]=H_r^[kaμ,ρ]=0. SinceGis contin- uous, we get by dominated convergence fora∈Aμ, that

nlim→∞

_∞

0

d

H_r^[kaμn,ρ], H_r^[kaμ,ρ] dr=0.

This implies that fora∈Aμ, lim_n→∞D([k_aμ_n, ρ],[k_aμ, ρ])=0.

LetC0(Mf(E))be the set of real continuous functions defined on(Mf(E), D) such that lim_n→∞F (μ_n)=0 whenever lim_n→∞D(0, μn)= +∞, that is whenever limn→∞μn,1 = ∞.

Corollary 3.6 Let(I^∗, ρ)be a random variable with values inR+,×Mf(E), such that the distribution ofI^∗has no atom (i.e.P(I^∗=x)=0 for allx∈R₊). LetF ∈ C0(Mf(E)). Then the functionF:μ→E[F ([k_I∗μ, ρ])]belongs toC0(Mf(E)).

Proof LetQbe the distribution ofI^∗and(R_a(dρ), a≥0)be a measurable version of the conditional law ofρknowingI^∗:Q(da)R_a(dρ)is the distribution of(I^∗, ρ).

ForF∈C0(Mf(E)), we have E[F ([kI^∗μ, ρ])] =

Ra[F ([kaμ, ρ])]Q(da).

Let(μ_n, n≤1)be a sequence converging toμ. From Lemma3.5, for alla∈Aμ, limn→∞D([k_aμ_n, ρ],[k_aμ, ρ])=0. Since the complement ofAμis at most count- able and sinceQ(da)has no atoms, we get thatQ(da)-a.s. limn→∞F ([kaμn, ρ])= F ([kaμ, ρ]), for anyρ∈Mf(E). NoticeF is bounded. By dominated convergence, we get that limn→∞E[F ([k_I∗μ_n, ρ])] =E[F ([k_I∗μ, ρ])]. Thus the function F is continuous.

Notice that the total mass of [k_I∗μ, ρ] is equal to (μ,1 −I^∗)₊+ ρ,1. If limn→∞μn,1 = ∞, then a.s. we have limn→∞F ([kI^∗μn, ρ])=0. By the domi- nated convergence Theorem, we get lim_n→∞F(μ_n)=0.

In conclusion, we get thatF∈C₀(Mf(E)).

3.2 Feller Property: Proof of Theorem2.1

Let(P_t, t≥0)be the transition semi-group of the Markov processρonMf(E).

RecallC0(Mf(E))is the set of real continuous functions defined on(Mf(E), D) such that limn→∞F (μ_n)=0 whenever limn→∞D(0, μn)= +∞, that is whenever limn→∞μn,1 = ∞.

Let us recall that (see e.g. [8, Proposition III.2.4]) ρ is a Feller process if and only if

(i) IfF∈C0(Mf(E)), then for everyt≥0,P_tF ∈C0(Mf(E)).

(ii) For everyF∈C0(Mf(E)), for everyμ∈Mf(E), limt→0P_tF (μ)=F (μ).

To prove condition (i), we state the next lemma.

Lemma 3.7 For everyt >0, the distribution ofI_t has no atom.

Proof Let us denote byτ =(τ_r, r≥0)the right-continuous inverse of the process (−It, t≥0). We know (cf. [3, Chap. VII]) that the processτ is a subordinator with Laplace exponentψ⁻¹. As limλ→∞ψ (λ)/λ= ∞, we have that limλ→∞ψ⁻¹(λ)/λ

= 0. The process τ has no drift. It is moreover strictly increasing since the process−I_t is continuous. It is easy to check that ifP(−I_t=x) >0 for somex >0, then we haveP(−I_t=x, τ_x=t ) >0 andP(τ_−It =t ) >0. This is in contradiction to the fact thatτ has no drift (see Theorem III.4 in [3]). Hence the lemma is proved.

LetF ∈C0(Mf(E))and lett≥0. Then, as the distribution of the random variable It has no atoms (see Lemma3.7), we can apply Corollary3.6withI^∗= −It and get the functionF:μ→Eμ[F (ρ_t^μ)]belongs toC0(Mf(E)).

It remains to prove condition (ii). Let us remark that the exploration process is right-continuous at t=0. Indeed, for every continuous function f on E bounded

by 1, we have,Pμ-a.s.

|ρ_t, f − μ, f| =k_−Itμ, ρ_t⁰ , f

− μ, f

=−μ−k_−Itμ, f + ρ_t⁰, f

H₋^μ_I

t+ ·

≤ μ−k_−Itμ,1 + ρ_t⁰,1

≤ −I_t+X_t−I_t.

And the right continuity ofρ at 0 follows from the right-continuity ofXandI at 0.

LetF ∈C0(Mf(E)). As the functionF is bounded, we get by dominated conver- gence that, for everyμ∈Mf(E),

Eμ[F (ρ_t)] −−→

t→0 F (μ).

4 Proof of Theorem2.2

4.1 Other Properties of the Exploration Process

In this section we recall some properties of the exploration process.

4.1.1 Poisson Decomposition

Letμ∈Mf(E). We decompose the path ofρ underPμaccording to excursions of the total mass ofρabove its minimum. More precisely, we denote by(α_i, β_i), i∈I the excursion intervals ofX−I away from 0. For everyi∈I,t∈(α_i, β_i), we have ρ_αi=k_−Iαiμ=k_−Iαiρ_t. We defineρⁱ by the formulaρ_tⁱ=ρ_(α⁰

i+t )∧βior equivalently ifμ∈Mf:

k₋I_αiμ, ρⁱ_t

=ρ_{(αi+t )∧βi}.

The local time of X−I at level 0 is given by−I, and if(τ_r, r≥0)is the right continuous inverse ofI, then the process(τr, r≥0)is a subordinator with Laplace exponentψ⁻¹(cf. [3, Chap. VII]). Excursion theory forX−Iensures the following result.

Lemma 4.1 Letμ∈Mf(E). The point measure

i∈Iδ_(−I

αi,ρⁱ)is underPμa Pois- son point measure with intensitydrN[dρ].

Let F be a non-negative measurable function defined on R₊ ×Mf(E)× D([0,∞),Mf(E)), where D([0,∞),Mf(E)) stands for the Skorohod space of càd-làg paths inMf(E). We have

Eμ

i∈I

F

αi, ρα_i, ρⁱ =Eμ

_∞

0

drN[F (v, krμ, ρ)]|v=τ_r

. (9)

4.1.2 The Dual Process and Representation Formula

We shall need theMf-valued processη=(η_t, t≥0)defined underNby η_t(dr)=

0<s≤t

Xs−<I_t^s

X_s−I_t^s δ_Hs(dr).

The process η is the dual process of ρ under N (see Corollary 3.1.6 in [4]). Let σ=inf{s >0;ρ_s=0}denote the length of the excursion. Using this duality, it easy to check (see the proof of Lemma 3.2.2 in [4]), that for any bounded measurable functionF defined onMf,

N σ

0

e^{−ψ (γ )t}F (ρt)

=N σ

0

e^−γηt,1F (ρt)

. (10)

We recall the Poisson representation of (ρ, η)under N. Let N(dx d du) be a Poisson point measure on[0,+∞)³with intensity

dx π(d )1_[0,1](u)du.

For everya >0, let us denote byMa the law of the pair(μ_a, ν_a)of finite measures onR+defined by: forf ∈B+(R+)

μ_a, f =

N(dx d du)1_[0,a](x)u f (x), (11) νa, f =

N(dx d du)1_[0,a](x) (1−u)f (x). (12) We eventually setM=_+∞

0 dae^−α0aMa.

Proposition 4.2 For every non-negative measurable functionF onM²_f, N

σ 0

F (ρ_t, η_t) dt

=

M(dμ dν)F (μ, ν).

4.2 Computation of the Resolvent

Letf be a bounded function defined onR₊of classC¹with bounded first derivative such thatf (∞)=limt→∞f (t )exists. By convention, we putf(∞)=0.

We set forμ∈Mf(E)

F_λ(μ)=e^−μ,f

λ−ψ (f (H^μ))+f(H^μ). (13) Letγ=ψ⁻¹(λ). We shall compute

U_λ(F_λ)(μ)=Eμ

_∞

0

dte^{−ψ (γ )t−ρt,f}

ψ (γ )−ψ (f (H_t))+f(H_t) .

We define the functionf_r by f_r(·)=f (· +H_r^μ), where H_r^μ=H^krμ and use the conventionx+ ∞ =xfor allx∈E. Using notations of Sect.4.1.1and (9), we have

U_λ(F_λ)(μ)=Eμ

i∈I

e^{−ψ (γ )αi} _βi−αi

0

dte^{−ψ (γ )t−[k−Iαiμ,ρti],f}.

×

ψ (γ )−ψ f_−Iαi

H^ρit +f_−I

αi

H^ρit

=Eμ

_∞

0

dre^{−ψ (γ )τr}N σ

0

dte^{−ψ (γ )t−[krμ,ρt],f}

×

ψ (γ )−ψ (f_r(H_t))+f_r(H_t) .

Recall (τ_r, r ≥0) is a subordinator with Laplace exponent ψ⁻¹ and (10). Since [krμ, ρt], f = krμ, f + ρt, fr, we get

U_λ(F_λ)(μ)= _∞

0

dre^{−γ r}e^−krμ,fN

× σ

0

dte^{−γηt,1−ρt,fr}

ψ (γ )−ψ (fr(Ht))+f_r(Ht) . We deduce from Proposition4.2that

N σ

0

dte^{−γηt,1−ρt,fr}(ψ (γ )−ψ (f_r(H_t))+f_r(H_t))

= _∞

0

dae^−α0a

ψ (γ )−ψ (f_r(a))+f_r(a)

×exp

− _a

0

dx ₁

0

du

(0,∞)

π(d )

1−e^{−γ (1−u) −ufr(x)}

= _∞

0

da

ψ (γ )−ψ f

a+H_r^μ +f

a+H_r^μ

×e^{−0adx01du ψ(γ (1−u)+uf (x+Hrμ))}. We set fory∈E

(y)= _∞

0

da

ψ (γ )−ψ (f (a+y))+f(a+y) e^−g(a,y), where g(a, y)=_a

0 dxψ (f (x+y))−ψ (γ )

f (x+y)−γ with the convention ^{ψ (v)}_v⁻₋^{ψ (v)}_v =ψ(v), so that

U_λ(F_λ)(μ)= _∞

0

dre^{−γ r}e^−krμ,f

H_r^μ . (14)