MichelBenaïm OlivierRaimond Self-interactingdiffusionsIV:Rateofconvergence ∗

El e c t ro nic

Journ a l of

Pr

ob a b il i t y

Vol. 16 (2011), Paper no. 66, pages 1815–1843.

Journal URL

http://www.math.washington.edu/~ejpecp/

Self-interacting diffusions IV: Rate of convergence

Michel Benaïm

Institut de Mathématiques Université de Neuchâtel,

Rue Emile-Argand 11, CH-2000 Neuchâtel, michel.benaim@unine.ch

Olivier Raimond

Laboratoire Modal’X, Université Paris Ouest, 200 avenue de la République 92000 Nanterre, France

olivier.raimond@u-paris10.fr

Abstract

Self-interacting diffusions are processes living on a compact Riemannian manifold defined by a stochastic differential equation with a drift term depending on the past empirical measureµt of the process. The asymptotics ofµt is governed by a deterministic dynamical system and under certain conditions(µt)converges almost surely towards a deterministic measureµ^∗(see Benaïm, Ledoux, Raimond (2002) and Benaïm, Raimond (2005)). We are interested here in the rate of convergence ofµt towardsµ^∗. A central limit theorem is proved. In particular, this shows that greater is the interaction repelling faster is the convergence.

Key words:Self-interacting random processes, reinforced processes.

AMS 2010 Subject Classification:Primary 60K35; Secondary: 60H10, 62L20, 60F05.

Submitted to EJP on July 29, 2009, final version accepted July 18, 2011.

∗We acknowledge financial support from the Swiss National Science Foundation Grant 200021-103625/1

1 Introduction

Self-interacting diffusions

Let M be a smooth compact Riemannian manifold and V : M × M → R a sufficiently smooth mapping¹. For all finite Borel measureµ, letVµ:M→Rbe the smooth function defined by

Vµ(x) = Z

M

V(x,y)µ(d y). Let(e_α) be a finite family of vector fields onM such thatP

αe_α(e_αf)(x) = ∆f(x), where∆is the Laplace operator onM ande_α(f)stands for the Lie derivative of f alonge_α. Let(B^α)be a family of independent Brownian motions.

A self-interacting diffusion on M associated to V can be defined as the solution to the stochastic differential equation (SDE)

d X_t=X

α

e_α(X_t)◦d B^α_t − ∇(Vµ_t)(X_t)d t

whereµt=¹_tRt

0δX_sdsis the empirical occupation measure of(X_t).

In absence of drift (i.e V = 0), (X_t) is just a Brownian motion on M but in general it defines a non Markovian process whose behavior at time t depends on its past trajectory through µt. This type of process was introduced in Benaim, Ledoux and Raimond (2002) ([3]) and further analyzed in a series of papers by Benaim and Raimond (2003, 2005, 2007) ([4], [5] and [6]). We refer the reader to these papers for more details and especially to [3]for a detailed construction of the process and its elementary properties. For a general overview of processes with reinforcement we refer the reader to the recent survey paper by Pemantle (2007) ([16]).

Notation and Background

We letM(M)denote the space of finite Borel measures onM,P(M)⊂ M(M)the space of proba- bility measures. IfIis a metric space (typically,I=M,R⁺×Mor[0,T]×M) we letC(I)denote the space of real valued continuous functions onI equipped with the topology of uniform convergence on compact sets. The normalized Riemann measure on M will be denoted byλ.

Let µ ∈ P(M) and f : M → Ra nonnegative or µ−integrable Borel function. We write µf for R f dµ, and fµ for the measure defined as fµ(A) = R

Af dµ. We let L²(µ) denote the space of functions for which µ|f|² < ∞, equipped with the inner product 〈f,g〉_µ = µ(f g) and the norm kfk_µ=p

µf². We simply write L² forL²(λ).

Of fundamental importance in the analysis of the asymptotics of(µt)is the mappingΠ:M(M)→ P(M)defined by

Π(µ) =ξ(Vµ)λ (1)

1The mappingV_x:M→Rdefined byV_x(y) =V(x,y)isC²and its derivatives are continuous in(x,y).

whereξ:C(M)→C(M)is the function defined by ξ(f)(x) = e^−f(x)

R

Me^−f(y)λ(d y). (2)

In[3], it is shown that the asymptotics ofµtcan be precisely related to the long term behavior of a certain semiflow onP(M)induced by the ordinary differential equation (ODE) onM(M):

µ˙=−µ+ Π(µ). (3)

Depending on the nature ofV, the dynamics of (3) can either be convergent or nonconvergent lead- ing to similar behaviors for{µt}(see[3]). WhenV is symmetric, (3) happens to be aquasigradient and the following convergence result holds.

Theorem 1.1 ([5]). Assume that V is symmetric, i.e. V(x,y) =V(y,x). Then the limit set of {µt} (for the topology of weak* convergence) is almost surely a compact connected subset of

Fix(Π) ={µ∈ P(M):µ= Π(µ)}.

In particular, ifFix(Π) is finite then(µt) converges almost surely toward a fixed point ofΠ. This holds for a generic functionV (see[5]). Sufficient conditions ensuring thatFix(Π)has cardinal one are as follows:

Theorem 1.2([5],[6]). Assume that V is symmetric and that one of the two following conditions hold (i) Up to an additive constant V is a Mercer kernel: For some constant C, V(x,y) =K(x,y) +C,

and for all f ∈L²,

Z

K(x,y)f(x)f(y)λ(d x)λ(d y)≥0.

(ii) For all x ∈M,y∈M,u∈T_xM,v∈T_yM

Ric_x(u,u) +Ric_y(v,v) +Hess_x,yV((u,v),(u,v))≥K(kuk²+kvk²)

where K is some positive constant. HereRic_x stands for the Ricci tensor at x andHess_x,y is the Hessian of V at(x,y).

ThenFix_(Π)reduces to a singleton{µ^∗}andµt→µ^∗with probability one.

As observed in [6] the condition (i) in Theorem 1.2 seems well suited to describe self-repelling diffusions. On the other hand, it is not clearly related to the geometry of M. Condition(ii) has a more geometrical flavor and is robust to smooth perturbations (of M and V). It can be seen as a Bakry-Emery type condition for self interacting diffusions.

In[5], it is also proved that every stable (for the ODE (3)) fixed point ofΠhas a positive probability to be a limit point forµt; and any unstable fixed point cannot be a limit point forµt.

Organisation of the paper Letµ^∗∈Fix_(Π). We will assume that

Hypothesis 1.3. µtconverges a.s. towardsµ^∗.

In this paper we intend to study the rate of this convergence. Let

∆t=e^t/2(µe^t−µ^∗).

It will be shown that, under some conditions to be specified later, for all g= (g₁, . . . ,g_n)∈C(M)ⁿ the process

∆sg₁, . . . ,∆sg_n,V∆s

s≥t converges in law, as t → ∞, toward a certain stationary Ornstein-Uhlenbeck process(Z^g,Z)onRⁿ×C(M). This process is defined in Section 2. The main result is stated in section 3 and some examples are developed. It is in particular observed that a strong repelling interaction gives a faster convergence. The section 4 is a proof section.

In the following K (respectively C) denotes a positive constant (respectively a positive random constant). These constants may change from line to line.

2 The Ornstein-Uhlenbeck process ( Z

, Z ) .

For a more precise definition of Ornstein-Uhlenbeck processes on C(M) and their basic properties, we refer the reader to the appendix (section 5). Throughout all this section we letµ∈ P(M)and g= (g₁, ...,g_n)∈C(M)ⁿ. Forx ∈M we setV_x :M →R^{defined byV}x(y) =V(x,y).

2.1 The operatorG_µ

Leth∈C(M)and letG_µ,h:R×C(M)→Rbe the linear operator defined by

G_µ,h(u,f) =u/2+Cov_µ(h,f), (4) where Cov_µis the covariance on L²(µ), that is the bilinear form acting onL²×L²defined by

Cov_µ(f,h) =µ(f h)−(µf)(µh). We define the linear operatorG_µ:C(M)→C(M)by

G_µf(x) =G_µ,Vx(f(x),f) = f(x)/2+Cov_µ(Vx,f). (5) It is easily seen that kG_µfk_∞≤(2kVk_∞+1/2)kfk_∞. In particular,G_µ is a bounded operator. Let {e^{−t Gµ}}denote the semigroup acting on C(M) with generator−G_µ. From now on we will assume the following:

Hypothesis 2.1. There exists κ > 0 such that µ << λ with k^dµ_dλk_∞ < ∞, and such that for all f ∈L²(λ),〈G_µf,f〉_λ≥κkfk²_λ.

Let

λ(−G_µ) = lim

t→∞

log(ke^{−t Gµ}k)

t .

This limit exists by subadditivity. Then

Lemma 2.2. Hypothesis 2.1 implies thatλ(−G_µ)≤ −κ <0.

Proof : For all f ∈L²(λ), d

d tke^{−t Gµ}fk²_λ=−2〈G_µe^{−t Gµ}f,e^{−t Gµ}f〉λ≤ −2κke^{−t Gµ}fkλ.

This implies thatke^{−t Gµ}fk_λ≤e^−κtkfk_λ. Denote by g_t the solution of the differential equation d

d tg_t(x) =Cov_µ(Vx,g_t)

with g₀ = f ∈C(M). Note that e^{−t Gµ}f = e^−t/2g_t. It is straightforward to check that (using the fact that k^dµ_dλk_∞ < ∞) ^d

d tkg_tkλ ≤ Kkg_tkλ with K a constant depending only on V and µ. Thus sup_t∈[0,1]kg_tk_λ≤Kkfk_λ. Now, since for allx ∈M andt∈[0, 1]

d d tg_t(x)

≤Kkg_tk_λ≤Kkfk_λ, we havekg₁k_∞≤Kkfk_λ. This implies thatke^−Gµfk_∞≤Kkfk_λ. Now for allt>1, and f ∈C(M),

ke^{−t Gµ}fk_∞ = ke^−Gµe^{−(t−1)Gµ}fk_∞ ≤ Kke^{−(t−1)Gµ}fk_λ

≤ Ke^−κ(t−1)kfk_λ ≤ Ke^−κtkfk_∞. This implies thatke^{−t Gµ}k ≤Ke^−κt, which proves the lemma. QED TheadjointofG_µ is the operator onM(M)defined by the relation

m(G_µf) = (G_µ^∗m)f for allm∈ M(M)and f ∈C(M). It is not hard to verify that

G_µ^∗m= 1

2m+ (V m)µ−(µ(V m))µ. (6)

2.2 The generatorA_µ and its inverseQ_µ

LetH²be the Sobolev space of real valued functions onM, associated with the normkfk²_H=kfk²_λ+ k∇fk²_λ. Since Π(µ) and λ are equivalent measures with continuous Radon-Nykodim derivative, L²(Π(µ)) =L²(λ). We denote byK_µthe projection operator, acting onL²(Π(µ)), defined by

K_µf = f −Π(µ)f. We denote byA_µthe operator acting onH² defined by

A_µf = 1

2∆f − 〈∇Vµ,∇f〉.

Note that for f andhinH² (denoting〈·,·〉the Riemannian inner product onM)

〈A_µf,h〉_Π(µ)=−1 2 Z

〈∇f,∇h〉(x)Π(µ)(d x). For all f ∈C(M)there existsQ_µf ∈H²such thatΠ(µ)(Q_µf) =0 and

f −Π(µ)f =K_µf =−A_µQ_µf. (7)

It is shown in[3]thatQ_µf isC¹ and that there exists a constantK such that for all f ∈C(M)and µ∈ P(M),

kQ_µfk_∞+k∇Q_µfk_∞≤Kkfk_∞. (8) Finally, note that for f andhin L²,

Z

〈∇Q_µf,∇Q_µh〉(x)Π(µ)(d x) =−2〈A_µQ_µf,Q_µh〉_Π(µ)=2〈f,Q_µh〉_Π(µ). (9)

2.3 The covarianceC_µ^g

We letCb_µdenote the bilinear continuous formCb_µ:C(M)×C(M)→R^{defined by} Cb_µ(f,h) =2〈f,Q_µh〉_Π(µ).

This form is symmetric (see its expression given by (9)). Note also that for some constantKdepend- ing onµ,|Cb_µ(f,h)| ≤Kkfk_∞× khk_∞.

We let C_µ denote the mapping C_µ : M × M → R ^{defined by C}_µ(x,y) = Cb_µ(Vx,V_y). Let ˜M = {1, . . . ,n} ∪M andC_µ^g: ˜M×M˜ →Rbe the function defined by

C_µ^g(x,y) =







Cb_µ(g_x,g_y) for x,y ∈ {1, . . . ,n}, C_µ(x,y) for x,y∈M,

Cb_µ(V_x,g_y) for x∈M, y ∈ {1, . . . ,n}. ThenC_µ andC_µ^gare covariance functions (as defined in subsection 5.2).

In the following, when n= 0, ˜M = M and C_µ^g = C_µ. When n ≥ 1, C(M˜) can be identified with Rⁿ×C(M).

Lemma 2.3. There exists a Brownian motion onRⁿ×C(M)with covariance C_µ^g. Proof : Since the argument are the same forn≥1, we just do it forn=0. Let

d_Cµ(x,y) := p

C_µ(x,x)−2C_µ(x,y) +C_µ(y,y)

= k∇Q_µ(V_x−V_y)k_Π(µ) ≤ KkV_x−V_yk_∞

where the last inequality follows from (8). Thend_Cµ(x,y)≤K d(x,y). Thus d_Cµ satisfies (30) and we can apply Theorem 5.4 of the appendix (section 5). QED

2.4 The process(Z^g,Z)

LetG_µ^g:Rⁿ×C(M)→Rⁿ×C(M)be the operator defined by G_µ^g=

‚ I_n/2 A^g_µ 0 G_µ

Œ

(10) where I_n is the identity matrix onR^{n andAg}_µ ^:C(M)→Rⁿ is the linear map defined byA^g_µ(f) =

Cov_µ(f,g₁), . . . , Cov_µ(f,g_n) .

SinceG_µ^gis a bounded operator, for any lawν onRⁿ×C(M), there exists ˜Z= (Z^g,Z)an Ornstein- Uhlenbeck process of covarianceC_µ^g and drift−G_µ^g, with initial distribution given byν (using Theo- rem 5.6). More precisely, ˜Z is the unique solution of

¨ d Z_t = dW_t−G_µZ_td t d Z_t^gi = dW_t^gi −€

Z_t^gi/2+Cov_µ(Zt,g_i)Š

d t, i=1, . . . ,n (11) where ˜Z₀is aRⁿ×C(M)-valued random variable of lawν and ˜W= (W^g,W)is aRⁿ×C(M)-valued Brownian motion of covariance C_µ^g independent of ˜Z. In particular, Z is an Ornstein-Uhlenbeck process of covarianceC_µ and drift−G_µ. Denote byP^g,_t ^µthe semigroup associated to ˜Z. Then Proposition 2.4. Assume hypothesis 2.1. Then there existsπ^g,µthe law of a centered Gaussian variable inRⁿ×C(M), with varianceVar(π^g,µ)where for(u,m)∈Rⁿ× M(M),

Var_(π^g,µ₎₍u,m) := E^€₍mZ_∞+〈u,Z_∞^g〉)²Š

= Z _∞

0

Cb_µ(f_t,f_t)d t with f_t =e^−t/2P

iu_ig_i+V m_t,and where m_tis defined by m_tf =m₀(e^{−t Gµ}f) +

Xn

i=1

u_i Z t

0

e^−s/2Cov_µ(g_i,e^{−(t−s)Gµ}f)ds. (12)

Moerover,

(i) π^g,µ is the unique invariant probability measure ofP_t.

(ii) For all bounded continuous function ϕ on Rⁿ × C(M) and all (u,f) ∈ Rⁿ × C(M), lim_t→∞P^g,µ_{t ϕ(}u,f) =π^g,µϕ.

Proof : This is a consequence of Theorem 5.7. To apply it one can remark thatG_µ^g is an operator like the ones given in example 5.11.

The varianceVar(π^g,µ)is given byVar(π^g,µ)(ν) =R_∞

0 〈ν,e^−sGµgC_µ^ge^s(Gµg)∗ν〉dsforν= (u,m)∈Rⁿ× M(M) =C(M˜)^∗. ThusVar_(π^g,µ₎₍u,m) =R_∞

0 Cb_µ(f_t,f_t)d t with f_t =P

iu_t(i)g_i+V m_t and where (u_t,m_t) =e^{−t(Gµg)∗}(u,m). Now

(G_µ^g)^∗=

‚ I/2 0 (A^g_µ)^∗ (G_µ)^∗

Œ

and(A^g_µ)^∗u=P

iu_i(gi−µg_i)µ. Thusu_t=e^−t/2uandm_t is the solution withm₀=mof d m_t

d t =−e^−t/2 X

i

u_i(g_i−µgi)

!

µ−(G_µ)^∗m_t. (13) Note that (13) is equivalent to

d

d t(mtf) =−e^−t/2Cov_µ X

i

u_ig_i,f

!

−m_t(G_µf) for all f ∈C(M), andm₀=m. From which we deduce that

m_t=e^{−t G∗µ}m₀− Z t

0

e^−s/2e^{−(t−s)G∗µ} X

i

u_i(g_i−µgi)µ

! ds which implies the formula form_t given by (12). QED

An Ornstein-Uhlenbeck process of covarianceC_µ^g and drift −G_µ^g will be called stationarywhen its initial distribution isπ^g,µ.

3 A central limit theorem for µ

We state here the main results of this article. We assumeµ^∗∈Fix(Π)satisfies hypotheses 1.3 and 2.1. Set∆t=e^t/2(µe^t−µ^∗), D_t=V∆t andD_t+·= (D_t+s)_s≥0. Then

Theorem 3.1. D_t+· converges in law, as t → ∞, towards a stationary Ornstein-Uhlenbeck process of covariance C_µ∗ and drift−G_µ∗.

Forg∈C(M)ⁿ, we setD_t^g= (∆tg,D_t)andD_t+·^g = (D^g_t+s)_s≥0. Then

Theorem 3.2. D_t+·^g converges in law towards a stationary Ornstein-Uhlenbeck process of covariance C_µ^g∗ and drift−G_µ^g∗.

DefineCb:C(M)×C(M)→Rthe symmetric bilinear form defined by

Cb(f,h) = Z _∞

0

Cb_µ∗(f_t,h_t)d t, (14) with (h_t is defined by the same formula, withhin place of f)

f_t(x) =e^−t/2f(x)− Z t

0

e^−s/2Cov_µ∗(f,e^{−(t−s)Gµ∗}V_x)ds. (15) Corollary 3.3. ∆_tg converges in law towards a centered Gaussian variable Z_∞^g of covariance

E[Z_∞^giZ_∞^gj] =Cb(g_i,g_j).

Proof : Follows from theorem 3.2 and the calculus ofVar(π^g,µ)(u, 0). QED

3.1 Examples

3.1.1 Diffusions

Suppose V(x,y) = V(x), so that (X_t) is just a standard diffusion on M with invariant measure µ^∗= _λ^{e x p(−}_exp(−V^V)λ₎.

Let f ∈C(M). Sincee^{−t Gµ∗}1=e^−t/21, f_t defined by (15) is equal toe^−t/2f. Thus

Cb(f,g) =2µ^∗(fQ_µ∗g). (16) Corollary 3.3 says that

Theorem 3.4. For all g ∈ C(M)ⁿ, ∆^g_t converges in law toward a centered Gaussian variable (Z_∞^g1, . . . ,Z_∞^gn), with covariance given by

E(Z_∞^giZ_∞^gj) =2µ^∗(giQ_µ∗g_j).

Remark 3.5. This central limit theorem for Brownian motions on compact manifolds has already been considered by Baxter and Brosamler in[1]and[2]; and by Bhattacharya in[7]for ergodic diffusions.

3.1.2 The caseµ^∗=λandV symmetric.

Suppose here thatµ^∗ = λ and that V is symmetric. We assume (without loss of generality since Π(λ) =λimplies thatVλis a constant function) thatVλ=0.

Since V is compact and symmetric, there exists an orthonormal basis (e_α)_α≥0 in L²(λ) and a se- quence of reals(λ_α)_α≥0 such thate₀is a constant function and

V =X

α≥1

λ_αe_α⊗e_α.

Assume that for allα, 1/2+λ_α> 0. Then hypothesis 2.1 is satisfied, and the convergence of µt

towardsλholds with positive probability (see[6]).

Let f ∈C(M) and f_t defined by (15), denoting f^α = 〈f,e_α〉_λ and f_t^α = 〈f_t,e_α〉_λ, we have f_t⁰ = e^−t/2f⁰and forα≥1,

f_t^α = e^−t/2f^α−λ_αe^{−(1/2+λα)t}

‚e^λαt−1 λ_α

Œ f^α

= e^{−(1/2+λα)t}f^α.

Using the fact thatCb_λ(f,g) =2λ(fQ_λg), this implies that

Cb(f,g) =2X

α≥1

X

β≥1

1

1+λ_α+λ_β〈f,e_α〉λ〈g,e_β〉λλ(e_αQ_λe_β). This, with corollary 3.3, proves

Theorem 3.6. Assume hypothesis 1.3 and that 1/2+λ_α > 0 for all α. Then for all g ∈ C(M)ⁿ,

∆^g_t converges in law toward a centered Gaussian variable (Z_∞^g1, . . . ,Z_∞^gn), with covariance given by E(Z_∞^giZ_∞^gj) =Cb(g_i,g_j).

In particular,

E_(Z∞^eαZ_∞^eβ) = 2

1+λ_α+λ_βλ(e_αQ_λe_β).

When allλ_αare positive, which corresponds to what is named a self-repelling interaction in[6], the rate of convergence ofµt towardsλis bigger than when there is no interaction, and the bigger is the interaction (that is largerλ_α’s) faster is the convergence.

4 Proof of the main results

We assume hypothesis 1.3 and µ^∗ satisfies hypothesis 2.1. For convenience, we choose for the constantκin hypothesis 2.1 a constant less than 1/2. In all this section, we fix g = (g1, ...,g_n) ∈ C(M)ⁿ.

4.1 A lemma satisfied byQ_µ

We denote byX(M)the space of continuous vector fields on M, and equip the spacesP(M)and X(M)respectively with the weak convergence topology and with the uniform convergence topology.

Lemma 4.1. For all f ∈ C(M), the mapping µ 7→ ∇Q_µf is a continuous mapping fromP(M) in X(M).

Proof : Letµandν be inM(M), and f ∈C(M). Seth=Q_µf. Then f =−A_µh+ Π(µ)f and k∇Q_µf − ∇Q_νfk_∞ = k − ∇Q_µA_µh+∇Q_νA_µhk_∞

= k∇h+∇Q_νA_µhk_∞

≤ k∇(h+Q_νA_νh)k_∞+k∇Q_ν(A_µ−A_ν)hk_∞. Since∇(h+Q_νA_νh) =0 and(A_µ−A_ν)h=〈∇V_µ−ν,∇h〉, we get

k∇Q_µf − ∇Q_νfk∞≤Kk〈∇V_µ−ν,∇h〉k∞. (17) Using the fact that (x,y) 7→ ∇V_x(y) is uniformly continuous, the right hand term of (17) con- verges towards 0, whend(µ,ν)converges towards 0, d being a distance compatible with the weak convergence. QED

4.2 The process∆

Seth_t=Vµ_t andh^∗=Vµ^∗. Recall∆_t=e^t/2(µ_e^t−µ^∗)andD_t(x) =V∆_t(x) = ∆_tV_x. Then(D_t)is a continuous process taking its values inC(M)andD_t=e^t/2(h_e^t−h^∗).

To simplify the notation, we set K_s = K_µs, Q_s =Q_µs andA_s =A_µs. Let(M_t^f)_t≥1 be the martingale defined byM_t^f =P

α

Rt

1 e_α(Q_sf)(X_s)d B^α_s. The quadratic covariation ofM^f andM^h(with f andhin C(M)) is given by

〈M^f,M^h〉t= Z t

1

〈∇Q_sf,∇Q_sh〉(Xs)ds.

Then for allt≥1 (with ˙Q_t= _{d t}^dQ_t) ,

Q_tf(X_t)−Q₁f(X1) =M_t^f + Z t

1

Q˙_sf(X_s)ds− Z t

1

K_sf(X_s)ds.

Thus

µtf = 1 t

Z t

1

K_sf(Xs)ds+1 t

Z t

1

Π(µs)f ds+1 t

Z 1

0

f(Xs)ds

= −1 t

‚

Q_tf(X_t)−Q₁f(X1)− Z t

1

Q˙_sf(X_s)ds

Œ

+ M_t^f t +1

t Z t

1

〈ξ(h_s),f〉λds+1 t

Z 1

0

f(X_s)ds.

For f ∈C(M)(using the fact thatµ^∗f =〈ξ(h^∗),f〉_λ),∆_tf =P5

i=1∆ⁱ_tf with

∆¹_tf = e^−t/2 −Q_e^tf(X_e^t) +Q₁f(X1) + Z e^t

1

Q˙_sf(X_s)ds

!

∆²_tf = e^−t/2M_e^ft

∆³_tf = e^−t/2 Z e^t

1

〈ξ(h_s)−ξ(h^∗)−Dξ(h^∗)(h_s−h^∗),f〉_λds

∆⁴_tf = e^−t/2 Z e^t

1

〈Dξ(h^∗)(h_s−h^∗),f〉λds

∆⁵_tf = e^−t/2 Z 1

0

f(X_s)ds−µ^∗f

! .

ThenD_t=P5

i=1Dⁱ_t, whereDⁱ_t=V∆ⁱ_t. Finally, note that

〈Dξ(h^∗)(h−h^∗),f〉_λ=−Cov_µ∗(h−h^∗,f). (18) 4.3 First estimates

We recall the following estimate from[3]: There exists a constantK such that for allf ∈C(M)and t>0,

kQ˙_tfk_∞≤ K tkfk_∞.

This estimate, combined with (8), implies that for f andhinC(M),

〈M^f −M^h〉t≤Kkf −hk_∞×t and that

Lemma 4.2. There exists a constant K depending onkVk_∞such that for all t≥1, and all f ∈C(M) k∆¹_tfk_∞+k∆⁵_tfk_∞≤K×(1+t)e^−t/2kfk_∞, (19) which implies that((∆¹+ ∆⁵)t+s)s≥0 and((D¹+D⁵)t+s)s≥0 both converge towards0(respectively in M(M)and in C(R⁺×M)).

We also have

Lemma 4.3. There exists a constant K such that for all t≥0and all f ∈C(M), E[(∆²_tf)²] ≤ Kkfk²_∞,

|∆³_tf| ≤ Kkfk_λ×e^−t/2 Z t

0

kD_sk²_λds,

|∆⁴_tf| ≤ Kkfk_λ×e^−t/2 Z t

0

e^s/2kD_sk_λds.

Proof : The first estimate follows from

E_[(∆²_tf)²] =e^−tE_[(M_e^ft)²] =e^−tE_[〈M^f〉e^t] ≤ Kkfk²_∞. The second estimate follows from the fact that

kξ(h)−ξ(h^∗)−Dξ(h^∗)(h−h^∗)k_λ=O(kh−h^∗k²_λ).

The last estimate follows easily after having remarked that

|〈Dξ(h^∗)(h_s−h^∗),f〉|=|Cov_µ∗(h_s−h^∗,f)| ≤Kkfk_λ× kh_s−h^∗k_λ. This proves this lemma. QED

4.4 The processes∆⁰ and D⁰

Set∆⁰= ∆²+ ∆³+ ∆⁴andD⁰=D²+D³+D⁴. For f ∈C(M), set ε_t^f =e^t/2〈ξ(h_e^t)−ξ(h^∗)−Dξ(h^∗)(h_e^t−h^∗),f〉λ. Then

d∆⁰_tf =−∆⁰_tf

2 d t+d N_t^f +ε_t^fd t+〈Dξ(h^∗)(D_t),f〉_λd t where for all f ∈C(M),N^f is a martingale. Moreover, for f andhinC(M),

〈N^f,N^h〉t= Z t

0

〈∇Q_e^sf(Xe^s),∇Q_e^sh(Xe^s)〉ds.

Then, for all x,

d D⁰_t(x) =−D⁰_t(x)

2 d t+d M_t(x) +εt(x)d t+〈Dξ(h^∗)(D_t),V_x〉λd t

whereM is the martingale inC(M)defined byM(x) =N^Vx andεt(x) =ε^V_t^x. We also have G_µ∗(D⁰)t(x) = D⁰_t(x)

2 − 〈Dξ(h^∗)(D⁰_t),V_x〉_λ. DenotingL_µ∗= L_−Gµ∗ (defined by equation (32) in the appendix (section 5)),

d L_µ∗(D⁰)t(x) = d D⁰_t(x) +G_µ∗(D⁰)t(x)d t and we have

L_µ∗(D⁰)t(x) =M_t(x) + Z t

0

ε⁰_s(x)ds withε⁰_s(x) =ε⁰_sV_x where for all f ∈C(M),

ε⁰_sf =ε_s^f +〈Dξ(h^∗)((D¹+D⁵)s),f〉λ. Using lemma 5.5,

D⁰_t=L⁻_µ∗¹(M)_t+ Z t

0

e^{−(t−s)Gµ∗}ε⁰_sds. (20) Denote ∆tg = (∆tg₁, . . . ,∆tg_n), ∆⁰_tg = (∆⁰_tg₁, . . . ,∆⁰_tg_n) , N^g = (N^g1, . . . ,N^gn) and ε⁰_tg = (ε⁰_tg₁, . . . ,ε⁰_tg_n). Then, denoting L_µ^g_∗=L_−G^g

µ∗ (withG_µ^g_∗ defined by (10)) we have L_µ^g_∗(∆⁰g,D⁰)t= (N_t^g,M_t) +

Z t

0

(ε⁰sg,ε⁰_s)ds

so that (using lemma 5.5 and integrating by parts) (∆⁰_tg,D⁰_t) = (L_µ^g_∗)⁻¹(N^g,M)t+

Z t

0

e^−(t−s)G

g

µ∗(ε⁰sg,ε⁰_s)ds. (21) Moreover

(L_µ^g_∗)⁻¹(N^g,M)t=

Nb_t^g1, . . . ,Nb_t^gn,L⁻_µ∗¹(M)t

, where

Nb_t^gi =N_t^gi− Z t

0

‚N_s^gi

2 +Cb_µ∗(L⁻_µ∗¹(M)s,g_i)

Πds.