Self-interacting diffusions IV: Rate of convergence

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Self-interacting diffusions IV: Rate of convergence

Michel Benaïm, Olivier Raimond

To cite this version:

Michel Benaïm, Olivier Raimond. Self-interacting diffusions IV: Rate of convergence. 2009. �hal-

00408248�

Self-interacting diffusions IV: Rate of convergence ^∗

Michel Bena¨ım

Universit´e de Neuchˆatel, Suisse Olivier Raimond

Universit´e Paris Ouest Nanterre la d´efense, France July 30, 2009

Abstract

Self-interacting diffusions are processes living on a compact Riemannian man- ifold defined by a stochastic differential equation with a drift term depending on the past empirical measure µ

of the process. The asymptotics of µ

is governed by a deterministic dynamical system and under certain conditions (µ

) converges almost surely towards a deterministic measure µ

(see Bena¨ım, Ledoux, Raimond (2002) and Bena¨ım, Raimond (2005)). We are interested here in the rate of convergence of µ

towards µ

. A central limit theorem is proved. In particular, this shows that greater is the interaction repelling faster is the convergence.

∗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 mapping

. For all finite Borel measure µ, let V µ : M → R be the smooth function defined by

V µ(x) = Z

M

V (x, y)µ(dy).

Let (e

) be a finite family of vector fields on M such that X

α

e

(e

f )(x) = ∆f(x),

where ∆ is the Laplace operator on M and e

(f ) stands for the Lie derivative of f along e

. Let (B

) 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)

dX

= X

α

e

(X

) ◦ dB

− ∇ (V µ

)(X

)dt.

where

µ

= 1 t

Z

0

δ

ds is the empirical occupation measure of (X

).

In absence of drift (i.e V = 0), (X

) 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 trajectories through µ

. This type of process was intro- duced in Benaim, Ledoux and Raimond (2002) (hence after referred as [3]) and further analyzed in a series of papers by Benaim and Raimond (2003, 2005, 2007) (hence after referred as [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 pro- cesses with reinforcement we refer the reader to the recent survey paper by Pemantle (2007) ([15]).

1The mapping Vx :M → Rdefined by Vx(y) = V(x, y) is C² and its derivatives are continuous in (x, y)

Notation and Background

Standing Notation We let M (M ) denote the space of finite Borel mea- sures on M, P (M) ⊂ M (M ) the space of probability measures. If I is a metric space (typically, I = M, R

× M or [0, T ] × M ) we let C(I) denote the space of real valued continuous functions on I equipped with the topology of uniform convergence on compact sets. When I is compact and f ∈ C(I) we let k f k = sup

| f (x) | . The normalized Riemann measure on M will be denoted by λ.

Let µ ∈ P (M) and f : M → R a nonnegative or µ − integrable Borel function. We write µf for R

f dµ, and f µ for the measure defined as f µ(A) = R

A

f dµ. We let L

(µ) denote the space of such functions for which µ | f |

< ∞ , equipped with the inner product

h f, g i

= µ(f g) and the norm

k f k

= p µf

. We simply write L

for L

(λ).

Of fundamental importance in the analysis of the asymptotics of (µ

) is the mapping Π : M (M ) → P (M ) defined by

Π(µ) = ξ(V µ)λ (1)

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

R

M

e

λ(dy) . (2)

In [3], it is shown that the asymptotics of µ

can be precisely related to the long term behavior of a certain semiflow on P (M ) induced by the ordinary differential equation (ODE) on M (M ) :

˙

µ = − µ + Π(µ). (3)

Depending on the nature of V, the dynamics of (3) can either be convergent

or nonconvergent leading to similar behaviors for { µ

} (see [3]). When V is

symmetric, (3) happens to be a quasigradient and the following convergence

result hold.

Theorem 1.1 ([5]) Assume that V is symmetric. i.e. V (x, y) = V (y, x).

Then the limit set of { µ

} (for the topology of weak* convergence) is almost surely a compact connected subset of

Fix (Π) = { µ ∈ P (M ) : µ = Π(µ) } .

In particular, if Fix (Π) is finite then (µ

) converges almost surely toward a fixed point of Π. This holds for a generic function V (see [5]).

Sufficient conditions ensuring that Fix (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, That is V (x, y) = K(x, y) + C

and Z

K(x, y)f (x)f (y)λ(dx)λ(dy) ≥ 0 for all f ∈ L

.

(ii) For all x ∈ M, y ∈ M, u ∈ T

M, v ∈ T

M

Ric

(u, u) + Ric

(v, v) + Hess

V ((u, v), (u, v)) ≥ K( k u k

+ k v k

) where K is some positive constant. Here Ric

stands for the Ricci tensor at x and Hess

is the Hessian of V at (x, y).

Then Fix (Π) reduces to a singleton { µ

} and µ

→ 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

of Π has a positive probability to be a limit point for µ

; and any unstable

fixed point cannot be a limit point for µ

.

Organisation of the paper

Let µ

∈ Fix (Π). We will assume that

Hypothesis 1.3 µ

converges a.s. towards µ

. Sufficient conditions are given by Theorem 1.2

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

∆

= e

(µ

− µ

).

It will be shown that, under some conditions to be specified later, for all g = (g

, . . . , g

) ∈ C(M )

the process

[∆

g

, . . . , ∆

g

, V ∆

]

converges in law, as t → ∞ , toward a certain stationary Ornstein-Uhlenbeck process (Z

, Z ) on R

× 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 conver- gence. The section 4 is a proof section. The appendix, section 5, contains general material on random variables and Ornstein-Uhlenbeck processes on C(M ).

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

2 The Ornstein-Uhlenbeck process (Z ^g , Z).

Throughout all this section we let µ ∈ P (M ). For x ∈ M we set V

: M → R defined by V

(y) = V (x, y).

2.1 The operator G

Let g ∈ C(M) and let G

: R × C(M ) → R be the linear operator defined by

G

(u, f ) = u/2 + Cov

(g, f ), (4)

where Cov

is the covariance on L

(µ), that is the bilinear form acting on L

× L

defined by

Cov

(f, g) = µ(f g) − (µf )(µg).

We define the linear operator G

: C(M) → C(M ) by

G

f (x) = G

(f (x), f ) (5)

= f (x)/2 + Cov

(V

, f ).

It is easily seen that k G

f k ≤ (2 k V k + 1/2) k f k . In particular, G

is a bounded operator. Let { e

} denotes the semigroup acting on C(M ) with generator − G

. From now on we will assume the following:

Hypothesis 2.1 There exists κ > 0 and b λ ∈ P (M ) such that µ << b λ with k

k

< ∞ , λ and b λ are equivalent measures with k

k

< ∞ and k

k

< ∞ , and such that for all f ∈ L

( b λ),

h G

f, f i

≥ κ k f k

. Let

λ( − G

) = lim

t→∞

log( k e

k )

t .

This limit exists by subadditivity. Then

Lemma 2.2 Hypothesis 2.1 implies that λ( − G

) ≤ − κ < 0.

Proof : For all f ∈ L

( b λ), d

dt k e

f k

= − 2 h G

e

f, e

f i

≤ − 2κ k e

f k

. This implies that k e

f k

≤ e

k f k

.

Denote by g

the solution of the differential equation dg

dt = Cov

(V

, g

)

with g

= f , where f ∈ C(M ). Note that e

f = e

g

. It is straightfor- ward to check that (using the fact that k

k

< ∞ )

d

dt k g

k

≤ K k g

k

with K a constant depending only on V and µ. Thus sup

t∈[0,1]

k g

k

≤ K k f k

. Now, since for all x ∈ M and t ∈ [0, 1]

d dt g

(x)

≤ K k g

k

≤ K k f k

,

we have k g

k ≤ K k f k

. This implies that k e

f k ≤ K k f k

. Now for all t > 1, and f ∈ C(M),

k e

f k = k e

e

f k

≤ K k e

f k

≤ Ke

k f k

≤ Ke

k f k

.

This implies that k e

k ≤ Ke

, which proves the lemma. QED The adjoint of G

is the operator on M (M ) defined by the relation

m(G

f ) = (G

m)f

for all m ∈ M (M ) and f ∈ C(M ). It is not hard to verify that G

m = 1

2 m + (V m)µ − (µ(V m))µ. (6)

2.2 The generator A

and its inverse Q

Let H

be the Sobolev space of real valued functions on M , associated with the norm k f k

= k f k

+ k∇ f k

. Since Π(µ) and λ are equivalent measures with continuous Radon-Nykodim derivative, L

(Π(µ)) = L

(λ) := L

. We denote by K

the projection operator, acting on L

(Π(µ)), defined by

K

f = f − Π(µ)f.

We denote by A

the operator acting on H

defined by A

f = 1

2 ∆f − h∇ V µ, ∇ f i . Note that for f and g in L

,

h A

f, g i

= − 1 2

Z

h∇ f, ∇ g i (x)Π(µ)(dx) where h· , ·i denotes the Riemannian inner product on M.

For all f ∈ C(M) there exists Q

f ∈ H

such that Π(µ)( Q

f) = 0 and f − Π(µ)f = K

f = − A

Q

f. (7) Note that if P

denotes the semigroup with generator A

, then

Q

f = Z

0

P

K

f dt.

Since there exists p

( · , · ) such that P

f (x) =

Z

M

p

(x, y)f(y)Π(µ)(dy), we have

Q

f (x) = Z

M

q

(x, y)f (y)Π(µ)(dy) where

q

(x, y ) = Z

0

(p

(x, y) − 1)dt.

Then, as shown in [3], Q

f is C

and there exists a constant K such that for all f ∈ C(M) and µ ∈ P (M),

k Q

f k

≤ K k f k

(8) k∇ Q

f k

≤ K k f k

. (9) Finally, note that for f and g in L

,

Z

h∇ Q

f, ∇ Q

g i (x)Π(µ)(dx) = − 2 h A

Q

f, Q

g i

(10)

= 2 h f, Q

g i

.

2.3 The covariance C

We let C b

denote the bilinear continuous form C b

: C(M) × C(M ) → R defined by

C b

(f, g) = 2 h f, Q

g i

.

This form is symmetric (see its expression given by (10)). Note also that for some constant depending on µ,

| C b

(f, g) | ≤ K k f k × k g k .

We let C

denote the mapping C

: M × M → R defined by C

(x, y) = C b

(V

, V

).

Then C

is a covariance function (or a Mercer kernel), i.e. it is continuous, symmetric and P

i,j

λ

λ

C

(x

, x

) ≥ 0.

2.4 The process Z

We now define an Ornstein-Uhlenbeck process on C(M ) of covariance C

and drift − G

. This heavily relies on the general construction given in the appendix.

A Brownian motion on C(M) with covariance C

is a C(M)-valued stochas- tic process W = { W

}

such that

(i) W

= 0;

(ii) t 7→ W

is continuous;

(iii) For every finite subset S ⊂ R × M, { W

(x) }

is a centered Gaussian random vector;

(iv) E [W

(x)W

(y)] = (s ∧ t)C

(x, y ).

Lemma 2.3 There exists a Brownian motion on C(M ) with covariance C

. Proof : Let

d

(x, y ) :=

q

C

(x, x) − 2C

(x, y) + C

(y, y)

= k∇ Q

(V

− V

) k

≤ K k V

− V

k

where the last inequality follows from (9). Then d

(x, y ) ≤ Kd(x, y )

and the result follows from Proposition 5.8 and Remark 5.7 in the appendix.

QED

We say that a C(M )-valued process Z is an Ornstein-Uhlenbeck process of covariance C

and drift − G

if

Z

= Z

− Z

0

G

Z

ds + W

(11) where

(i) W is a C(M )-valued Brownian motion of covariance C

; (ii) Z

is a C(M )-valued random variable;

(iii) W and Z

are independent.

Note that we can think of Z as a solution to the linear SDE dZ

= dW

− G

Z

dt.

It follows from section 5.3 in the appendix that such a process exists and defines a Markov process. Furthermore

Proposition 2.4 Under hypothesis 2.1,

(i) (Z

) converges in law toward a C(M )-valued random variable Z

; (ii) Z

is Gaussian, in the sense that for every finite set S ⊂ M, { Z

(x) }

is a centered Gaussian random vector;

(iii) Let π

denotes the law of Z

. Then π

is characterized by its variance Var (π

) : M (M) → R ,

m 7→ E ((mZ

)

),

and for all m ∈ M , Var (π

)(m) =

Z

0

Z

M×M

C

(x, y )m

(dx)m

(dy)dt

= Z

0

C b

(V m

, V m

)dt where

m

= e

m.

Proof : This follows from Proposition 5.16 in the appendix. Example 5.18 shows that assertion (iii) of this proposition is satisfied. QED

2.5 The process Z

.

For g = (g

, . . . , g

) ∈ C(M )

, let ˜ M = { 1, . . . , n } ∪ M be the disjoint union of { 1, . . . , n } and M, and C

: ˜ M × M ˜ → R be the function defined by

C

(x, y) =

 



C b

(g

, g

) for x, y ∈ { 1, . . . , n } , C

(x, y) for x, y ∈ M,

C b

(V

, g

) for x ∈ M, y ∈ { 1, . . . , n } . Then C

is a Mercer kernel (see section 5.2).

A Brownian motion on R

× C(M ) with covariance C

is a R

× C(M)- valued stochastic process (W

, W ) = { (W

, . . . , W

, W

) }

such that:

(i) W = { W

}

is a C(M )-valued Brownian motion with covariance C

; (ii) For every finite subset S ⊂ R × M, { W

, W

(x) }

is a centered

Gaussian random vector;

(iii) E (W

W

) = (s ∧ t) C b

(g

, g

) and E (W

(x)W

) = (s ∧ t) C b

(V

, g

).

Lemma 2.5 There exists a Brownian motion on R

× C(M) with covariance C

.

Proof : Let ˜ d be the distance on ˜ M defined by d(x, y) = ˜

 



1

for x, y ∈ { 1, . . . , n } , d(x, y ) for x, y ∈ M,

d(x, x

) + 1 for x ∈ M, y ∈ { 1, . . . , n }

where x

is some arbitrary point in M. This makes ˜ M a compact metric space, and it is easy to show that the function C

verifies hypothesis 5.6 (use the proof of Lemma 2.3). The result follows by application of Proposition 5.8. QED

Let now be Z

= (Z

, . . . , Z

) ∈ R

denote the solution to the SDE dZ

= dW

− (Z

/2 + Cov

(Z

, g

)) dt, i = 1, . . . , n (12) where (W

, W ) is as above and Z = (Z

) is given by (11).

The following result generalizes Proposition 2.4.

Proposition 2.6 Under hypothesis 2.1,

(i) The process (Z

, Z

) converges in law toward a centered R

× C(M) valued Gaussian random variable (Z

, Z

).

(ii) Let π

denotes the law of (Z

, Z

). Then π

is characterized by its variance

Var (π

) : R

× M (M ) → R , (u, m) 7→ E (mZ

+ h u, Z

i )

; and for all u ∈ R

, m ∈ M (M ),

Var (π

)(u, m) = Z

0

C b

(f

, f

)dt with

f

= e

X

i

u

g

+ V m

, and where m

is defined by

m

f = m

(e

f) + X

i=1

u

Z

0

e

Cov

(g

, e

f)ds. (13)

Proof : Let G

: R

× C(M ) → R

× C(M) be the operator defined by G

=

I/2 A

0 G

(14)

where A

: C(M ) → R

is the linear map defined by A

(f) =

Cov

(f, g

), . . . , Cov

(f, g

) .

Then (Z

, Z) is a C( ˜ M)-valued Ornstein-Uhlenbeck process of covariance C

and drift − G

. It is not hard to verify that under hypothesis 2.1, the assumptions of Proposition 5.16 hold, so that (Z

, Z

) converges in law to- ward a centered R

× C(M ) valued Gaussian random variable (Z

, Z

) with variance

Var (π

)(u, m) = Z

0

C b

(f

, f

)dt with f

= P

i

u

(i)g

+ V m

and where (u

, m

) = e

(u, m). Now (G

)

=

I/2 0 (A

)

(G

)

and (A

)

u = P

i

u

(g

− µg

)µ. Thus u

= e

u and dm

dt = − (A

)

u

− (G

)

m

Thus m

is the solution with m

= m of dm

dt = − e

X

i

u

(g

− µg

)

!

µ − G

m

(15) Note that (15) is equivalent to

d

dt (m

f ) = − e

Cov

X

i

u

g

, f

!

− m

(G

f) for all f ∈ C(M), and m

= m. From which we deduce that

m

= e

m

− Z

0

e

e

X

i

u

(g

− µg

)µ

! ds

which implies the formula for m

given by (13). QED

For further reference we call (Z

, Z) an Ornstein-Uhlenbeck process of co-

variance C

and drift − G

. It is called stationary when its initial distribution

is π

.

3 A central limit theorem for µ _t

We state here the main results of this article. We assume µ

∈ Fix (Π) satisfies hypotheses 1.3 and 2.1. Set ∆

= e

(µ

− µ

), D

= V ∆

and D

= { D

: s ≥ 0 } . Then

Theorem 3.1 D

converges in law, as t → ∞ , towards a stationary Ornstein- Uhlenbeck process of covariance C

and drift − G

.

For g = (g

, . . . , g

) ∈ C(M )

, we set D

= (∆

g, D

) and D

= { D

: s ≥ 0 } . Then

Theorem 3.2 (D

)

) converges in law towards a stationary Ornstein- Uhlenbeck process of covariance C

and drift − G

.

Define C b : C(M ) × C(M) → R the symmetric bilinear form defined by C(f, g) = b

Z

0

C b

(f

, g

)dt, (16) with (g

is defined by the same formula, with g in place of f )

f

(x) = e

f(x) − Z

0

e

Cov

(f, e

V

)ds. (17)

Corollary 3.3 ∆

g converges in law towards a centered Gaussian variable Z

of covariance

E [Z

Z

] = C(f, g). b

Proof : Follows from theorem 3.2 and the calculus of Var (π

)(u, 0). QED

3.1 Examples

3.1.1 Diffusions

Suppose V (x, y) = V (x), so that (X

) is just a standard diffusion on M with invariant measure µ

=

.

Let f ∈ C(M ). Then f

defined by (17) is equal to (using e

1 = e

1) = e

f. Thus

C(f, g) = 2µ b

(f Q

g). (18)

Corollary 3.3 says that

Theorem 3.4 For all g ∈ C(M )

, ∆

converges in law toward a centered Gaussian variable (Z

, . . . , Z

), with covariance given by

E (Z

Z

) = 2µ

(g

Q

g

).

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 case µ

= λ and V symmetric.

Suppose here that µ

= λ and that V is symmetric. We assume (without loss of generality since Π(λ) = λ implies that V λ is a constant function) that V λ = 0.

Since V is compact and symmetric, there exists an orthonormal basis (e

)

in L

(λ) and a sequence of reals (λ

)

such that e

is a constant function and

V = X

α≥1

λ

e

⊗ e

.

Assume that for all α, 1/2 + λ

> 0. Then hypothesis 2.1 holds with b λ = λ, and the convergence of µ

towards λ holds with positive probability (see [6]).

Let f ∈ C(M) and f

defined by (17), denoting f

= h f, e

i

and f

= h f

, e

i

, we have f

= e

f

and for α ≥ 1,

f

= e

f

− λ

e

e

− 1 λ

f

= e

f

. Using the fact that

C b

(f, g) = 2λ(f Q

g ), this implies that

C(f, g) = 2 b X

α≥1

X

β≥1

1

1 + λ

+ λ

h f, e

i

h g, e

i

λ(e

Q

e

).

This, with corollary 3.3, proves

Theorem 3.6 Assume hypothesis 1.3 and that 1/2 + λ

> 0 for all α. Then for all g ∈ C(M )

, ∆

converges in law toward a centered Gaussian variable (Z

, . . . , Z

), with covariance given by

E (Z

Z

) = C(g b

, g

).

In particular,

E (Z

Z

) = 2

1 + λ

+ λ

λ(e

Q

e

).

Note that when all λ

are positive, which corresponds to what is named a self-repelling interaction in [6], the rate of convergence of µ

towards λ is bigger than when there is no interaction, and the bigger is the interaction (that is larger λ

’s) faster is the convergence.

4 Proof of the main results

We assume hypothesis 1.3 and µ

satisfies hypothesis 2.1. It is possible to choose κ in hypothesis 2.1 such that κ < 1/2. In the following κ will denote such constant. Note that we have λ( − G

) < − κ. Such κ exists when hypothesis 2.1 holds.

4.1 A lemma satisfied by Q

We denote by X (M ) the space of continuous vector fields on M , and equip the spaces P (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

f is a continuous mapping from P (M ) in X (M ).

Proof : Let µ and ν be in M (M ), and f ∈ C(M ). Set g = Q

f . Then f = − A

g + Π(µ)f and

k∇ Q

f − ∇ Q

f k

= k − ∇ Q

A

g + ∇ Q

A

g k

= k∇ g + ∇ Q

A

g k

≤ k∇ (g + Q

A

g ) k

+ k∇ Q

(A

− A

)g k

since ∇ (g + Q

A

g ) = 0 and (A

− A

)g = h∇ V

, ∇ g i , we get

k∇ Q

f − ∇ Q

f k

≤ K kh∇ V

, ∇ g ik

. (19) Using the fact that (x, y) 7→ ∇ V

(y) is uniformly continuous, the right hand term of (19) converges towards 0, when d(µ, ν) converges towards 0, d being a distance compatible with the weak convergence. QED

4.2 The process ∆

Set h

= V µ

and h

= V µ

. Recall ∆

= e

(µ

− µ

) and D

= V ∆

. Note that D

(x) = ∆

V

.

To simplify the notation, we set K

= K

, Q

= Q

and A

= A

. Let (M

)

be the martingale defined by

M

= X

α

Z

1

e

( Q

f)(X

)dB

.

The quadratic covariation of M

and M

(with f and g in C(M)) is given by

h M

, M

i

= Z

1

h∇ Q

f, ∇ Q

g i (X

)ds.

Then for all t ≥ 1 (with ˙ Q

=

Q

) , Q

f(X

) − Q

f(X

) = M

+

Z

1

Q ˙

f(X

)ds − Z

1

K

f (X

)ds.

Thus

µ

f = 1 t

Z

1

K

f (X

)ds + 1 t

Z

1

Π(µ

)f ds + 1 t

Z

0

f (X

)ds

= − 1 t

Q

f (X

) − Q

f (X

) − Z

1

Q ˙

f(X

)ds

+ M

t + 1

t Z

1

h ξ(h

), f i

ds + 1 t

Z

0

f (X

)ds.

Note that (D

) is a continuous process taking its values in C(M) and that D

= e

(h

− h

). For f ∈ C(M ) (using the fact that µ

f = h ξ(h

), f i

),

∆

f = X

i=1

∆

f (20)

with

∆

f = e

− Q

f(X

) + Q

f(X

) + Z

1

Q ˙

f(X

)ds

!

∆

f = e

M

∆

f = e

Z

1

h ξ(h

) − ξ(h

) − Dξ(h

)(h

− h

), f i

ds

∆

f = e

Z

1

h Dξ(h

)(h

− h

), f i

ds

∆

f = e

Z

0

f (X

)ds − µ

f

.

Then D

= P

i=1

D

, where D

= V ∆

. Finally, note that

h Dξ(h

)(h − h

), f i

= − Cov

(h − h

, f). (21)

4.3 First estimates

We recall some estimates from [3]: There exists a constant K such that for all f ∈ C(M ) and t > 0,

k Q

f k

≤ K k f k

k∇ Q

f k

≤ K k f k

k Q ˙

f k

≤ K

t k f k

. These estimates imply in particular that

h M

− M

i

≤ K k f − g k

× t and that

Lemma 4.2 There exists a constant K depending on k V k

such that for all t ≥ 1, and all f ∈ C(M )

k ∆

f k

+ k ∆

f k

≤ K × (1 + t)e

k f k

, (22)

which implies that ((∆

+ ∆

)

)

and ((D

+ D

)

)

both converge

towards 0 (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 ≥ 0 and all f ∈ C(M ),

E [(∆

f)

] ≤ K k f k

,

| ∆

f | ≤ K k f k

× e

Z

0

k D

k

ds,

| ∆

f | ≤ K k f k

× e

Z

0

e

k D

k

ds.

Proof : The first estimate follows from

E [(∆

f)

] = e

E [(M

)

] = e

E [ h M

i

]

≤ e

Z

1

k∇ Q

f k

ds

≤ K k f k

.

The second estimate follows from the fact that

k ξ(h) − ξ(h

) − Dξ(h

)(h − h

) k

= O( k h − h

k

).

The last estimate follows easily after having remarked that

|h Dξ(h

)(h

− h

), f i| = | Cov

(h

− h

, f ) |

≤ K k f k

× k h

− h

k

≤ K k f k

× s

k D

k

. This proves this lemma. QED

4.4 The processes ∆

and D

Set ∆

= ∆

+ ∆

+ ∆

and D

= D

+ D

+ D

. For g ∈ C(M), set ǫ

= e

h ξ(h

) − ξ(h

) − Dξ(h

)(h

− h

), g i

.

Then

d∆

g = − ∆

g

2 dt + dN

+ ǫ

dt + h Dξ(h

)(D

), g i

dt

where for all g ∈ C(M), N

is a martingale. Moreover, for f and g in C(M), h N

, N

i

=

Z

0

h∇ Q

f(X

), ∇ Q

g (X

) i ds.

Then, for all x, dD

(x) = − D

(x)

2 dt + dM

(x) + ǫ

(x)dt + h Dξ(h

)(D

), V

i

dt

where M is the martingale in C(M ) defined by M (x) = N

and ǫ

(x) = ǫ

. We also have

G

(D

)

(x) = D

(x)

2 − h Dξ(h

)(D

), V

i

.

Denoting L

= L

(defined by equation (32) in the appendix), this implies that

dL

(D

)

(x) = dD

(x) + G

(D

)

(x)dt

= dM

(x) + h Dξ(h

)((D

+ D

)

), V

i

dt + ǫ

(x)dt Thus

L

(D

)

(x) = M

(x) + Z

0

ǫ

(x)ds with ǫ

(x) = ǫ

V

where for all f ∈ C(M ),

ǫ

f = ǫ

+ h Dξ(h

)((D

+ D

)

), f i

. Using lemma 5.10,

D

= L

(M)

+ Z

0

e

ǫ

ds. (23) For g = (g

, . . . , g

) ∈ C(M)

, we denote ∆

g = (∆

g

, . . . , ∆

g

), N

= (N

, . . . , N

) and ǫ

g = (ǫ

g

, . . . , ǫ

g

). Then, denoting L

= L

µ∗

(with G

defined by (14)) we have

L

(∆

g, D

)

= (N

, M

) + Z

0

(ǫ

g, ǫ

)ds

so that (using lemma 5.10 and integrating by parts) (∆

g, D

) = (L

)

(N

, M )

+

Z

0

e

(ǫ

g, ǫ

)ds. (24) Moreover

(L

)

(N

, M)

=

N b

, . . . , N b

, L

(M )

,

where

N b

= N

− Z

0

N

2 + C b

(L

(M )

, g

)

ds.

4.5 Estimation of ǫ

4.5.1 Estimation of k L

(M )

k

Lemma 4.4 (i) For all α ≥ 2, there exists a constant K

such that for all t ≥ 0,

E [ k L

(M)

k

]

≤ K

.

(ii) a.s. there exists C with E [C] < ∞ such that for all t ≥ 0, k L

(M )

k

≤ C(1 + t).

Proof : Since k L

(M )

k

≤ K k L

(M )

k

, we estimate k L

(M )

k

. We have

dL

(M )

= dM

− G

L

(M)

dt.

Let N be the martingale defined by N

=

Z

0

* L

(M)

k L

(M)

k

, dM

+

bλ

.

We have h N i

≤ Kt for some constant K. Then

d k L

(M)

k

= 2 k L

(M)

k

dN

− 2 h L

(M)

, G

L

(M )

i

dt

+ d

Z

h M(x) i

λ(dx) b

. Note that there exists a constant K such that

d dt

Z

h M (x) i

b λ(dx)

≤ K