A Galves

E Löcherbach

E Orlandi

Perfect simulation of infinite range Gibbs measures and coupling with their finite range approximations

Keywords: Coupling, perfect simulation, d-distance, Gibbs measure, infinite range interactions, finite range approximations, interacting particle systems 82B20, 60K35, 60G60, 62M40

In this paper we address the questions of perfectly sampling a Gibbs measure with infinite range interactions and of perfectly sampling the measure together with its finite range approximations. We solve these questions by introducing a perfect simulation algorithm for the measure and for the coupled measures. The algorithm works for general Gibbsian interaction under requirements on the tails of the interaction. As a consequence we obtain an upper bound for the error we make when sampling from a finite range approximation instead of the true infinite range measure.

Introduction

Consider a Gibbs measure with an infinite range potential and its finite range approximation obtained by truncating the interaction at a certain range. Can we construct a perfect sampling of the Gibbs measure with an infinite range potential? Further, if we make a local inspection of a perfect sampling of the finite range approximation, how often does it coincide with a sample from the original infinite range measure? We address these questions by introducing a new perfect simulation algorithm for the original Gibbs measure and a coupled perfect simulation algorithm for the original measure together with its finite range approximation.

By a perfect simulation algorithm we mean a function mapping a family of independent uniform random variables on the set of configurations having as law the infinite range Gibbs measure. By a coupled perfect simulation algorithm we mean a function mapping a family of independent uniform random variables on the set of pairs of configurations having as first marginal the original infinite range measure and as second marginal the finite range approximation. These functions are defined through two coupled Glauber dynamics having as reversible measures the original Gibbs measure and its finite range approximation, respectively. The perfect simulation algorithm produces coupled samples of the two measures.

The main difficulty in this constructive approach is that we have to deal with infinite range spin flip rates. This difficulty is overcome by a decomposition of the spin flip rates of the Glauber dynamics as a convex sum of local spin flip rates.

Our results are obtained under high temperature conditions and assuming that the interaction decays fast enough. We do not need any monotonicity property of the infinite range Gibbs measure. Under these conditions, we show that the algorithm that we propose stops almost surely, after a finite number of steps. Moreover, we give an upper bound, uniform in space, for the probability of local discrepancy between the coupled realizations of the two measures. This is the content of Theorem 1. As a corollary, Theorem 4 shows that the same upper bound holds for Ornstein's d-distance between the two measures.

An upper bound for the d-distance slightly worse than the one obtained in Theorem 4 can be obtained under less restrictive assumptions, by applying Dobrushin's contraction method. We refer the reader to the series of papers of Dobrushin (1968[START_REF] Dobrushin | Prescribing a system of random variables by conditional distributions[END_REF] as well as to [START_REF] Presutti | Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics[END_REF] for a recent nice self-contained presentation of the subject. However, the contraction method is not constructive and does not provide an explicit sampling procedure for the measures.

The approach followed in the present paper was suggested by a recent work of [START_REF] Galves | Perfect simulation and finitary coding for multicolor systems with interactions of infinite range[END_REF]. There, the flip rates of an infinite range interacting multicolor system were decomposed as convex combinations of local range flip rates. With respect to this work, the novelty of what follows is that the decomposition is explicit and more suitable in view of the intended coupling result. The idea of decomposing flip rates can be traced back to [START_REF] Ferrari | Ergodicity for spin systems with stirrings[END_REF] in which something with the same flavor was done for a finite range interacting particle system and to [START_REF] Ferrari | Cesàro mean distribution of group automata starting from measures with summable decay[END_REF] in which the probability transition of a stochastic chain of infinite order was decomposed as a convex combination of finite order probability transitions. We refer to [START_REF] Galves | Perfect simulation and finitary coding for multicolor systems with interactions of infinite range[END_REF] for more details and related literature.

This paper is organized as follows. In Section 2, we present the basic notations and the main result, Theorem 1. In Section 3, we introduce the Glauber dynamics and the associated processes. The representation of the spin flip rates as convex combination of local rates is given in Theorem 2. In Section 4.1 we introduce the backward sketch process which is the basis of the perfect simulation algorithm and collect the main properties of this process in Proposition 3. Section 4.2 presents the coupled perfect simulation algorithm for the couple of measures. In Section 5 we prove Theorem 1. We conclude the paper in Section 6 discussing how certain features of high temperature Gibbs systems can be obtained applying our construction. In particular we give results on Ornstein's d-distance between the infinite range and the finite range Gibbs measure, and on the decay of correlations. These results are obtained under requirements stronger than the one needed using the Dobrushin contraction method, since they are based on our explicit coupled perfect simulation procedure. and rhythmic pattern identification in natural languages (grant number 03/09930-9) and CNRS-FAPESP project Probabilistic phonology of rhythm. Ce travail a bénéficié d'une aide de l'Agence Nationale de la Recherche portant la référence ANR-08-BLAN-0220-01.

Definitions and main results

Let A = {-1, 1} and S = A Z d be the set of spin configurations. We endow S with the product sigma algebra. We define on Z d the L 1 norm, i = d k=1 |i k |. Configurations will be denoted by Greek letters σ, η, . . . . A point i ∈ Z d will be called a site. For any i ∈ Z d , notations σ(i) and σ i will be used in order to denote the value of the configuration σ at site i. By extension, for any subset V ⊂ Z d , σ(V ) and σ V ∈ A V denote the restriction of the configuration σ to the set of positions in V. If V ⊂ Z d is a finite subset, we denote V ⋐ Z d . In this case, we write |V | for the cardinal of V. Finally, for any i ∈ Z d and σ ∈ S, we shall denote σ i the modified configuration σ i (j) = σ(j), for all j = i, and

σ i (i) = -σ(i). Definition 1 An interaction is a collection {J B , B ⋐ Z d } of real numbers indexed by finite subsets B ⋐ Z d , |B| ≥ 2, which satisfies sup i∈Z d B:i∈B |J B | < ∞.
(2.1)

For finite B ⊂ Z d and σ ∈ S let χ B (σ) = i∈B σ(i).
Definition 2 A probability measure µ on S is said to be a Gibbs measure relative to the interaction {J B , B ⋐ Z d } if for all i ∈ Z d and for any fixed ζ ∈ S, a version of the conditional probability µ({σ :

σ(i) = ζ(i)|σ(j) = ζ(j) for all j = i}) is given by µ({σ : σ(i) = ζ(i)|σ(j) = ζ(j) for all j = i}) = 1 1 + exp(-2 B:i∈B J B χ B (ζ)) . (2.2)
To define the Gibbs measure µ [L] relative to the interaction truncated at range L we set

B i (L) = {j ∈ Z d : j -i ≤ L}. (2.3)
Then µ [L] is the probability measure on S such that for all i ∈ Z d and for any fixed ζ ∈ S, a version of the conditional probability µ [L] ({σ :

σ(i) = ζ(i)|σ(j) = ζ(j) for all j = i}) is given by µ [L] ({σ : σ(i) = ζ(i)|σ(j) = ζ(j) for all j = i}) = 1 1 + exp(-2 B:i∈B,B⊂B i (L) J B χ B (ζ))
.

We consider the interaction J β B = β J B where β is a positive parameter. The associated Gibbs measures will be denoted by µ and µ [L] , omitting to write the explicit dependence on β. We denote by (X(i), i ∈ Z d ), (X [L] (i), i ∈ Z d ) two random configurations on S distributed according to µ and µ [L] , respectively.

All processes that we consider in this paper will be constructed as functions of an independent family of uniform random variables (U n (i), n ∈ N, i ∈ Z d ). Denote (Ω, A, P ) the probability space on which is defined the independent family of uniform random variables.

Definition 3 A perfect sampling algorithm of the measure µ is a map F from [0, 1] N×Z d to {-1, +1} Z d , such that F (U n (j), n ∈ N, j ∈ Z d ) has distribution µ.
A coupled perfect sampling algorithm for the pair ((X(i),

X [L] (i)), i ∈ Z d ) is a map G from [0, 1] N×Z d to {-1, +1} Z d ×Z d , such that G(U n (j), n ∈ N, j ∈ Z d ) has marginals (X(i), i ∈ Z d ) and (X [L] (i), i ∈ Z d ).
Since F produces an unbiased sample from the target distribution µ, the algorithm is denoted "perfect".

Definition 4

The perfect sampling algorithm F of the measure µ is said to stop almost surely after a finite number of steps if for every site i ∈ Z d , there exists a finite subset F (i) of Z d and a finite random variable

N (i) ST OP ≥ 1 such that if U ′ n (j) = U n (j) for all j ∈ F (i) , 1 ≤ n ≤ N (i) ST OP , (2.4
)

then F (U n (j), n ∈ N, j ∈ Z d )(i) = F (U ′ n (j), n ∈ N, j ∈ Z d )(i). (2.5) Note that (2.5) implies that F (U n (j), n ∈ N, j ∈ Z d )(i) depends only on U n (j) for 1 ≤ n ≤ N (i)
ST OP . The following theorem is our main result.

Theorem 1 Assume that

sup i∈Z d k |B i (k)|   B:i∈B,B⊂B i (k),B ⊂B i (k-1) |J B |   < ∞. ( 2

.6)

1. There exists β c > 0 such that for any β < β c there is a perfect sampling algorithm of µ. This algorithm stops after a finite number of steps almost surely and

sup i∈Z d E(N (i) ST OP ) ≤ 1 γ , (2.7)
where γ is given in (4.37).

2. Moreover, for any β < β c , there exists a coupled perfect sampling algorithm for the pair ((X(i),

X [L] (i)), i ∈ Z d ) satisfying sup i∈Z d P (X(i) = X [L] (i)) ≤ 1 γ sup i∈Z d 1 -e -β P B:i∈B,B ⊂B i (L) |J B | . (2.8)
Remark 1 The constant β c is the solution of equation (5.39) in Section 5.

Remark 2 In case of an interaction defined through a pairwise potential {J(i, j), (i, j) ∈ Z d × Z d } with J(i, i) = 0 for i ∈ Z d , condition (2.6) of Theorem 1 reads as

sup i∈Z d k |B i (k)|   j: i-j =k |J(i, j)|   < ∞.
In this case, sup

i∈Z d P (X(i) = X [L] (i)) ≤ 1 γ sup i∈Z d 1 -e -β P j: i-j >L |J(i,j)| .
The above perfect sampling algorithms are based on a coupled construction of two processes having µ and µ [L] as reversible measures. These two processes will be introduced in the next section. The definition of the algorithms can only be given in Section 4. The proof of Theorem 1 is given in Section 5.

Glauber dynamics

We now introduce a Glauber dynamics having µ as reversible measure. This is an interacting particle system (σ t (i), i ∈ Z d , t ∈ R) taking values in S. Sometimes we will also use the short notation (σ t ) t for the interacting particle system. To describe the process, we need some extra notation.

For any i ∈ Z d , σ ∈ {-1, 1} Z d , and β > 0, we define c i (σ), which is the rate at which the spin i flips when the system is in the configuration σ, as

c i (σ) = exp -β B:i∈B J B χ B (σ) . (3.9)
The generator G of the process (σ t ) t is defined on cylinder functions f : S → R as follows

G f (σ) = i∈Z d c i (σ)[f (σ i ) -f (σ)] . (3.10)
Observe that the condition (2.6) of Theorem 1 implies that sup i B:i∈B

|B| |J B | < ∞,
where |B| designs the cardinal of the set B. Hence the rates c i (σ) are uniformly bounded in i and σ, and sup

i∈Z d j∈Z d sup σ |c i (σ) -c i (σ j )| < ∞. (3.11)
Therefore, Theorem 3.9 of Chapter 1 of [START_REF] Liggett | Interacting particles systems[END_REF] implies that G is the generator of a Markov process (σ t ) t on SS. By construction, the process (σ t ) t is reversible with respect to the Gibbs measure µ corresponding to the interaction

J β B = β J B .
For any ℓ ≥ 1, we now define a Glauber dynamics having µ [ℓ] as reversible measure. More precisely, we consider the Markov process (σ

[ℓ] t ) t on SS having generator G [ℓ] f (σ) = i∈Z d c [ℓ] i (σ)[f (σ i ) -f (σ)], (3.12)
where the rates c

[ℓ]

i are given by

c [ℓ] i (σ) = r [ℓ] i exp   -β B:i∈B,B⊂B i (ℓ) J B χ B (σ)   (3.13) and r [ℓ] i = e -β P B:i∈B,B ⊂B i (ℓ) |J B | .
Let us stress the fact that compared to the usual definition of a Glauber dynamics, an extra factor r

[ℓ]
i appears in the definition of the rates c

[ℓ]
i . This extra factor is important in view of the intended coupling of both processes σ t and σ

[ℓ] t (see Theorem 2 below) and does not change the equilibrium behavior of the process as is shown in the next proposition.

Proposition 1 The process (σ [ℓ]

t ) t is reversible with respect to the Gibbs measure µ [ℓ] relative to the pairwise interaction J β truncated at range ℓ.

Proof By construction, we have that

c [ℓ] i (σ) c [ℓ] i (σ i ) = µ [ℓ] (σ i ) µ [ℓ] (σ) .
Thus, by Proposition 2.7 of Chapter IV of [START_REF] Liggett | Interacting particles systems[END_REF], it follows that (σ

[ℓ]
t ) t is reversible with respect to µ [ℓ] .

•

The main tool to prove Theorem 1 is a coupled construction of the processes (σ

t (i), i ∈ Z d , t ∈ R) and (σ [ℓ] t (i), i ∈ Z d , t ∈ R) for ℓ = L, using the basic family of uniform random variables (U n (i), n ∈ N, i ∈ Z d ).
This coupling is based on a decomposition of the spin flip rates c i (σ) as a convex combination of local range spin flip rates.

To present this decomposition we need the following notation. We define

M i = 2e β P B:i∈B |J B | ,
and the sequence (λ i (k)) k≥0 as follows

λ i (k) =      e -2β P B:i∈B |J B | if k = 0, e -β P B:i∈B,B ⊂B i (1) |J B | -e -2β P B:i∈B |J B | if k = 1, e -β P B:i∈B,B ⊂B i (k) |J B | -e -β P B:i∈B,B ⊂B i (k-1) |J B | if k > 1.
Finally, for all σ ∈ S and for any k ≥ 2, we define the update probabilities

p [k] i (-σ(i)|σ) = 1 M i e -β P B:i∈B,B⊂B i (k-1) J B χ B (σ) e -β P B:i∈B,B⊂B i (k),B ⊂B i (k-1) J B χ B (σ) -e -β P B:i∈B,B⊂B i (k),B ⊂B i (k-1) |J B | 1 -e -β P B:i∈B,B⊂B i (k),B ⊂B i (k-1) |J B |
,

and for k = 1, p [1] i (-σ(i)|σ) = 1 M i e -β P B:i∈B,B⊂B i (1) J B χ B (σ) -e -β P B:i∈B,B⊂B i (1) |J B | 1 -e -2β P B:i∈B,B⊂B i (1) |J B | e -β P B:i∈B,B ⊂B i (1) |J B | .
To obtain a probability measure on A, we define for all k ≥ 1,

p [k] i (σ(i)|σ) = 1 -p [k] i (-σ(i)|σ).
Finally, for k = 0, we define

p [0] i (1) = p [0] i (-1) = 1 2 . (3.14)
Since this last probability does not depend on the site i we omit the subscript i. Note that, by construction, for any integer k ≥ 1, the probabilities p

[k]
i (a|σ) depend only on σ(B i (k)). Now we state the decomposition theorem.

Theorem 2 Let us assume that the uniform summability condition (2.1) holds.

1. The sequence (λ i (k)) k≥0 defines a probability distribution on the set of positive integers {0, 1, ...}.

2. For any ℓ ≥ 1, σ ∈ S, the following decomposition holds

c [ℓ] i (σ) = M i λ i (0) 1 2 + ℓ k=1 λ i (k)p [k] i (-σ(i)|σ) . (3.15)
3. For any σ ∈ S, the following decomposition holds

c i (σ) = M i λ i (0) 1 2 + ∞ k=1 λ i (k)p [k] i (-σ(i)|σ) . (3.16)
Proof Item 1 follows directly from the definition and (2.1).

To prove item 2, set

c [0] i (σ) = 1 2 M i λ i (0)
and observe that

c [ℓ] i (σ) = ℓ k=1 c [k] i (σ) -c [k-1] i (σ) + c [0] i (σ),
where the differences c

[k] i (σ) -c [k-1] i (σ) are always positive. Since by definition c [k] i (σ) -c [k-1] i (σ) = M i λ i (k) p [k] i (-σ(i)|σ) ,
we obtain the decomposition stated in item 2.

To prove item 3, observe that the uniform summability condition (2.1) implies that

lim k→∞ c [k] i (σ) = c i (σ) , (3.17) since r [k] i → 0 for k → ∞.
Thus, taking into account item 2, we obtain the desired decomposition.

•

To present the coupling, it is convenient to rewrite the generator of the Glauber dynamics given in (3.10) as

G f (σ) = i∈Z d a∈A c i (a|σ)[f (σ i,a ) -f (σ)] , (3.18)
where σ i,a (j) = σ(j), for all j = i, and σ i,a (i) = a, and where

c i (a|σ) = c i (σ) if a = -σ(i), M i -c i (σ) if a = σ(i) . (3.19)
In a similar way, for any integer ℓ ≥ 1, we define the rates

c [ℓ] i (a|σ) = c [ℓ] i (σ) if a = -σ(i), M i ℓ k=0 λ i (k) -c [ℓ] i (σ) if a = σ(i) (3.20)
and the generator

G [ℓ] f (σ) = i∈Z d a∈A c [ℓ] i (a|σ)[f (σ i,a ) -f (σ)] . (3.21)
This amounts to include in the generators the rates of invisible jumps in which the spin i is updated with the same value it had before. Obviously the generator defined in (3.10) (and in (3.12)) defines the same stochastic dynamics as the generator given in (3.18) (and in (3.21), respectively).

With this representation, we have the following corollary of Theorem 2.

Corollary 1

G f (σ) = i∈Z d a∈A k≥0 M i λ i (k)p [k] i (a|σ)[f (σ i,a ) -f (σ)] ,
and for any ℓ ≥ 1,

G [ℓ] f (σ) = i∈Z d a∈A ℓ k=0 M i λ i (k)p [k] i (a|σ)[f (σ i,a ) -f (σ)] .
From now on, we fix ℓ = L. The decomposition given in Corollary 1 suggests the following construction of the Glauber dynamics having G and G [L] as infinitesimal generator.

For each site i ∈ Z d consider a Poisson point process N i having rate M i . The Poisson processes corresponding to distinct sites are all independent. If, at time t, the Poisson clock associated to site i rings, we choose a range k with probability λ i (k) independently of everything else. Then we update the value of the configuration at this site by choosing a symbol a with probability p

[k] i (a|σ(B i (k))) depending only on the configurations inside the set B i (k). It is clear that using this decomposition, we can construct both processes σ t and σ

[L]
t in a coupled way, starting from any initial configuration.

Actually we can do better than this. We can make a perfect simulation of the pair of measures µ and µ [L] which are the invariant probability measures of the processes having generators G and G [L] respectively.

Let us explain how we sample the configuration at a fixed site i ∈ Z d under the measure µ.

The simulation procedure has two stages. In the first stage we determine the set of sites whose spins influence the spin at site i under equilibrium. We call this stage backward sketch procedure. It is done by climbing up from time 0 back to the past a reverse time Poisson point process with rate M i until the last time the Poisson clock rung. At that time, choose a range k with probability λ i (k). If k = 0, we decide the value of the spin with probability 1 2 independently of anything else. If k is different from zero, we restart the above procedure from any of the sites j ∈ B i (k). The procedure stops whenever each site involved in this backward time evolution has chosen a range 0.

When this occurs, we can start the second stage, in which we go back to the future assigning spins to all sites visited during the first stage. We call this procedure forward spin assignment procedure. This is done from the past to the future by using the update probabilities p

[k] i starting at the sites which ended the first procedure by choosing range 0. For each one of these sites a spin is chosen by tossing a fair coin. The values obtained in this way enter successively in the choice of the values of the spins depending on a neighborhood of range greater or equal to 1.

In the case of the measure µ [L] we use the same procedure, but only considering the choices of k which are smaller or equal to L. These procedures will be described formally in the next section.

4 Perfect simulation of the measure µ and coupling

Backward sketch procedure

Before describing formally the two algorithms described at the end of the last section, let us define the stochastic process which is behind the backward sketch procedure. Our aim is to define, for every site i ∈ Z d , a process (C

(i) s ) s≥0 taking values in the set of finite subsets of Z d , such that C (i)
s is the set of sites at time -s whose spins affect the spin of site i at time t = 0. This is done as follows.

For each i ∈ Z d , denote by . . . T i -2 < T i -1 < T i 0 < 0 < T i 1 < T i 2 < .
. . the occurrence times of the rate M i Poisson point process N i on the real line. The Poisson point processes associated to different sites are independent. To each point T i n associate an independent mark K i n according to the probability distribution (λ i (k)) k≥0 . As usual, we identify the Poisson point processes and the counting measures through the formula

N i [s, t] = n∈Z 1 {s≤T i n ≤t} .
The backward point process starting at time 0, associated to site i ∈ Z d is defined as

T (i,0) n = -T i -n+1 , for any n ≥ 1. (4.22)
We also define the associated marks

K (i,0) n = K i -n+1 (4.23)
and for any k ≥ 0, the backward k-marked Poisson point process starting at time 0 as

N (i,k) [s, u] = n∈Z 1 {s≤T (i,0) n ≤u} 1 {K (i,0) n =k} . (4.24)
To define the backward sketch process we need to introduce a family of transformations {π (i,k) , i ∈ Z d , k ≥ 0} on F(Z d ), the set of finite subsets of Z d , defined as follows. For any unitary set {j} and k ≥ 1,

π (i,k) ({j}) = B i (k), if j = i, {j}, otherwise. (4.25)
For k = 0, we define

π (i,0) ({j}) = ∅, if j = i, {j}, otherwise. (4.26)
For any set finite set F ⊂ Z d , we define similarly

π (i,k) (F ) = ∪ j∈F π (i,k) ({j}). ( 4

.27)

The backward sketch process starting at site i at time 0 will be denoted by (C

(i) s ) s≥0 . The set C (i)
s is the set of sites at time -s whose spins affect the spin of site i at time t = 0. The evolution of this process is defined through the following equation. C (i) 0 = {i}, and

f (C (i) s ) = f (C (i) 0 ) + k≥0 j∈Z d s 0 [f (π (j,k) (C (i) u-)) -f (C (i) u-)] N (j,k) (du), (4.28)
where f : F(Z d ) → R is any bounded cylinder function. This family of equations characterizes completely the time evolution (C (i) s ) s≥0 . In a similar way, for the truncated process, we define its associated backward sketch process by

f (C [L],(i) s ) = f (C [L],(i) 0 ) + L k=0 j∈Z d s 0 [f (π (j,k) (C [L],(i) u- )) -f (C [L],(i) u- )] N (j,k) (du), (4.29)
where we use the same Poisson point processes N (j,k) , 0 ≤ k ≤ L, as in (4.28).

The following proposition summarizes the properties of the family of processes defined above.

Proposition 2 For any site

i ∈ Z d , C (i) s and C [L],(i) s
, s ∈ R + , are Markov jump processes having as infinitesimal generator,

Lf (C) = i∈C k≥1 M i λ i (k)[f (C ∪ B i (k)) -f (C)] + λ i (0)[f (C \ {i}) -f (C)],
(4.30)

with initial condition at time t = 0, C (i) 0 = {i}, and

L [L] f (C) = i∈C L k=1 M i λ i (k)[f (C ∪ B i (k)) -f (C)] + λ i (0)[f (C \ {i}) -f (C)], (4.31)
with initial condition at time t = 0, C

[L],(i) 0

= {i}, respectively. Here f : F(Z d ) → R is any bounded cylindrical function.

Let

T (i)

ST OP = inf{s : C (i) s = ∅} be the relaxation time. We introduce the sequence of successive jump times T (i) n , n ≥ 1, of processes N (j,k) whose jumps occur in (4.28). Let

T (i) 1 = T (i,0) 1
and define successively for n ≥ 2

T (i) n = inf{t > T (i) n-1 : ∃j ∈ C (i) T (i) n-1 , ∃k : N (j,k) (] T (i) n-1 , t]) = 1}. (4.32) Now we put C (i) n = C (i) T (i) n . (4.33)
Finally, notice that

N (i)
ST OP of Definition 4 is equal to

N (i) ST OP = inf{n : C (i) n = ∅}.
This is the number of steps of the backward sketch process. For the algorithm to be successful, it is crucial to show that both relaxation time

T (i)
ST OP and the number of steps

N (i)
ST OP are finite. This is the content of next theorem.

Proposition 3 Under the requirement

sup i∈Z d k≥1 |B i (k)|λ i (k) < 1, (4.34)
we have uniformly for any i ∈ Z d , P (T

(i)
ST OP > t) ≤ e -γt (4.35) and E(N

(i) ST OP ) ≤ 1 γ , (4.36)
where γ = 1sup

i∈Z d k≥1 |B i (k)|λ i (k). (4.37)
The proof of Proposition 3 will be given at the end of Section 5 below.

Algorithms

We now go to the crucial point of precisely defining the maps F and G whose existence is claimed in Theorem 1. These maps are implicitly defined by the backward sketch procedure and the forward spin assignment procedure. Let us give the algorithmic like description of these procedures. The following variables will be used.

• N is an auxiliary variables taking values in the set of non-negative integers {0, 1, 2, . . .}

• N (i)
ST OP is a counter taking values in the set of non-negative integers {0, 1, 2, . . .}

• N

[L],(i)
ST OP is a counter taking values in the set of non-negative integers {0, 1, 2, . . .}

• I is a variable taking values in Z d • I [L] is a variable taking values in Z d • K is a variable taking values in {0, 1, . . .} • K [L] is a variable taking values in {0, 1, . . .} • B is an array of elements of Z d × {0, 1, . . .} • B [L] is an array of elements of Z d × {0, 1, . . .}
• C is a variable taking values in the set of finite subsets of Z d

• C [L] is a variable taking values in the set of finite subsets of Z d

• W is an auxiliary variable taking values in A

• X is a function from Z d to A ∪ {∆}, where ∆ is some extra symbol that does not belong to A

• X [L] is a function from Z d to A ∪ {∆}, where ∆ is some extra symbol that does not belong to A

We will present the backward sketch procedure only for G, the coupled perfect simulation algorithm. The backward sketch procedure for F can be immediately deduced ignoring the steps from 11. to 18. concerning the variables with superscript L.

Algorithm 1 Backward sketch procedure 1. Input: {i}; Output:

N (i) ST OP , N [L],(i) ST OP , B, B [L] 2. N ← 0, N (i) ST OP ← 0, N [L],(i) ST OP ← 0, B ← ∅, B [L] ← ∅, C ← {i}, C [L] ← {i} 3. WHILE C = ∅ 4. N ← N + 1 5. Choose independent random variables S (j,k) N
having exponential distribution of parameter M j λ j (k) for any j ∈ C and for any k ≥ 0 6. Choose (I, K) = arg min{S

(j,k) N , j ∈ C, k ≥ 0} 7. IF K = 0, C ← C \ {I} 8. ELSE C ← C ∪ B I (K) 9. ENDIF 10. B(N ) ← (I, K) 11. IF C [L] = ∅ 12. N [L],(i) ST OP ← N 13. Choose (I [L] , K [L] ) = arg min{S (j,k) N , j ∈ C, 0 ≤ k ≤ L} 14. IF K [L] = 0, C [L] ← C [L] \ {I [L] } 15. ELSE C [L] ← C [L] ∪ B I [L] (K [L] ) 16. ENDIF 17. B [L] (N ) ← (I [L] , K [L] ) 18. ENDIF 19. ENDWHILE 20. N (i) ST OP ← N 21. RETURN N (i) ST OP , N [L],(i) ST OP , B, B [L] If B = B [L]
, then we use the following Forward spin assignment procedure to sample X(i) = X [L] (i). This is the algorithmic translation of the ideas presented in the last paragraph of Section 4. If B = B [L] , then we use the Forward spin assignment procedure twice independently, starting with input

N (i)
ST OP , B in order to sample X(i) and starting with input

N [L],(i) ST OP , B [L] in order to sample X [L] (i).
Algorithm 2 Forward spin assignment procedure 1. Input:

N (i) ST OP , B; Output: {X(i)} 2. N ← N (i) ST OP 3. X(j) ← ∆ for all j ∈ Z d 4. WHILE N ≥ 1 5. (I, K) ← B(N ).
6. IF K = 0 choose W randomly in A according to the probability distribution

P (W = v) = p [0] I (v) = 1 2
7. ELSE choose W randomly in A according to the probability distribution

P (W = v) = p [K] I (v|X) 8. ENDIF 9. X(I) ← W 10. N ← N -1 11. ENDWHILE 12. RETURN {X(i)}
Remark 3 At a first look to steps 5 and 6 of Algorithm 1, the reader might think that the simulation of an infinite number of exponential variables is necessary in order to perform the algorithm. Actually, it is sufficient to simulate a finite number of finite valued random variables, see [START_REF] Knuth | The complexity of nonuniform random number generation. Algorithms and complexity[END_REF] and also Section 10 of [START_REF] Galves | Perfect simulation and finitary coding for multicolor systems with interactions of infinite range[END_REF].

Proof of Theorem 1

This section is devoted to the proof of Theorem 1. For the convenience of the reader, we start by recalling the following theorem of [START_REF] Galves | Perfect simulation and finitary coding for multicolor systems with interactions of infinite range[END_REF].

Theorem 3 Under the requirement (4.34), sup

i∈Z d k≥1 |B i (k)|λ i (k) < 1,
the Gibbs measure µ is the unique invariant probability measure of the process (σ t ) t . The output X(i) obtained using successively Algorithms 1 and 2 given in Section 4 is a perfect sampling of the Gibbs measure µ. Finally, the estimate (4.35) holds :

P (T (i) ST OP > t) ≤ e -γt .
The same result holds true also for X [L] obtained using successively Algorithms 1 and 2 but now with restriction that only ranges k ≤ L are considered. We first check that the assumption (2.6) of Theorem 1 implies condition (4.34) of Theorem 3.

Lemma 1 Under the conditions of Theorem 1, condition (4.34) is satisfied.

Proof For any k ≥ 0, write for short

S >k i = B:i∈B,B ⊂B i (k) |J B |. For each i ∈ Z d we have that ∞ k=1 λ i (k)|B i (k)| ≤ 2dλ i (1) + ∞ k=2 |B i (k)| e -βS >k i -e -βS >k-1 i = 2de -βS >1 i 1 -e -βS >0 i e -β P B:i∈B,B⊂B i (1) |J B | + ∞ k=2 |B i (k)|e -βS >k i 1 -e -β P B:i∈B,B⊂B i (k),B ⊂B i (k-1) |J B | ≤ 2de -βS >1 i 1 -e -βS >0 i e -β P B:i∈B,B⊂B i (1) |J B | + β sup i   ∞ k=2 |B i (k)| B:i∈B,B⊂B i (k),B ⊂B i (k-1) |J B |   .
(5.38)

Under the conditions of Theorem 1, this last expression is finite. Therefore condition (4.34) reads as

2d sup i e -βS >1 i 1 -e -βS >0 i e -β P B:i∈B,B⊂B i (1) |J B | + β sup i   ∞ k=2 |B i (k)| B:i∈B,B⊂B i (k),B ⊂B i (k-1) |J B |   < 1.
Finally, let β c be the solution of

2d sup i e -βS >1 i 1 -e -βS >0 i e -β P B:i∈B,B⊂B i (1) |J B | + β sup i   ∞ k=2 |B i (k)| B:i∈B,B⊂B i (k),B ⊂B i (k-1) |J B |   = 1. (5.39)
This concludes the proof.

• Proof of Theorem 1.
Proof of item 1. Theorem 3 combined with Lemma 1 proves that the output of the perfect sampling algorithm is indeed a sampling of µ. Next we prove (2.7). Define

L (i) n = |C (i) n |, the cardinal of the set C (i)
n after n steps of the algorithm. Let (D

(i) n ) n≥0,i∈Z d be i.i.d. random variables, independent of the process, taking values in {-1, 0, 1, 2, . . .} such that for k ≥ 1 P (D (i) n = |B i (k)| -1) = λ i (k), (5.40)
and for k = 0

P (D (i) n = -1) = λ i (0). ( 5
P (I (i) n = k|C (i) n-1 ) = M k j∈C (i) n-1 M j .
Put

S (i) n = n k=1 D I (i) k k .
Note that by construction, S

n + nγ is a super-martingale. Then a very rough upper bound is

L (i) n ≤ 1 + S (i) n as long as n ≤ V (i) ST OP , (5.42)
where

V (i)
ST OP is defined as

V (i) ST OP = min{k : S (i) k = -1}.

By construction

N (i) ST OP ≤ V (i)
ST OP . Fix a truncation level N. Then by the stopping rule for super-martingales, we have that

E(S (i) V (i) ST OP ∧N ) + γE(V (i) ST OP ∧ N ) ≤ 0. But notice that E(S (i) V (i) ST OP ∧N ) = -1 • P (V (i) ST OP ≤ N ) + E(S (i) N ; V (i) ST OP > N ). On {V (i) ST OP > N }, S (i)
N ≥ 0, hence we have that E(S (i)

V (i) ST OP ∧N ) ≥ -P (V (i) ST OP ≤ N ). We conclude that E(V (i) ST OP ∧ N ) ≤ 1 γ P (V (i) ST OP ≤ N ). Now, letting N → ∞, we get E(V (i) ST OP ) ≤ 1 γ ,
and therefore E(N

(i) ST OP ) ≤ 1 γ .
Proof of item 2. We have to show that the coupled perfect sampling algorithm G defined in Section 4 achieves the bound (2.8). For that sake we introduce the following stopping times. For any site j ∈ Z d and any t ≥ 0, let

T (j,t) 1 = inf{T (j,0) n > t} -t
be the first jump of the Poisson point process (T (j,0) n

) n shifted by time t.

Let us call T (i)

L the first time that a range of order k > L has been chosen. T

L is given by

T (i) L = inf{t > 0 : j∈C (i) t k>L N (j,k) ([t, t + T (j,t) 1
[) ≥ 1}. (5.43) Recall that in order to construct µ and µ [L] , we use the same Poisson point processes for all ranges k ≤ L. Thus we have that

P (X(i) = X [L] (i)) ≤ P (T (i) L ≤ T (i)
ST OP ).

(5.44)

By Lemma 2 given below we conclude the proof of Theorem 1.

•

Lemma 2 P (T (i) L ≤ T (i) ST OP ) ≤ sup i∈Z d 1 -e -β P B:i∈B,B ⊂B i (L) |J B | 1 γ . Proof Put α i (k) = ℓ≤k λ i (ℓ).
Given the structure of the interaction, we have for k ≥ 2, α i (k) = e Then we have

P (T (i) L ≤ T (i) ST OP ) ≤ n≥1 E P N (j,k) (] T (i) n-1 , T (i) n ]) = 1 for j ∈ C (i) n-1 , k > L C (i) n-1 , N (i) ST OP > n -1 = n≥1 E    j∈C (i) n-1 M j (1 -α j (L)) k∈C (i) n-1 M k 1 {N (i) ST OP >n-1}    ≤ n≥1 E    j∈C (i) n-1 M j δ(L) k∈C (i) n-1 M k 1 {N (i) ST OP >n-1}    = δ(L)E(N (i) ST OP ).
In the above calculus we used that given

C (i) n-1 , T (i) n defined in (4.32) is a jump of N (j,k) with probability M j λ j (k) k∈C (i) n-1 M k .
Summing over all possibilities k > L gives the term (M j (1α j (L)))/(

Final discussion

It is worth to note that Theorem 1 immediately provides an upper bound for the d-distance of µ and µ [L] . The d-distance is defined as follows.

Definition 5 Given two probability measures µ and ν on S, a coupling between µ and ν is a probability measure on S × S having as first and second marginals µ and ν, respectively.

The set of all couplings between µ and ν will be denoted M(µ, ν).

In the next definition, the elements of the product space S × S will be denoted by ((σ

1 (i), σ 2 (i)), i ∈ Z d ).
Definition 6 The distance d between two probability measures ν 1 and ν 2 on S is defined as

d(ν 1 , ν 2 ) = inf Q∈M(ν 1 ,ν 2 ) sup i∈Z d Q(σ 1 (i) = σ 2 (i)) .
This definition naturally extends Ornstein's d-distance to the space of non-homogeneous random fields. As a corollary of Theorem 1 we obtain Theorem 4 Under the assumptions of Theorem 1 we have

d(µ, µ [L] ) ≤ C sup i∈Z d 1 -e -β P B:i∈B,B ⊂B i (L) |J B | , (6.45)
where C is the same constant as in (2.8).

Upper bounds on the d-distance can as well be obtained by using a contraction argument. This approach was introduced in Dobrushin (1968[START_REF] Dobrushin | Prescribing a system of random variables by conditional distributions[END_REF] to prove uniqueness of the infinite volume Gibbs measure. Applied to our context, it implies the existence of a good coupling through compactness arguments.

For a nice and self contained presentation of this method, we refer the reader to Presutti (2009), see also Chapter 8 of Georgii. Applying Dobrushin's method, we obtain the following upper bound on Ornstein's d-distance.

Theorem 5 Assume that β sup i B:i∈B

|J B | = r < 1. ( 6

.46)

Then there exist unique infinite volume Gibbs measures µ and µ [L] and they satisfy

d(µ, µ [L] ) ≤ β 1 -r   sup i∈Z d B:i∈B,B ⊂B i (L) |J B |   . (6.47)
Proof The proof of Theorem 5 can be achieved following the ideas in Dobrushin (1968[START_REF] Dobrushin | Prescribing a system of random variables by conditional distributions[END_REF], exposed in Chapter 3 of [START_REF] Presutti | Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics[END_REF]. •

Observe that condition (6.46) is weaker than the one requested in Theorem 1. In particular, this condition is too weak to guarantee that our perfect simulation procedure of µ and µ [L] stops after a finite number of steps. As a counterpart of the less restrictive assumption (6.46), the upper bound (6.47) is not as good as the upper bound (6.45).

However, from our point of view, the important fact is that the contraction method of Dobrushin is not constructive and does not provide an explicit sampling procedure for both measures. To provide an explicit coupled perfect simulation procedure for the measures is precisely the goal of the present paper.

Further features of high temperature region can be deduced by our constructive approach. We discuss as an example the decay of correlations between localized observables as the regions of localization are moved away from each other.

This decay of correlations can be described in terms of an associated random walk that we introduce now. Suppose that µ is translational invariant. In this case, the sequence (λ i (k)) k does not depend on i, and we shall write λ(k) = λ i (k), for any k ≥ 0. Let ξ 1 , ξ 2 , . . . Finally, consider the following left-continuous random walk S n = ξ 1 + . . . + ξ n , for n ≥ 0. Note that S n is an upper bound on the total number of sites whose spins affect the spin at site i under equilibrium, after n steps of the algorithm. Under condition (4.34), E(ξ 1 ) = -γ < 0, hence S n drifts to -∞. This implies that M = sup n S n , the maximum of the random walk, is finite almost surely.

The following theorem is an immediate consequence of our construction and of results of [START_REF] Korshunov | On distribution tail of the maximum of a random walk[END_REF] concerning the maximum of a random walk. In order to state the theorem, we have to recall the notion of locally power functions. where γ is given in (4.37).

2. Suppose that ρ > 0, ϕ(ρ) < 1, that P (ξ 1 + ξ 2 ≥ n) ∼ cP (ξ 1 ≥ n) as n → ∞ for some constant c and suppose that e ρn P (ξ 1 ≥ n) is locally power. Then as n → ∞ Then the same comparison argument that leads to (5.42) yields

P (M ≥ |B 0 (n)|) ∼ ρE(e ρM
P (C (i) ∩ C (j) = ∅) ≤ P (M + M ′ ≥ R) ≤ 2P (M ≥ R/2).
This concludes the proof of (6.48).

Item 1 is a consequence of Theorem 1 of [START_REF] Korshunov | On distribution tail of the maximum of a random walk[END_REF], see [START_REF] Korshunov | On distribution tail of the maximum of a random walk[END_REF].

Remark 4

 4 In[START_REF] Galves | Perfect simulation and finitary coding for multicolor systems with interactions of infinite range[END_REF] the above result was proved for the minimal decomposition of the spin flip rates. The same proof works for the decomposition of the spin flip rates c i (σ) considered in Section 4. Note that in contrary to the decomposition considered in[START_REF] Galves | Perfect simulation and finitary coding for multicolor systems with interactions of infinite range[END_REF], the decomposition in Section 4 is explicit in terms of the interaction.

  ,B ⊂B i (k) |J B | and thus 1α i (L) ≤ sup i∈Z d 1e -β P B:i∈B,B ⊂B i (L) |J B | . Call this last quantity δ(L) = sup i∈Z d 1e -β P B:i∈B,B ⊂B i (L) |J B | .

  be i.i.d. random variables with common distributionP (ξ 1 = -1) = λ(0), P (ξ 1 = |B 0 (k)| -1) = λ(k), k ≥ 1.Put ϕ(λ) = E(e λξ 1 ) and let ρ = sup{λ > 0 : ϕ(λ) ≤ 1}.

Definition 7 ATheorem 6

 76 function f : R + → R + is called locally power if for every fixed t ≥ 0,f (x + t) ∼ f (x) as x → ∞, i.e. lim x→∞ f (x+t) f (x) = 1. Take two cylinder functions f and g having support ∆ f ⋐ Z d and ∆ g ⋐ Z d . Assume the distance d(∆ f , ∆ g ) = R. Under the assumption (4.34) we have |µ(f g)µ(f )µ(g)| ≤ 2c f c g (|∆ f | + |∆ g |) P (M ≥ R/2),(6.48)wherec f = sup i∈∆ sup σ∈S |f (σ)f (σ i )| (6.49)and c g is similarly defined. Concerning the decay of P (M ≥ R/2), the following holds.1. Suppose that ρ = 0 and that P (ξ1 + ξ 2 ≥ n) ∼ 2P (ξ 1 ≥ n) as n → ∞. Then as n → ∞, P (M ≥ |B 0 (n)|) ∼ 1 γ ∞ k=n+1 λ(k)|B 0 (k)| -|B 0 (n)| k>n+1 λ(k)

3 .Remark 5

 35 Suppose that ρ > 0, ϕ(ρ) = 1 and ϕ ′ (ρ) < +∞. ThenP (M ≥ |B 0 (n)|) ∼ ce -ρnas n → ∞, where c is some positive constant. Using the same type of upper bounds as in the proof of Lemma 1, note that the expression∞ k=n+1 λ(k)|B 0 (k)|that defines the decay of correlations can be upper bounded byβ ,B⊂B 0 (k),B ⊂B 0 (k-1) |J B |   |B 0 (k)|,which yields the tail of the converging series of condition (2.6).Proof We start by proving (6.48). It is sufficient to prove it for functions f and g having support ∆ f = {i} and ∆ g = {j}, R = ij > 0. For any site k ∈ Z d , let C (k) number of sites whose spins affect the spin of site k under equilibrium (recall (4.33)). Then it is evident that|µ(f g)µ(f )µ(g)| ≤ c f c g (|∆ f | + |∆ g |)P (C (i) ∩ C (j) = ∅).(6.51)

( 1 -

 1 First notice that F (t) = P (ξ 1 > t) is constant on intervals [|B 0 (n)| -1, |B 0 (n + 1)| -1[. Then evidently F (|B 0 (n)| -1) = 1α(n), where α n = λ 0 + . . . + λ n . Therefore, ∞ |B 0 (n)| F (t)dt = ∞ k=n α(k)) [|B 0 (k + 1)| -|B 0 (k)|] -(1α(n)) = ∞ k=n+1 λ(k)|B 0 (k)| -(1α(n))[|B 0 (n)| + 1] = ∞ k=n+1 λ(k)|B 0 (k)| -[|B 0 (n)| + 1] k>n λ(k) .

  Poisson clock rings at the n-th jump of the process C(i) . Recall that I

	independent given the sequence (C	(i) n ) n such that	(i) n are conditionally
				.41)
	Note that under assumption (2.6), the condition (4.34) holds (see Lemma 1). Hence,
	sup i∈Z d E(D	(i)

1 ) ≤ -γ < 0. For any i ∈ Z d , let us call I (i)

n the index of the site whose

Acknowledgments. The authors thank L. Bertini, P. Collet, P. Ferrari, R. Fernández, D. Gabrielli and E. Presutti for helpful discussions. We thank the Institut Henri Poincaré -Centre Emile Borel (workshop Mécanique Statistique, Probabilités et Systèmes de Particules 2008) for hospitality and support. The authors have been partially supported by Prin07: 20078XYHYS (E.O.), by CNPq 305447/2008-4 (A.G.), ANR-08-BLAN-0220-01 (E.L.) and Galileo 2008-09 (E.L. and E.O.). This paper is part of PRONEX/FAPESP's project Stochastic behavior, critical phenomena This integral ∞ |B 0 (n)| F (t)dt is the quantity that determines the asymptotic behavior of P (M ≥ |B 0 (n)|) as it is shown in Theorem 1 of Korshunov (1997). Items 2 and 3 are a consequence of Theorem 2 and Theorem 3 (ii) of Korshunov (1997).

•


Camille Tardif
email: camille.tardif@uni.lu

Lyapunov spectrum of a relativistic stochastic flow in the Poincaré group

Keywords: Relativistic processes. Lévy processes in Lie groups. Poincaré group. Lyapunov spectrum. Hyperbolic dynamics. AMS Subject Classification: 37H15, 60G51, 83A05

published or not. The documents may come     Lyapunov spectrum of a relativistic stochastic flow in the Poincaré group.

Introduction

In 1966 Dudley [START_REF] Dudley | Lorentz-invariant Markov processes in relativistic phase space[END_REF] defined a class of relativistic processes with Lorentzian-covariant dynamics in the framework of special relativity. Such a process ξ t with values in Minkowski space-time R 1,d , is differentiable and has velocity smaller than the speed of light. So it can be parametrized by its proper time, which amounts to impose to the velocity ξt to be an element of the unit pseudo sphere H d of R 1,d . The restriction to the tangent space of H d of Minkowski ambient pseudometric turns H d into a Riemannian manifold of constant negative curvature. The invariance of the process ( ξt , ξ t ) by the natural action of the set of Lorentz transforms on H d × R 1,d imposes to the laws of ξt to be invariant by the action of the isometries of H d . Among this class of relativistic processes, there is essentially only one which is continuous. It corresponds to the case where ξt is a Riemannian Brownian motion in the hyperbolic space and in this case ( ξt , ξ t ) is called Dudley diffusion. Forty years after this seminal work, Franchi and Le Jan [START_REF] Franchi | Relativistic diffusions and Schwarzschild geometry[END_REF] extended Dudley diffusion to the framework of any Lorentz manifold. They defined relativistic processes with Lorentzian-covariant dynamics on generic Lorentzian manifolds by rolling without slipping a Dudley diffusion on the unit tangent space. They studied the asymptotic behavior of such diffusion in the Schwarzschild space-time. Bailleul [START_REF] Bailleul | Poisson boundary of a relativistic diffusion[END_REF] succeeded to compute the Poisson boundary of Dudley diffusion in Minkowski space-time and showed that it coincides with the causal boundary of R 1,d . The asymptotic behavior of relativistic diffusions was investigated in other non flat Lorentzian manifolds ( [START_REF] Angst | Etude de diffusions à valeurs dans des variétés lorentziennes[END_REF], [START_REF] Franchi | Relativistic diffusion in Gödel's universe[END_REF]) with the aim of describing how the asymptotic behavior of the diffusion reflects the asymptotic geometry of the manifold.

In this work we ask a new question concerning these processes dealing with the asymptotic behaviour of some stochastic flow associated to it. As Brownian motion on a Riemannian manifold , the relativistic diffusion [START_REF] Franchi | Relativistic diffusions and Schwarzschild geometry[END_REF] is obtained by projecting a diffusion process with values in the orthonormal frame bundle, solution of a stochastic differential equation. This SDE generates a stochastic flow which, in our Lorentzian framework, consists in a stochastic perturbation of the geodesic flow. Existence and computation, for example, of the Lyapunov spectrum and stable manifolds of these flows may be investigated in the same way as it was done by Carverhill and Elworthy [START_REF] Carverhill | Lyapunov exponents for a stochastic analogue of the geodesic flow[END_REF] for the canonical stochastic flow in the Riemannian framework. The main difficulty to study the flow of relativistic processes comes from the fact that the orthonormal frame bundle of a Lorentz manifold is never compact. Nevertheless in this article we provide a study of the asymptotic dynamics of the stochastic flow generated by Dudley processes in the Minkowski space-time (without restricting ourselves to the diffusion case). Precisely, in this framework, the orthonormal frame bundle is identified with the Poincaré group G := P SO(1, d) ⋉ R 1,d and denoting φ t the left invariant stochastic flow associated to one of Dudley's processes in G we obtain the description of the Lyapunov spectrum and the stable manifolds of ϕ t . Precisely we obtain the following two results.

Theorem 1 (Lyapunov spectrum). There exist a constant α > 0 and two asymptotic random Lie sub-algebras V - ∞ ⊂ V 0 ∞ of Lie( G) such that for some norm • on Lie( G) and X ∈ Lie( G) we have for almost every trajectory

1 t log dϕ t (Id)( X) ϕt(Id) -→ t→+∞      α if X ∈ Lie( G) \ V 0 ∞ 0 if X ∈ V 0 ∞ \ V - ∞ -α if X ∈ V - ∞ \ {0}
Theorem 2 (Stable manifolds). Denote by V - ∞ := exp(V - ∞ ) and d the distance associated to a left invariant and Ad(SO(d))-invariant Riemanian metric in G. Then for any two distinct points g′ and g in G we have

• If g′ ∈ gV - ∞ then 1 t log d (ϕ t (g), ϕ t (g ′ )) -→ t→+∞ -α. • If g′ / ∈ gV - ∞ then lim inf t→∞ d (ϕ t (g), ϕ t (g ′ )) > 0.
We begin by constructing, in section 2, Dudley processes as projections of left Lévy processes on the Poincaré group G, identified with the orthonormal frame bundle of the Minkowski spacetime. These Lévy processes are solutions of stochastic integral equations and induce a left invariant stochastic flow ϕ t in G. In section 3 we find the asymptotic behavior of Dudley processes and exhibit the asymptotic random variables (θ ∞ , λ ∞ ) ∈ S d-1 × R * + . Finally in section 4 we prove Theorems 1 and 2 and explicit the projection of the stable manifold in H d × R 1,d by showing that it corresponds to a skew product of a horosphere by a line.

Note that stochastic flows generated by Lévy processes on semi-simple Lie groups were intensively studied by Liao ([12], [START_REF] Liao | Stable manifolds for stochastic flows induced by Lévy processes on Lie groups[END_REF], [START_REF] Liao | Lévy processes in Lie groups[END_REF]). But his results cannot be used directly here since our Lévy processes lie in the Poincaré group which is not semi-simple. Moreover in our work we suppose only that the Levy measure is integrable at infinity whereas Liao [START_REF] Liao | Lévy processes in Lie groups[END_REF] request the entire integrability of it.

Our work is also strongly inspirited by the work of Bailleul and Raugi [START_REF] Bailleul | Where does randomness lead in spacetime?[END_REF] where the authors used Raugi's methods [START_REF] Raugi | Fonctions harmoniques sur les groupes localement compacts à base dénombrable[END_REF] to find the Poisson boundary of Dudley diffusion.

Dudley processes and their lift in the Poincaré group

We present in this section the geometrical framework of special relativity and define a natural class of relativistic Markov processes with Lorentzian-covariant dynamics introduced by Dudley in [START_REF] Dudley | Lorentz-invariant Markov processes in relativistic phase space[END_REF]. They are obtained by projecting left Lévy processes with values in the Poincaré group and are described by two parameters: a diffusion coefficient σ ∈ R and a jump intensity Lévy measure ν on R * + .

Minkowski space-time and Poincaré group

The Minkowski space-time R 1,d is R × R d endowed with the Lorentz quadratic form q defined by

∀ξ = (ξ 0 , ξ 1 , . . . , ξ d ) ∈ R × R d , q(ξ) = ξ 0 2 -ξ 1 2 -• • • -ξ d 2 .
We denote by ξ := (ξ 1 , . . . , ξ d ) t the space component of ξ.

Set Q = Diag(1, -1, . . . , -1)
the matrix of q in the canonical basis (e 0 , e 1 , . . . , e d ). Time orientation is given by the constant vector field e 0 and some ξ ∈ R 1,d is said to be future oriented when q(ξ, e 0 ) > 0. A path γ s in R 1,d is said to be time-like when it is differentiable almost everywhere and q( γs ) > 0 and q( γs , e 0 ) > 0. The Poincaré group is the group of affine q-isometries which preserve orientation and time-orientation. It is the semi-direct product connected group

G := P SO(1, d) ⋉ R 1,d
where G := P SO(1, d) denotes the group of linear q-isometries which preserve orientation and time-orientation. An element g = (g, ξ) ∈ G is made up of its linear part g ∈ G and its translation part ξ. We identify G with the sub-group of G which fixes 0. By this way, we identify R 1,d with the homogeneous space G/G. The identity element of G and G is denoted by Id (thus for us Id = (Id, 0)). At g = (g, ξ) ∈ G we associate the affine frame ((g(e 0 ), g(e 1 ), . . . , g(e d )); ξ) of R 1,d and G is identified with the bundle of q-orthonormal, oriented and time-oriented, frames over R 1,d . We denote by π :

G -→ R 1,d g = (g, ξ) -→ ξ
the projection associated to this trivial fibration and G = π-1 {0}. The canonical basis being fixed we identify G with the matrix group

G = g ∈ SL(R d+1 ), gQg t = Q, q(g(e 0 ), e 0 ) > 0 ,
and its Lie algebra is

Lie(G) = X ∈ M d+1 (R), XQ -QX t = 0 = 0 b t b C , b ∈ R d , C ∈ M d (R) s.t C = -C t .
We have Lie(

G) = Lie(G) × R 1,d and for X = (X, x), Y = (Y, y) ∈ Lie( G) [ X, Y ] = ([X, Y ], Xy -Y x).
We identify Lie(G) with Ker(d Id π) and its elements are vertical for the fibration π. We set

V i := e 0 e t i + e i e t 0 i = 1, . . . , d V ij := [V i , V j ] = e i e t
je j e t i j > i.

Moreover we set

H 0 := (0, e 0 ) ∈ Lie( G)
which is horizontal for the fibration π. Notation For X ∈ Lie( G) we denote by X l the left invariant vector field in G associated. Denote by K the subgroup of G made of the rotations of R d . We have

K := 1 0 0 k , k ∈ SO(d) ,
and K is also the stabilizer of e 0 under the action of G on R 1,d . The homogeneous space G/K can be identified with the orbit of e 0 under the action of G which is the unit pseudo sphere

H d := {ξ ∈ R 1,d , q(ξ) = 1, ξ 0 > 0}
and is a Riemannian manifold of constant negative curvature when its tangent space is endowed with the restriction of q on it. For r ∈ R + and θ ∈ S d-1 ⊂ R d , define

S(r, θ) := exp r d i=1 θ i V i = cosh(r) sinh(r)θ t sinh(r)θ Id + (cosh(r) -1)θθ t .
Each g ∈ G can be decomposed in polar form g = S(r(g), θ(g))R where R ∈ K.

Dudley processes

In this paragraph we define the relativistic processes introduced by Dudley in [START_REF] Dudley | Lorentz-invariant Markov processes in relativistic phase space[END_REF]. These processes enjoy two natural properties:

• they are G-invariant i.e their dynamics are invariant by a change of q-orthonormal frame

• their trajectories in R 1,d are time-like: they are almost everywhere differentiable,and the tangents vectors are time-like and time oriented.

First remark that no Markov processes with values in R 1,d is G-invariant. Indeed, the law at some time t > 0 of such process starting at 0 would be a G-invariant probability measure in R 1,d which is necessary trivial by the following lemma.

Lemma 1. The only G-invariant probability measure in R 1,d is the Dirac measure at 0.

Proof. Let µ be a G-invariant probability measure in R 1,d . First suppose that the support of µ is not contained in the q-orthogonal hyperplane of some light-like line {u(e 0 + θ), u ∈ R} ( θ := d i=1 θi e i ∈ S d-1 ). So there exist a compact set C in the complement of this hyperplane such that µ(C) > 0. For g = S(r, θ), r > 0 and ξ = (ξ 0 , ξ) ∈ R 1,d , denoting • the Euclidean norm in R 1,d we have

g(ξ) 2 = 2q(g(ξ), e 0 ) 2 -q(g(ξ)) = 2q(ξ, g -1 (e 0 )) 2 -q(ξ) = 2 cosh(r)ξ 0 -sinh(r) θ • ξ 2 -q(ξ) = 2 cosh(r)(ξ 0 -θ • ξ) + e -r θ • ξ 2 -q(ξ) = 2 cosh(r)q(ξ, e 0 + θ) + e -r θ • ξ 2 -q(ξ).
Since C is a compact set in the complement of the q-orthogonal hyperplan to {u(e 0 + θ), u ∈ R} then inf ξ∈C |q(ξ, e 0 + θ)| > 0 and thus r can be chosen such that inf ξ∈C g(ξ) is arbitrary large. Thus g can be chosen such that C and g(C) are disjoint. Furthermore q(g(ξ), e 0 + θ) = q(ξ, g -1 (e 0 + θ)) = e r q(ξ, e 0 + θ) and thus g(C) belongs to the complement of the hyperplan q-orthogonal to e 0 + θ. By iteration we can find a sequence g k = S(r k , θ), k ∈ N such that the compact sets g k (C) are pairwise disjoint. Thus we obtain a contradiction writing 1

≥ µ(∪ k g k (C)) = k µ(g k (C)) = k µ(C) = +∞.
Now if the support of µ is contained in some hyperplan tangent to the light-cone and is not restricted to 0, we can find a compact set C in this hyperplan with 0 / ∈ C and µ(C) > 0 and we can choose a rotation R ∈ K such that R(C) is not in the hyperplan. Thus µ(C) = µ(R(C)) = 0 and it gives a contradiction. So we proved that µ is necessary the Dirac measure in {0}.

Thus R 1,d = G/G cannot be the space of states of some non-trivial G-invariant Markov process. But G-homogeneous spaces of the form G/ K where K is a compact subgroup of G have some K-invariant probability measure and may be endowed with some G-invariant Markov processes (see [START_REF] Liao | Lévy processes in Lie groups[END_REF] or [START_REF] Hunt | Semi-groups of measures on Lie groups[END_REF] ). The smallest spaces of states we can consider correspond to the maximal compact sub-group of G. Thus it is natural to consider the space of states G/K ≃ H d × R 1,d . The group K is seen as the subgroup of G which stabilize 0 and e 0 under the action of G on R 1,,d .

We denote by π :

G -→ H d × R 1,d ≃ G/K the canonical projection ∀g = (g, ξ) ∈ G π(g) = (g(e 0 ), ξ).
The following Proposition exhibit all the relativistic processes in

H d × R 1,d .
It is essentially an application of a result of Liao ( Theorem 2.1 and 2.2 p 42 in [START_REF] Liao | Lévy processes in Lie groups[END_REF] ).

Proposition 1. The Markov processes on H d × R 1,d , starting at (ζ 0 , ξ 0 ), which are G-invariant and whose trajectories are time-like are of the form (ζ s , ξ s ) where ζ s is a G-invariant Markov process on H d and ξ t = ξ 0 + a t 0 ζ s ds; a being some positive constant. For such a process there exist σ > 0 and a measure ν on R + satisfying

+∞ 0 min(1, r 2 )ν(dr) < +∞, such that (ζ t , ξ t ) = π(g t ) in law where gt is a left Levy process on G starting at g0 s.t (ζ 0 , ξ 0 ) = π(g 0 ) of which generator L is defined by ∀f ∈ C 2 ( G) Lf (g) = aH l 0 f (g) + σ 2 2 d i=1 (V l i ) 2 f (g) + +∞ 0 S d f (gS(r, θ)) -f (g) -r1 r∈[0,1] d i=1 θ i V l i f (g) ν(dr)dθ.
Definition 1 (Dudley processes). When a = 1 then ξt = ζ t and ξ t is parametrized by its proper time, i.e q( ξt ) = 1. When moreover σ or ν is non trival we call ( ξt , ξ t ) a Dudley process and we consider exclusively these processes in the sequel. When ν = 0 ( ξt , ξ t ) is continuous and is called Dudley diffusion.

Remark 1. The process ξ t is differentiable and ξt is càdlàg.

Proof of Proposition 1. Let (ζ t , ξ t ) a Markov (Feller) process on H d × R 1,d starting at (ζ 0 , ξ 0 ),
which is G-invariant and whose trajectories are time-like. By choosing g0 such that (ζ 0 , ξ 0 ) = π(g 0 ) and considering the Markov process g-1 0 (ζ t , ξ t ) it remains to prove the proposition in the case where (ζ 0 , ξ 0 ) = (e 0 , 0).

Set

H i := (0, e i ) ∈ Lie( G), i = 1, . . . , d.
The family {H 0 , H i , V i , V ij } i<j∈{1,...,d} form an orthonormal basis of Lie( G) for an Ad(K)-invariant inner product on Lie( G) and Ker d Id π = {V ij } i<j∈{1,...,d} . We set

X 0 := H 0 X i := H i X d+i := V i i = 1, . . . , d. An h = (h, ξ) ∈ G can be decomposed in h = exp d i=0 x i ( h)X i exp 2d i=d+1 x i ( h)X i R where R ∈ K, x 0 ( h) = ξ 0 and for i = 1, . . . , d x i ( h) = ξ i and x d+i ( h) = r(h)θ i (h)
. By Theorem 2.1 and 2.2 p 42 of [START_REF] Liao | Lévy processes in Lie groups[END_REF], (ζ t , ξ t ) coincide in law with π(g t ) where gt is a left Levy process in G, starting at Id, which is K-right invariant and generated by

Lf (g) = 1 2 2d i,j=0 a ij X l i X l j f (g)+ 2d i=0 b i X l i f (g)+ G f (g h) -f (g) - 2d i=0 1 r(h)≤1 ξ i (h)≤1 x i ( h)X i f (g) Π(d h).
The matrice

A := (a ij ) is a positive symmetric , (b i ) i ∈ R 2d+1 and Π is a Levy measure invariant by K-conjugation in G. The right K-invariance of gt ensures that for all k ∈ K ≃ SO(d) , diag(1, k, k)Adiag(1, k, k) -1 = A. Thus, A is necessarily of the form   σ 0 0 0 σId σ ′ Id 0 σ ′ Id σId   ,
where σ, σ ≥ 0 and σσ ≥ (σ ′ ) 2 . Moreover, using again K-invariance it comes b i = 0 for i = 1, . . . , 2d and we set a := b 0 . The trajectories of gt projected in R 1,d need to be differentiable so the jump measure Π is supported on G and σ and σ are necessarily null. Since the trajectories are time-oriented we have a > 0. The push forward of Π by π is supported on G/K ≃ H d and is K-invariant. Thus Π can be chosen of the form

∀f ∈ C 0 (G) Πf = +∞ 0 S d-1 f (S(r, θ))ν(dr)dθ
where ν is a Levy measure on R * + ( i.e satisfying min(1, r 2 )ν(dr) < +∞).

Denote by g t the G-component of gt . Thus g t (e 0 ) = ξt and ξ t = t 0 g s (e 0 )ds. By definition g t is a G-valued left Levy process, K-right invariant, generated by L defined by

∀f ∈ C 2 (G) Lf (g) = σ 2 2 d i=1 (V l i ) 2 f (g)+ +∞ 0 S d f (gS(r, θ)) -f (g) -r1 r∈[0,1] d i=1 θ i V l i f (g) ν(dr)dθ.
Denote by Π the Levy measure supported on G defined by

Πf = +∞ 0 S d-1
f (S(r, θ))ν(dr)dθ.

Define U 0 := {g ∈ G, r(g) ≤ 1} which is a K invariant neighborhood of Id in G. For f ∈ C 2 (G) we have the following Itô formula (see [2]) for g t f (g t ) = f (Id) + σ d i=1 t 0 V l i f (g s -)dB i s + σ 2 2 t 0 d i=1 (V l i ) 2 f (g s -)ds + t 0 U0 (f (g s -h) -f (g s -)) Ñ (ds, dh) + t 0 U0 f (g s -h) -f (g s -) -r(h) d i=1 θ i (h)V l i f (g s -) dsΠ(dh) (1)
+ t 0 (U0) c (f (g s -h) -f (g s -)) N (ds, dh),
where B t is a Brownian motion of R d , N is a Poisson random measure on R×G of intensity measure dt ⊗ Π and Ñ (ds, dh) := N (ds, dh) -dsΠ(dh) is the compensated random measure associated.

Asymptotic random variables

In this section we determine the asymptotic behavior of π(g t ) = (g t (e 0 ), ξ t ) under an integrability condition on the jump intensity measure ν (Ass.1). Writing g t = n t a t k t in some Iwasawa decomposition of G we first prove (Prop.2), applying the Itô formula (1) and the law of large number, that the abelian term a t = exp(α t V 1 ) is positively contracting, αt t converges almost surely to a positive constant α depending explicitly on σ and ν. Next we prove (Prop.4) that the nilpotent term n t converges almost surely to an asymptotic random variable n ∞ and this convergence is exponentially fast with rate α. Then we investigate (Prop.5) the asymptotic behavior of ξ t in R 1,d . Geometrically, seen in the projective space, the H d -valued process g t (e 0 ) converges to a limit angle θ ∞ ∈ ∂H d ≃ S d-1 of which n ∞ is a stereographic projection. Moreover, the process ξ t is asymptotic to some affine hyperplane q-orthogonal to θ ∞ of which position is fixed by another asymptotic random variable λ ∞ ∈ R * + . Figure 1 sum up the asymptotic results.

H d

Asymptotic angle

Asymptotic affine hyperplane

λ ∞ > 0 θ ∞ ∈ S d-1 θ ∞ ∈ S d-1

Iwasawa decomposition in G

Although a polar decomposition of G was used to introduce g t (defining the K invariant measure Π), Iwasawa decomposition seems to be more adapted to describe its asymptotic dynamics.

Introduce briefly this decomposition. The maximal abelian subalgebra contained in Vect{V 1 , . . . , V d }, which is the orthogonal subspace of Lie(G) of K for the Killing form, is of dimension one. Let choose A := Vect{V 1 } one of them. The linear endomorphism ad(V 1 ) of Lie(G) is diagonalisable with eigenvalues -1, 0 and 1.

Set N = {X ∈ Lie(G), ad(V 1 )X = -X} N = {X ∈ Lie(G), ad(V 1 )X = X}
the eigenspace corresponding respectively to the eigenvalue -1 and 1. Explicitly

N = Vect{V i -V 1i , i = 2, . . . , d} and N = Vect{V i + V 1i , i = 2, . . . , d}.
The eigenspace corresponding to 0 is A ⊕ M where M is the sub algebra of elements of K which commute with the elements of A. Explicitly

M = Vect{V ij , i, j = 2 . . . d} =      0 0 0 0 0 0 0 0 C   , C ∈ so(d -1)    .
The subspace N is a nilpotent Lie algebra (even abelian since [N , N ] = 0). The corresponding Iwasawa decomposition of Lie(G) is Moreover, the mapping from N × A × K to G which maps (n, a, k) to nak is an analytic diffeomorphism. For g ∈ G we denote by g = (g) N (g) A (g) K its decomposition in Iwasawa coordinates.

Lie(G) = N ⊕ A ⊕ K.
To simplify notations set n t := (g t ) N , a t := (g t ) A and k t = (g t ) K , thus g t = n t a t k t . Note that we have other Iwasawa decompositions like Lie(G) = N ⊕ A ⊕ K (with G = N AK). Iwasawa and polar coordinates.

For g ∈ G written in polar form g = exp r d i=1 θ i (V i -V 1i ) R where R ∈ K, we have q(e 0 , g(e 0 )) = cosh(r). Now denoting by b ∈ R d-2 and u ∈ R such that (g) N = exp d i=2 b i (V i -V 1i
) and (g) A = exp (uV 1 ) we can compute explicitly q(e 0 , g(e 0 )) in terms of b and u and we obtain

cosh(r) = 1 + b 2 2 cosh(u) + b 2 2 sinh(u). (2)
Moreover u can be expressed in term of θ and r via

e u = cosh(r) + θ 1 sinh(r) (3)

Asymptotic behavior of the G-component

The aim of this section is to show that n t converges almost surely to an asymptotic N -valued random variable n ∞ and that the convergence is exponentially fast. This result, stated in Proposition 4, appears to be a consequence of the contracting property of a t . For this we need the following integrability condition on ν. The group G is semi simple and the tools used in this section are very closed from those of Liao [START_REF] Liao | Lévy processes in Lie groups[END_REF]. Nevertheless we present a self-contained proof in our specific framework and our results are established under a weaker assumption than the ones of [START_REF] Liao | Lévy processes in Lie groups[END_REF] ( see remark 3 ). Namely we suppose that the following integrability condition is satisfied.

Assumption 1. +∞ 1 rν(dr) < +∞.
The following proposition computes explicitly the linear drift of a t which appears to be positive. The proof is essentially a consequence of the law of large number.

Proposition 2. Let denote by α t the R-valued process such that a t = exp(α t V 1 ). Then the following convergence holds almost surely

α t t -→ t→∞ α > 0.
The positive constant α is

α := d -1 2 σ 2 + +∞ 0 r cosh(r) -sinh(r) sinh(r) ν(dr).
Proof. First define log : A → A ≃ R exp(uV 1 ) → uV 1 and apply Itô formula (1) to the smooth map f : g → log(g) A . Remark that for g, h ∈ G (gh) A = (g) A ((g) K h) A and

f (g s-h) -f (g s-) = log (k s-g) A .
Moreover

V l i f (g s-) = d dt log(g s-e tVi ) A t=0 = d dt log e tAd(ks-)Vi A t=0 = {Ad(k s-)V i } A (V l i ) 2 f (g s-) = d dt {Ad g s-e tVi K V i } A t=0 = d dt {Ad e tAd(ks-)Vi K Ad(k s-)V i } A t=0 = {[{Ad(k s-)V i } K , Ad(k s-)V i ]} A . For k ∈ K we have Ad(k)V i = d j=1 k ij V j where k = (k ij ) ∈ K ≃ SO(d). Then we compute d i=1 {[{Ad(k s-)V i } K , Ad(k s-)V i ]} A = d i=1 d j=1 d l=1 (k s-) ij (k s-) il {[{V j } K , V l ]} A , since {V j } K = V 1j (and 0 for j = 1 ) and [V 1j , V l ] = V 1 δ jl we get d i=1 {[{Ad(k s-)V i } K , Ad(k s-)V i ]} A = d j=2 d i=1 ((k s-) ij ) 2 V 1 = (d -1)V 1 .
Thus Itô formula (1) can be written

log(a t ) = M t + d -1 2 σ 2 t + t 0 U c 0 log(k s-h) A N (ds, dh) + t 0 U0 log(k s-h) A -r(h) d i=1 θ i (h){Ad(k s-)V i } A dsΠ(dh),
where

M t := σ d i=1 t 0 {Ad(k s-)V i } A dB i s + t 0 U0 (k s-h) A Ñ (ds, dh) is a martingale. Its bracket is σ 2 t 0 d i=1 {Ad(k s-)V i } A 2 ds + t 0 U0 log(k s-h) A 2 dsΠ(dh) = t σ 2 + U0 log(h) A 2 Π(dh)
and thus we obtain that almost surely

M t t -→ t→+∞ 0.
Moreover, making the change of variable h ′ = k s-hk -1 s-we obtain using the K-invariant by conjugation of Π

t 0 U0 log(k s-h) A -r(h) d i=1 θ i (h){Ad(k s-)V i } A dsΠ(dh) = t 0 U0 log(h ′ ) A -r(h ′ ) d i=1 θ i (k -1 s-h ′ ){Ad(k s-)V i } A Π(dh ′ )ds = t 0 U0 log(h ′ ) A -r(h ′ ) d i=1 d j=1 (k s-) ij θ j (h ′ )(k s-) i1 V 1 Π(dh ′ )ds = t 0 U0 log(h ′ ) A -r(h ′ )θ 1 (h ′ )V 1 Π(dh ′ )ds = t S d-1 1 0 log(S(r, θ)) A -rθ 1 V 1 ν(dr)dθ
By (3) we have log(S(r, θ)) A = log(cosh(r) + θ 1 sinh(r))V 1 and the previous term equals

t S d-1 1 0 log(cosh(r) + θ 1 sinh(r)) -rθ 1 ν(dr)dθV 1 = t 1 0 1 -1 log (cosh(r) + u sinh(r)) -ru du 2 ν(dr)V 1 = t 1 0 r cosh(r) -sinh(r) sinh(r) ν(dr)
It remains to consider the asymptotic behavior of the stochastic integral t 0 U c 0 log(k s-h) A N (ds, dh). We know that there exist T i jump times of a Poisson process of intensity measure Π(U c 0 ) and random variable h n i.i.d of law Π| U c 0 /Π(U c 0 ) and independent of (T i ) i such that

t 0 U c 0 log(k s-h) A N (ds, dh) = n,Tn≤t log(k T - n h n ) A .
Moreover, by invariance of Π under conjugation by K we check easily that the random variables

h ′ n := Ad(k T - n )h n are i.i.d of common law Π| U c 0 /Π(U c 0 ) and we have t 0 U c 0 log(k s-h) A N (ds, dh) = Nt n=1 log(h ′ n ) A ,
where N t is a Poisson process of intensity measure Π(U c 0 ) and independent of (h ′ n ) n . Moreover since +∞ 1 rν(dr) < +∞ then log(h ′ n ) A is integrable and the law of large number ensures that

1 t t 0 U c 0 log(k s-h) A N (ds, dh) -→ t→+∞ Π(U c 0 )E[log(h ′ n ) A ].
The proof is ended by checking that

E[log(h ′ n ) A ] = 1 Π(U c 0 ) +∞ 1 r cosh(r) -sinh(r) sinh(r) ν(dr)V 1 . Remark 2. When +∞ 1 rν(dr) = +∞ we obtain E[|log(h ′ n ) A |] = +∞. Nevertheless E [-min (log(h ′ n ) A , 0)] = +∞ 1 1 -1
min log(cosh(r + θ 1 sinh(r)), 0 dθ 1 2 ν(dr)

= +∞ 1 1 e -r -log(v) dv 1 sinh(r) ν(dr) = +∞ 1 1 2 sinh(r)
(1e -r (r + 1))ν(dr) < +∞. Now, applying a generalized law of large numbers we deduce that almost surely

1 t t 0 U c 0 log(k s -h) A N (ds, dh) -→ t→+∞ +∞,
and so

α t t -→ t→+∞ +∞.
The following proposition establishes that g t is bounded in expectation on a finite time interval. This result is used to prove the convergence of n t in the next Proposition.

Proposition 3. Fix T > 0. Then E sup t∈[0,T ] r (g t ) < +∞.
Proof. We cannot directly apply Itô formula to g → r(g) since it is not regular at Id. But we can find a smooth function r such that r ≥ r on U 0 and r = r on U c 0 . For such a function we have r(g t ) = r(Id) + mt + mt +

I t + J t + t 0 U c 0 (r(g s -h) -r(g s -)) N (ds, dh). ( 4
)
Where mt := σ t 0 V l i r(g s -)dB i s and mt := t 0 U0 (r(g s -h)r(g s -)) Ñ (ds, dh) are martingales,

I t := σ 2 t 0 d i=1 V l i 2 r(g s -)ds and J t := t 0 r∈[0,1] θ∈S d-1 r (g s -S(r, θ)) -r(g s -) -r d i=1 θ i V l i r(g s -) dθν(dr)ds
are processes with finite variation.

To prove the proposition we will bound the supremum on [0, T ] of each of these five terms by means of X l r ∞ and (X l ) 2 r ∞ for some X ∈ Vect{V i , i = 1, . . . , d}. Thus we need the following lemma.

Lemma 2. For X ∈ P = Vect{V i , i = 1, . . . , d} we have X l r ∞ < +∞ and (X l ) 2 r ∞ < +∞.

Proof of the lemma. U 0 being a compact set it suffices to show that the supremum is finite on U c 0 . Since r = r on U c 0 it remains to prove that sup g∈U c 0 |X l r(g)| < +∞ and sup g∈U c 0 |(X l ) 2 r(g)| < +∞. The polar decomposition of g can be written

g = k exp(r(g)V 1 )k for some k, k ∈ K. Setting x ∈ R d such that X = i x i V i we have r(g exp(sX)) = r exp(r(g)V 1 ) exp s d i=1 (kx) i V i .
Let θ ∈ S d-1 be such that (kx) i = x θ i . Then we compute explicitly, for g ∈ U c 0 :

r exp(r(g)V 1 ) exp s d i=1 (kx) i V i = (cosh) -1 cosh(r(g)) cosh(s x ) + θ 1 sinh(r(g)) sinh(s x ) = r(g) + s x θ 1 + s 2 2 x 2 cosh(r(g)) sinh(r(g)) (1 -(θ 1 ) 2 ) + O(s 3 ).
Then it comes that

X l r(g) = d ds r(g exp(sX)) s=0 = x θ 1 ,
and thus sup g∈U c 0 X l r(g) ≤ x . Moreover

(X l ) 2 r(g) = d 2 ds 2 r(g exp(sX)) s=0 = x 2 cosh(r(g)) sinh(r(g)) (1 -(θ 1 ) 2 )
and so sup g∈U c 0 (X l ) 2 r(g) ≤ 2 x 2 .

Return to the proof of Proposition 3. By Doob's norm inequalities (see [START_REF] Kallenberg | Foundations of modern probability[END_REF]) we obtain

E sup t∈[0,T ] | mt | 2 ≤ 4E ( mT ) 2 ≤ 4σ 2 E T 0 d i=1 V l i r(g s -) 2 ds ≤ 4σ 2 T d max i V l i r 2 ∞ < +∞.
and

E sup t∈[0,T ] | mt | 2 ≤ 4E ( mT ) 2 ≤ 4E T 0 r∈[0,1] θ∈S d-1 |r (g s -S(r, θ)) -r(g s -)| 2 ν(dr)dθds ≤ 8dT 1 0 r 2 ν(dr) max i V l i r 2 ∞ < +∞.
We have also

E sup t∈[0,T ] |I t | ≤ σ 2 T d max i (V l i ) 2 r ∞ < +∞.
Moreover applying a Taylor inequality to u ∈ [0, 1] → r(g s -S(ur, θ)) it comes r (g s -S(r, θ))r(g s -)r

d i=1 θ i V l i r(g s -) ≤ 1 2 r 2 sup θ∈S d-1 i θ i V l i 2 r ∞ < +∞
and thus

E sup t∈[0,T ] |J t | ≤ T 2 1 0 r 2 ν(dr) sup θ∈S d-1 i θ i V l i 2 r ∞ < +∞.
Finally, to bound the supremum of the last term of ( 4) we need the assumption 1. Indeed, since |r(g

s -S(r, θ)) -r(g s -)| ≤ r sup θ∈S d-1 i θ i V l i r ∞ we obtain E sup t∈[0,T ] t 0 U c 0 r(g s -h) -r(g s -)dsΠ(dh) ≤ T d +∞ 1 rν(dr) max i V l i r ∞ < +∞.
We can now state the main result of this section, the convergence of n t to some asymptotic random variable n ∞ and the speed of convergence. -valued process such that

n t = exp d i=2 b i-1 t (V i -V 1i ) .
Then b t converges almost surely to b ∞ exponentially fast with rate α, i.e

lim sup t→+∞ 1 t log b t -b ∞ ≤ -α. (5)
As a consequence n t converges almost-surely to n ∞ := exp

d i=2 b i-1 ∞ (V i -V 1i
) and defining h t := e -tαV1 n -1 ∞ g t we obtain

lim t→+∞ 1 t r (h t ) = 0 a.s. (6)
Proof. Denoting by [t] the integer part of t we decompose b t =

[t] j=1 (b j -b j-1 ) + b t -b [t] .
To prove that b t converge and (5) holds it is sufficient to verify that lim sup

j→∞ 1 j log sup s∈[0,1] b j -b j+s ≤ -α a.s. (7)
For j ∈ N and s ∈ [0, 1] we have g -1 j g j+s = k -1 j a -1 j n -1 j n j+s a j+s k j+s and thus

n -1 j n j+s = a j g -1 j g j+s k j N a -1 j . (8)
Denote by bj,s = ( bi

j,s ) i=2,...,d ∈ R d-1 such that g -1 j g j+s k j N = exp d i=2 bi j,s (V i -V 1i ) . Then, (8) implies that b j+s -b j = e -αj bj,s . (9)
Since g t is a Levy process, sup s∈[0,1] r(g -1 j g j+s ) j∈N are i.i.d random variables. Moreover, by Proposition 3 their common expectation E[sup s∈[0,1] r(g s )] is finite. Thus, applying the law of large number it comes

1 j sup s∈[0,1] r(g -1 j g j+s ) -→ j→∞ 0 a.s. (10)
Since r((g -1 j g j+s ) N ) ≤ 2r(g -1 j g j+s ) and, by (2), bj,s 2 = 2 cosh r g -1 j g j+s k j N -1 we deduce from (10) that for ε > 0 there exists j 0 such that for j > j 0 bj,s ≤ e εj .

(

Since, by Proposition 2, α j = jα + o(j) thus (7) follows from ( 9) and [START_REF] Liao | Stable manifolds for stochastic flows induced by Lévy processes on Lie groups[END_REF]. To finish the proof of the proposition we need to check [START_REF] Dudley | Lorentz-invariant Markov processes in relativistic phase space[END_REF]. We have

r(h t ) = r(e -αtV1 n -1 ∞ n t a t ) ≤ r(e -αtV1 a t ) + r(a -1 t n -1 ∞ n t a t ),
and r(e -αtV1 a t ) = |α t -αt| = o(t) and we obtain from (2) ( since

a -1 t n -1 ∞ n t a t ∈ N ), r(a -1 t n -1 ∞ n t a t ) = cosh -1 1 + e 2αt b t -b ∞ 2 2 .
So by [START_REF] Carverhill | Lyapunov exponents for a stochastic analogue of the geodesic flow[END_REF] we have also r(a -1 t n -1 t n ∞ a t ) = o(t) and ( 6) holds.

Remark 3.

• Without integrability condition, we can nevertheless show that g t satisfy the irreducibility and contraction conditions of [START_REF] Liao | Lévy processes in Lie groups[END_REF] and deduce that α t converges almost surely to +∞ and n t converges to n ∞ . Nevertheless, by remark 2, we obtain in the case where +∞ 1 rν(dr) = +∞, that almost surely αt t converges to +∞. • In [START_REF] Liao | Lévy processes in Lie groups[END_REF] the author uses a stronger hypothesis to prove the rate of convergence of a Lévy process in a semi-simple group. It corresponds in our case to assume that +∞ 0 rν(dr) < +∞.

Asymptotic behavior of the R 1,d -component

The linear endomorphism ξ → exp(V 1 )ξ is diagonalisable with eigenvalues -1, 0, +1. Denote by U -, U 0 and U + the respective eigenspaces. Explicitly U -= Vect{e 0e 1 }, U 0 = Vect{e 2 , . . . , e d } and U + = Vect{e 0 + e 1 }. For ξ ∈ R 1,d we denote by (ξ) -, (ξ) 0 and (ξ) + its projection on each eigenspace. Explicitly we obtain (ξ) -= -1 2 q(ξ, e 0 + e 1 )(e 0e 1 ), (ξ) + = 1 2 q(ξ, e 0e 1 )(e 0 + e 1 )

(ξ) 0 = d i=2 q(ξ, e i )e i .
Recall that by definition, ξ t = t 0 g s (e 0 )ds. The following proposition gives the asymptotic behavior of ξ t . Proposition 5. There exists an asymptotic random variable λ ∞ > 0 such that

(n -1 ∞ ξ t ) --→ t→+∞ λ ∞ (e 0 -e 1 ),
and moreover lim sup

t→+∞ 1 t log (n -1 ∞ ξ t ) --λ ∞ (e 0 -e 1 ) ≤ -α. (12)
We also have

lim sup t→+∞ 1 t log (n -1 ∞ ξ t ) 0 ≤ 0, and, lim sup t→+∞ 1 t log (n -1 ∞ ξ t ) + ≤ α. (13)
Proof. We have

- 1 2 q(n -1 ∞ ξ t , e 0 + e 1 ) = t 0 - 1 2 q(n -1
∞ n s a s (e 0 ), e 0 + e 1 )ds [START_REF] Raugi | Fonctions harmoniques sur les groupes localement compacts à base dénombrable[END_REF] and the integrand can be written

- 1 2 q(n -1 ∞ n s a s (e 0 ), e 0 + e 1 ) = - 1 2
q(e -αsV1 n -1 ∞ n s a s (e 0 ), e -αsV1 (e 0 + e 1 ))

= -1 2 e -αs q(h s (e 0 ), (e 0 + e 1 )),

where h s = e -αs n -1 ∞ g s as defined in Proposition 4. Denote by (r s , θs ) ∈ R + × S d-1 the polar decomposition of h s (e 0 ) ∈ H d . So r = r(h s ) and -1 2 e -αs q(h s (e 0 ), (e 0 + e 1 )) = 1 2 e -αs cosh(r s ) -θ1 s sinh(r s ) ∈ αs+rs) , e -(αs-rs) ].

1 2 [e -(
Proposition 4 ensures that rs = o(s) a.s, so fixing ε > 0 arbitrary small we can find s 0 > 0 such that for all s > s 0 the integrand of ( 14) is positive and bounded by e -(α-ε)s . This ensures the convergence of (n -1 ∞ ξ t ) -to λ ∞ (e 0e 1 ) with λ ∞ > 0. Moreover for t > s 0

- 1 2 q(n -1 ∞ ξ t , e 0 + e 1 ) -λ ∞ ≤ +∞ t e -(α-ε)s ds = 1 α -ε e -(α-ε)t ,
which prove [START_REF] Liao | Dynamical properties of Lévy processes in Lie groups[END_REF]. Now

(n -1 ∞ ξ t ) 0 = d i=2 t 0 q(n -1
∞ n s a s (e 0 ), e i )dse i , and for i = 2, . . . , d we have q(n -1 ∞ n s a s (e 0 ), e i ) = q(e -αsV1 n -1 ∞ n s a s (e 0 ), e i ) = θi s sinh(r s ) so |q(n -1

∞ n s a s (e 0 ), e i )| ≤ e rs and this ensures that lim sup t→+∞

1 t log (n -1 ∞ ξ t )0 ≤ 0. Moreover, (n -1 ∞ ξ t )+ = t 0 1 2 q n -1
∞ n s a s (e 0 ), e 0e 1 ds (e 0e 1 ) and 1 2 q(n -1 ∞ n s a s (e 0 ), e 0e 1 ) = 1 2 e αs q(h t (e 0 ), e 0e 1 ) = 1 2 e αs cosh(r s ) -θ1 s sinh(r s ) ∈ 1 2 [e αs-rs , e αs+rs ].

So (n -1 ∞ ξ t ) + ≤ (n -1 ∞ ξ s0 ) + + t s0
e (α+ε)s ds, and thus lim sup t→+∞

1 t log (n -1 ∞ ξ t ) + ≤ α.

Geometric description of the convergence

Denote by p : R 1,d \ {0} → P d R the projection onto the projective space of dimension d. The hyperboloïd H d is mapped onto the interior of a projective ball and its boundary (

∂H d ≃ S d-1
) is the image of the q-isotropy cone

∂H d := p ({ξ, q(ξ) = 0} \ {0}) .
From the relation q( ξt , n t (e 0 + e 1 )) = q(g t (e 0 ), n t (e 0 + e 1 )) = q(e 0 , a -1 t (e 0 + e 1 )) = e -αt -→ t→∞ 0,

we deduce that all limit points of p( ξt ) are q-orthogonal to θ ∞ := p(n ∞ (e 0 + e 1 )). Since the only point of p(H) which is q-orthogonal to θ ∞ is θ ∞ itself it comes that p( ξt ) converges to θ ∞ in P d R. Now, identifying P d R with its affine chart {ξ, ξ 0 = 1} we can consider that θ ∞ ∈ S d . From [START_REF] Applebaum | Lévy flows on manifolds and Lévy processes on Lie groups[END_REF] we deduce that r t → +∞ and since p( ξt ) = p(e 0 + θ t sinh(rt) cosh(rt) ) it comes that θ t converges to θ ∞ in S d . The two asymptotic random variables θ ∞ and n ∞ are linked by p(e 0 + θ ∞ ) = p(n ∞ (e 0 + e 1 ))

or more explicitly, b ∞ ∈ R d-1 (defined by n ∞ = exp d i=2 b i-1 ∞ (V 1 -V 1i ) ) is the stereographic projection of θ ∞ θ ∞ = 1 1 + b ∞ 2 1 -b ∞ 2 2b ∞ .
Concerning the asymptotic behavior of ξ t , Proposition 5 ensures that q(ξ t , n ∞ (e 0 + e 1 )) converges to λ ∞ . Thus geometrically ξ t is asymptotic to an affine hyperplan which is q-orthogonal to n ∞ (e 0 + e 1 ) (or e 0 + θ ∞ ) and passing by λ ∞ (e 0e 1 ) .

4 Lyapunov spectrum and stable manifolds

Lyapunov spectrum

The Levy process gt , with values in G and starting at some g, can be obtained by solving the following left invariant stochastic integro-differential equation in

G ∀f ∈ C 2 ( G), f (g t ) = f (g) + σ d i=1 t 0 V l i f (g s -) • dB i s + t 0 H l 0 (g s -)ds + t 0 U0 (f (g s -h) -f (g s -)) Ñ (ds, dh) + t 0 U0 f (g s -h) -f (g s -) -r(h) d i=1 θ i (h)V l i f (g s -) dsΠ(dh) (15)
+ t 0 (U0) c (f (g s -h) -f (g s -)) N (ds, dh).
This stochastic differential equation induces a stochastic flow ϕ t in G which maps g on the solution at time t and starting at g of (15). By left invariance ϕ t is also defined by

ϕ t : G -→ G g -→ gg t ,
where gt is starting at Id. Denote by • any norm on Lie( G) and by • g the left invariant (Finsler) metric associated in G on T g G. For v ∈ T g G we aim to investigate the asymptotic exponential rate of growth or decay of dϕ t (g)(v) ϕt(g) . Denote by L g the left translation by g in G. By left invariance of the flow, dϕ t

(g)(v) ϕt(g) = dϕ t (Id)( X) ϕt(Id) where X := (dL g ) -1 (v) ∈ T Id G = Lie( G). For g = (g, ξ) ∈ G and X, Y ∈ Lie( G) it comes Ad(g)( X) = (Ad(g)(X), gx -Ad(g)(X)ξ) (16) ad( Y )( X) = (ad(Y )(X), Y x -Xy) . (17)
The endomorphism X → ad(V 1 , 0)( X) is diagonalisable on Lie( G). Its eigenvalues are -1, 0, 1 and we denote by U -, U 0 and U + the eigenspaces associated. We can check that

X ∈ U + ⇐⇒ X ∈ N and x ∈ U + X ∈ U 0 ⇐⇒ X ∈ A ⊕ M and x ∈ U 0 X ∈ U -⇐⇒ X ∈ N and x ∈ U -.
Proof. All norms are equivalent and it suffices to check the inequalities for some particuliar norms. Let choose the following SO(d)-invariant euclidean norm on Lie( G) (X, x) := Tr(X t X) + x t x.

We obtain easily Ad(g, 0) = e r(g) .

Taking now (X, x) := Tr(X t X) + √ x t x we get

Ad(Id, ξ)(X, x) = Tr(X t X) + x -Xξ ≤ (X, x) + Xξ ≤ (X, x) + Tr(X t X) max i |ξ i | ≤ (X, x) 1 + max i |ξ i | .
Thus Ad(Id, ξ) ≤ 1 + α ξ for a constant α > 0 independant of ξ.

Stable manifolds

First, remark that V - ∞ and V 0 ∞ are Lie sub-algebras of Lie( G). Denote by

V - ∞ := exp(V - ∞ ), and V 0 ∞ := exp(V 0 ∞ ) the closed subgroup of G associated. Fix now a euclidean norm • on Lie( G) which is Ad(K)-invariant. Such a norm is of the form 0 b t b C , x := κ 2 b t b + β 2 Tr (C t C) γ 2 x t x + δ 2 (x 0 ) 2 ,
for some positive constants κ, β, γ and δ. We denote by d the distance in G associated to the left invariant Riemanian metric induced by • . To simplify notations, we denote by d(g, h) the distance between (g, 0) and (h, 0) for g, h ∈ G.

The following result shows that the stable manifold associated to ϕ t is ϕ 0 V - ∞ .

Theorem 2. Let g and g′ two distinct points in G.

• If g′ ∈ gV - ∞ then 1 t log d (ϕ t (g), ϕ t (g ′ )) -→ t→+∞ -α. • If g′ / ∈ gV - ∞ then lim inf t→∞ d (ϕ t (g), ϕ t (g ′ )) > 0.
The properties of d we need in the proof of Theorem 2 are sum up in the following proposition

Proposition 6. i) Left invariance ∀g, h ∈ G, d(g, gh ) = d(Id, h).
Thus d(Id, g-1 ) = d(Id, g) and triangularity inequality writes:

∀g, h ∈ G, d(Id, gh ) ≤ d(Id, g) + d(Id, h) ii) K-right invariance ∀g ∈ G and k ∈ K, d ((k, 0), g(k, 0)) = d(Id, g) iii) For X ∈ Lie( G) d Id, exp( X) ≤ X .
iv) There exists a neighborood O of 0 in Lie( G) and a constant C > 0 such that

∀ X ∈ O, C X ≤ d Id, exp( X) v) ∀(g, ξ) ∈ G d (Id, g) ≤ d (Id, (g, ξ)) vi) For g = S(r, θ)R and g ′ = S(r ′ , θ ′ )R ′ we have κ κ 2 + 2β 2 d (S(r, θ), S(r ′ , θ ′ )) ≤ d(g, g ′ )
vii) For all r ≥ 0 and θ ∈ S d-1 d (Id, S(r, θ)) = κr Proof of Proposition 6. i) and ii).The left and K-right invariance follows from the definition of the metric as being a left invariant Riemannian metric on G defined from an Ad(K)-invariant inner product on Lie( G). Inequality iii) is obtained remarking that the length of the path t ∈ [0, 1] → exp t X is equal to X . iv). Denote by exp : Lie( G) → G be the exponential map at Id induced by the metric • in G: for X ∈ Lie( G), exp( X) = γ X (1) where t ∈ [0, 1] → γ X (t) is the geodesic starting from Id in the direction X. The differential at 0 of exp is known to be identity and there exists a sufficient small neighborhood O ′ of 0 ∈ Lie( G) such that:

∀ X ∈ O ′ , X = d I, exp( X) . ( * )
Furthermore, the map exp -1 • exp can be defined in a neighborhood of 0 and its differential at 0 is the identity: exp -1 • exp( X) = X + o( X ). So we can find O neighborhood of 0 and C > 0 such that for all X ∈ O, C X ≤ exp v). Each path s ∈ [0, 1] → (g s , ξ s ) joining Id to (g, ξ) is of length 1 0 (g -1 s ġs , g -1 s ξ s ) ds which is greater than 1 0 (g -1 s ġs , 0) ds corresponding to the path s ∈ [0, 1] → (g s , 0) joining Id to (g, 0). vi). Consider a path s ∈ [0, 1] → S(r s , θ s )R s joining g to g ′ . We compute, using dot notation for d ds R -1 s S(r s , θ s ) -1 d ds (S(r s , θ s )R s ) = 0 ṙs θ t s + sinh(r s ) θs t ṙs θ s + sinh(r s ) θs (cosh(r s ) -1) θs θ t sθ s θt s + R -1 s Ṙs .

Its length l := 1 0 R -1 t S(r t , θ t ) -1 d dt S(r t , θ t )R t dt is larger than Moreover, the path s → S(r s , θ s ) which join S(r, θ) to S(r ′ , θ ′ ) is of length

≥ cosh(u) + (1θ t (mθ)) sinh(r) 2 we used (mθ) 1 = θ 1 (25) ≥ cosh(u). Return to the proof of Theorem 2. Second case: X 0 = 0 but x 0 = 0 . So X = (X + , x + + x 0 ) and explicitely exp( X) = (exp(X + ), x 0 + x + + X + x 0 2 ) = (Id, ξ)(exp(X + ), 0), where we have set ξ := x 0 + x + + X + x 0 2 . Thus g-1 t exp( Y )g t = h-1 t (e -tαV1 , 0) exp( X)(e tαV1 , 0) ht = (Id, h -1 t e -tαV1 ξ)(exp(Ad(h -1 t e -tαV1 )X), 0), and d(Id,g -1 t exp( Y )g t ) ≥ d Id, (Id, h -1 t e -tαV1 ξ)d(Id, (exp(Ad(h -1 t e -tαV1 )X), 0)).

As done previously in the first case, d(Id, (exp(Ad(h -1 t e -tαV1 )X), 0)) converges exponentially fast to 0 and it remains to prove that lim inf

Figure 1 :

 1 Figure 1: Asymptotic behavior of a Dudley diffusion

For X ∈

 ∈ Lie(G) we denote by {X} N (resp. {X} A and {X} K ) its projection in N (resp. A and K ) thus X = {X} N + {X} A + {X} K . Denoting by A := exp(A), N := exp(N ) the subgroup corresponding we obtain the corresponding Iwasawa decompositions of G G = N AK.

Proposition 4 .

 4 Denote by b t = (b i t ) i=2,...,d the R d-1

- 1 • 1 C

 11 exp( X) ≤ X . Taking O small enough so that exp -1 • exp(O) ⊂ O ′ , we can apply ( * ) to exp -1 • exp( X), thus yielding exp -1 • exp( X) = d I, exp( X) for every X ∈ O.

1 0κ 1 0κ

 11 ṙs θ s + sinh(r s ) θs ds = ( ṙs ) 2 + sinh(r s ) 2 θs 2 ds.

1 0κ 2 ≤ κ 2 + 2β 2 κ 1 0κ

 121 ( ṙs )2 + sinh(r s ) 2 θs 2 + β 2 2(cosh(r) -1) 2 θs 2 ds (21) ( ṙs )2 + sinh(r s ) 2 θs 2 ds. (22) Thus d(S(r, θ), S(r ′ , θ ′ )) ≤ κ 2 + 2β 2 κ l and taking the infimum over all the path joining g to g ′ we obtain vi). m = Id. Then we get d(Id, g -1 mg) = d(Id, S(r, -θ)mS(r, θ)) = d(S(r, θ), mS(r, θ)) = d(S(r, θ), S(r, mθ)m) ≥ κ κ 2 + 2β 2 d(S(r, θ), S(r, mθ)) by vi) of Proposition 6 ≥ κ κ 2 + 2β 2 d(S(r, θ), S(r, θ)m -1 )d(S(r, θ)m -1 , S(r, mθ)) = κ κ 2 + 2β 2 d(Id, m -1 )d(S(r, θ), mS(r, θ))= κ κ 2 + 2β 2 d(Id, m)d(Id, g -1 mg) Thus d(Id, g -1 mg) ≥ κ κ+ √ κ 2 +2β 2 d(Id, m) > 0.Suppose now a = Id. Let u = 0 such that a = exp(uV 1 ), then an explicit computation gives: cosh r(g-1 amg) = cosh(u) cosh(r) 2 -((mθ) 1 ) 2 sinh(r) 2sinh(r) 2 d i=2 θ i (mθ) i(23)= cosh(u) + cosh(u)(1 -((mθ) 1 ) 2 ) -d i=2 θ i (mθ) i sinh(r) 2

( 26 )

 26 Then by vi) and vii) of Proposition 6 it comesd(Id, g -1 amg) ≥ κ κ 2 + 2β 2 κu > 0.

t→∞dFigure 2 :

 2 Figure 2: π(V - ∞ ) is a skew-product of a horosphere with a line

Set g∞ := (n ∞ , λ ∞ (e 0e 1 )) ∈ G and V - ∞ := Ad(g ∞ ) U + , V 0 ∞ := Ad(g ∞ ) U 0 + U + . We denote by ht := e -tαV1 , 0 g-1 ∞ gt , where we recall that gt := ϕ t (Id) is starting at Id. Theorem 1. Let X ∈ Lie( G). For almost every trajectory

Proof. By left invariance of • , dϕ t (Id)( X) ϕt(Id) = Ad( gt -1 )( X) . We set, for g ∈ G Ad(g) := sup

Let X ∈ Lie( G). Writting g-1

Suppose for the moment that lim sup

and lim sup

Then we deduce from (18) that 1 t log Ad(g -1 t )( X) and 1 t log Ad e -tαV1 , 0 g-1 ∞ ( X) have the same limit when t goes to ∞. The linear isomorphism Ad e -tαV1 , 0 is diagonalisable with eigenvalues e -αt , 1 and e αt associated respectively to the eigenspaces U + , U 0 and U -. Decomposing Ad(g ∞ ) -1 ( X) in the direct sum U -⊕ U 0 ⊕ U + and using a Euclidean norm • on Lie( G) for which this decomposition is orthogonal, we deduce easily the theorem (note that the convergence is independant of the chosen norm).

Thus it remains to prove (19) and (20). We have ht = e -tαV1 , 0 g-1

Let ε > 0. By Proposition 4 we can find t 0 > 0 such that ∀t > t 0 r(h t ) ≤ εt and by Proposition 5 we have

Now using the following Lemma 3 we deduce easily (19) and (20).

Lemma 3. There exist positive constants α, β, γ such that for g ∈ G and ξ

)( Y ) converge to 0 exponentially fast with rate α and so for large t it evolves in O. Thus, using iii) and iv) of Proposition 6 we obtain the first point of the theorem as a direct consequence of Theorem 1.

Set

and write

Suppose first that X -= 0. Thus Y ∈ Lie( G)\V 0 ∞ and by Theorem 1 Ad(g -1 t ) Y converges to +∞ exponentially fast. Now suppose by contradiction that lim inf

Then we can find a s t such that d Id, exp Ad(g -1 st )( Y ) converges to 0 and for large t

Writting n = exp(Z), Z ∈ N and d(Id, h -1 t e -tαV1 ne tαV1 h t ) is dominated by e -tα+r(ht) Z (by Lemma 3 and iii) of Proposition 6) and converges exponentially fast to zero (recall that by Proposition 4 r(h t ) = o(t) a.s. ). Thus it remains to prove that lim inf d(Id, h -1 t e -tαV1 ame tαV1 h t ) > 0 to finish the proof in the first case. This is ensured by the following Lemma. Proof of lemma 4. Consider the polar decomposition g = S(r, θ)R. Suppose first that a = Id and Since X + x 0 2 ∈ U + we obtain directly that q(ξ) = q(x 0 ) which is negative since x 0 is supposed to be non zero. But

Projection on

the stable manifolds

We explicit here the projection of

∞ where (X, x) ∈ N × U + . We deduce, since in this case exp(X, x) = (exp(X), x), that an element of π(V - ∞ ) is of the form

where X lies in N and u ∈ R.

Since exp(X)(e 0 + e 1 ) = e 0 + e 1 for X ∈ N we obtain q(n ∞ exp(X)n -1 ∞ (e 0 ), n ∞ (e 0 + e 1 )) = q(n -1 ∞ e 0 , e 0 + e 1 ) = q(e 0 , n ∞ (e 0 + e 1 ))

and thus when X describes N then n ∞ exp(X)n -1 ∞ (e 0 ) draws the intersection between H d and the affine hyperplan passing by e 0 and q-orthogonal to n ∞ (e 0 + e 1 ). This submanifold of R 1,d is a paraboloïd of codimension 2 and is mapped by p (the projection onto the projective space) on a sphere tangent at ∂H d in θ ∞ and passing by p(e 0 ). It is called the horosphere tangent at θ ∞ and passing by e 0 and is denoted by H ∞ .

Moreover, since q(n ∞ exp(X)n -1 ∞ (e 0e 1 ), n ∞ (e 0 + e 1 )) = q(e 0e 1 , e 0 + e 1 ) = 0, we get that when X describes N then n ∞ exp(X)n -1 ∞ (e 0e 1 ) describes the intersection between the light cone {ξ, q(ξ) = 0} and the hyperplan passing by e 0e 1 and q-orthogonal to n ∞ (e 0 + e 1 ). Thus, when X describes N then Idn ∞ exp(X)n -1 ∞ (e 0e 1 ) draws a paraboloïd P ∞ in the hyperplan q-orthogonal to n ∞ (e 0 + e 1 ). For each ξ in the horosphere H ∞ corresponds a unique X ξ ∈ N such that ξ = n ∞ exp(X ξ )n -1 ∞ (e 0 ) and the one-to-one function ψ : ξ → Idn ∞ exp(X ξ )n -1 ∞ (e 0e 1 ) maps H ∞ on P ∞ . Then by (28), we obtain the following one-to-one map

and π(V - ∞ ) is a skew product of the line n ∞ (e 0 + e 1 ) with the horosphere H ∞ .