Transportation inequalities for stochastic differential equations of pure jumps

www.imstat.org/aihp 2010, Vol. 46, No. 2, 465–479

DOI:10.1214/09-AIHP320

© Association des Publications de l’Institut Henri Poincaré, 2010

Transportation inequalities for stochastic differential equations of pure jumps

Liming Wu

Laboratoire de Mathématiques Appliquées, CNRS-UMR 6620, Université Blaise Pascal, 63177 Aubière, France and Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, 100190 Beijing, China. E-mail:[email protected]

Received 15 September 2008; revised 3 April 2009; accepted 20 April 2009

Abstract. For stochastic differential equations of pure jumps, though the Poincaré inequality does not hold in general, we show thatW₁H transportation inequalities hold for its invariant probability measure and for its process-level law on right continuous paths space in theL¹-metric or in uniform metrics, under the dissipative condition. Several applications to concentration inequali- ties are given.

Résumé. Pour une équation différentielle stochastique de pur saut, bien que l’inégalité de Poincaré ne soit pas valide en général, nous pouvons quand même établir, sous la condition de dissipativité, des inégalités de transportW₁H pour sa mesure invariante et pour sa loi (au niveau de processus) sur l’espace des trajectoires càdlàg, muni de la métriqueL¹ou d’une métrique uniforme.

Quelques applications aux inégalités de concentration sont présentées.

MSC:60E15; 60H10; 60H07

Keywords:Transportation inequalities; Stochastic differential equations; Malliavin calculus

1. Introduction

1.1. Object

LetN (dt,du)(ω)=

kδ_(tk,uk)be a random Poisson point process onR⁺×U with intensity measure dt m(du), and N˜ =N−dt m(du)the compensated Poisson point process, where(U,U, m)is someσ-finite measure space andδ_x is the Dirac measure atx. The object of this paper is the following stochastic differential equation (SDE in short)

X_t=X₀+ _t

0

b(X_s)ds+ _t

0

U

σ (X_s−, u)N (ds,˜ du), (1.1)

whereX0 is some (random) initial point independent ofN (dt,du)and (C) the vector field b:R^d→R^d andR^d x→σ (x,·)∈L²(U, m;R^d)(the space of allm-square integrableR^d-valued measurable functions onU) are locally Lipschitzian.

Assume the following dissipative condition:

x−y, b(x)−b(y) +1

2

U

σ (x, u)−σ (y, u)²dm(u)≤ −K|x−y|², x, y∈R^d, (1.2) whereK∈Ris some constant,x, yis the Euclidean inner product and|x| :=√

x, x. IfX₀=x, the SDE (1.1) admits a unique solutionXt(x), which is right continuous and has left-limitXt−int, andEsup_s≤t|Xs|²<+∞for

eacht >0 [2]. We denote byP_t(x,dy)the distribution ofX_t(x)and(P_t)is the transition kernel semigroup of the Markov process(X_t). For anyf∈C_b²(R^d), by Itô’s formula, the generator of(X_t)is given by

Lf (x)=

b(x),∇f (x) +

U

f

x+σ (x, u)

−f (x)−

∇f (x), σ (x, u) m(du)

(∇being the gradient) and the correspondingcarré-du-champsoperator is given by Γ (f, f )(x):=1

2 L

f²

−2fLf

=1 2

U

f

x+σ (x, u)

−f (x)2

m(du).

A particular case:U=R^d∗:=R^d\ {0}, σ (x, u)=B(x)uwhereB(x)∈M(d×d)(the space ofd×d matrices). In such case the SDE (1.1) becomes

dXt=b(Xt)dt+B(Xt−)dLt, (1.3)

whereLt :=t 0

R^d∗uN (dt,˜ du) is a Lévy process of pure jumps with Lévy’s measurem(du). See Jacod [9] and references therein on this SDE driven by Lévy processes.

1.2. Several known results in the diffusion case WhenB(X_t)dLtis replaced by√

2 dWt in (1.3), where(W_t)is a standard Brownian motion valued inR^dandbisC¹, the dissipative condition (1.2) is equivalent to

∇^sb:=

1

2(∂x_jbi+∂x_ibj)

i,j=1,...,d

≤ −KI

(I being the identity matrix) in the order of definite positiveness of symmetric matrices, which is exactly the Bakry–

Emery’sΓ₂-condition [1]. AssumeK >0 from now on. Bakry–Emery’s criterion [1] says that the unique invariant probability measureμfor the semigroupP_twith generatorf+ b(x),∇f (x)satisfies the log-Sobolev inequality:

μ(flogf )−μ(f )logμ(f )≤ 1 2Kμ

|∇f|² f

, 0< f ∈C_b² R^d

(1.4) (μ(f ):=

fdμ), which is equivalent to the exponential decay in entropy:

H (νP_t|μ)≤e^−2KtH (ν|μ) ∀t >0, ν∈M₁ R^d

, (1.5)

whereM₁(·)denotes the space of probability measures on·,H (ν|μ)is the relative entropy ofνwith respect to (w.r.t.

in short)μ, defined by

H (ν|μ)= logdν

dμdν, ifνμ,

+∞, otherwise.

(1.6)

By Otto–Villani [12], the log-Sobolev inequality ofμimplies the following Talagrand’s transportation inequality W₂(ν, μ)²≤ 2

KH (ν|μ) ∀ν∈M₁ R^d

, (1.7)

which in turn implies the Poincaré inequalityKVarμ(f )≤μ(|∇f|²)or equivalently

Varμ(Ptf )≤e^−2KtVarμ(f ). (1.8)

HereW_p(ν, μ):=W_p,|·|(ν, μ)and theL^p-Wasserstein distanceW_p,d(ν₁, ν₂)between two probability measuresν₁, ν₂ on a metric space(E, d), for 1≤p≤ +∞, is defined by

W_p,d(ν₁, ν₂):=inf

X,YX−Yp, (1.9)

where the infinimum is taken for all probability spaces(Ω,F,P)and all couples ofE-valued random variablesX, Y defined thereon, of lawν₁andν₂, respectively.

For the SDE (1.3) withL_treplaced byW_t (but with non-constant diffusion coefficientB(x)), Djellout, Guillin and Wu [5], Theorem 5.6, proved that under condition

x−y, b(x)−b(y) +1

2B(x)−B(y)²

HS≤ −K|x−y|², x, y∈R^d (1.10)

(·HSis the Hilbert–Schmidt norm), Talagrand’sT₂-inequality (1.7) continues to hold with the constant 2/Kreplaced by sup_x∈Rd,|z|=1|σ (x)z|²/K, and the lawPxof the solutionX_[0,T](x)satisfies onC([0, T],R^d),

1 TW_1,d

L1(Q,Px)²≤W_2,d

L2(Q,Px)²≤2 sup_x∈Rd,|z|=1|σ (x)z|²

K² H (Q|Px), (1.11)

wheredL^p(γ1, γ2):=(T

0 |γ1(t)−γ2(t)|^pdt )^1/pfor two pathsγ1, γ2indexed by[0, T]. Those inequalities are sharp as shown in [5]. See the textbooks of Ledoux [11] and Villani [14] about transportation inequalities and their wide applications.

1.3. Poincaré inequality:A counter-example in jumps case

Almost everything changes in the pure jumps case, drastically. For instance, Gross’ log-Sobolev inequality for the Wiener measure no longer holds for the Poisson process; the equivalence between the log-Sobolev inequality for the invariant measureμand the exponential convergence in entropy (1.5) is lost [16]. Below we give a simple counter- example for which the Poincaré inequality does not hold.

Example 1.1 (Ornstein–Uhlenbeck process driven by Poisson process). Letd=1, dXt= −X_tdt+dN˜_t, X₀=x,

where N_t is a Poisson process with parameter 1 and N˜_t :=N_t −t. Its solution is given by X_t(x)=e^−tx + t

0e^−(t−s)dN˜_s.Its unique invariant measure μis the law of _∞

0 e^−tdN˜_t,that is,an infinitely divisible law of pure jumps with the Lévy measurem_∞(dz)=1(0,1)(z)¹_zdz.

The Poincaré inequality that is equivalent to the exponential decay(1.8)inL²(μ)reads now as:

Varμ(f )≤cPf,−Lfμ=cP

RΓ (f )dμ=c_P 2

R

f (x+1)−f (x)2

dμ(x), (1.12)

which does not hold for periodic smooth non-constant functionsf with period1.

The failure of the Poincaré inequality for this example is rooted at the fact that the Lévy measureδ₁ of N˜_t is too degenerate. For the linear equation dXt=AX_t+dLt (A∈M(d×d))satisfying (1.2), under some strong non- degenerate condition on the Lévy measuremofLt, some positive results are known about the Poincaré inequality (Röckner–Wang [15]) and the stronger Φ-entropy inequality (Gentil–Imbert [6]). WhenL_t contains the Gaussian noise, Röckner–Wang [15] proved stronger functional inequalities even in the infinite dimension setting.

In the non-linear drift (b) case, actually very little is known about functional inequalities to the knowledge of the author.

1.4. Purpose and organization

Though the Poincaré inequality does not hold in the pure jumps case in general, but some kind of transportation inequalities can be saved. That is the purpose of this paper.

This paper is organized as follows. In the next section after recalling Gozlan–Léonard’s characterization for gen- eralW1H-transportation inequalities, we first stateW1H inequalities for the kernelPt, the invariantμas well as for the law of the processX_[0,T]in theL¹-metric (Theorem2.2). Several applications are given for showing the sharp- ness and usefulness of thoseW₁H-transportation inequalities. A generalization to the uniform metrics is also stated (Theorem2.11) and explained.

In Section3, by means of the Malliavin calculus on the Poisson space and Klein–Ma–Privault’s forward-backward martingale method, we show a crucial concentration inequality for functionals on the Poisson space. Furthermore some estimates on the SDE (1.1) under (1.2) are established. Having those preparations we prove quite easily Theorems2.2 and2.11, respectively in Sections4and5.

We keep the notations in this introduction.

2. Main results and applications

2.1. Gozlan–Léonard’s characterization forW1H transportation inequality Letμ∈M₁(E)be fixed, whereEis a metric space with metricd.

Lemma 2.1 (Gozlan–Léonard [7]). Letα:R⁺→ [0,+∞]be a non-decreasing left continuous convex function with α(0)=0.The following properties are equivalent:

(i) theW₁H inequality below holds α

W_1,d(ν, μ)

≤H (ν|μ) ∀ν∈M₁(E); (α-W1H)

(ii) for everyf:(E, d)→Rbounded and Lipschitzian withfLip≤1,

e^{λ(f−μ(f ))}dμ≤e^α∗(λ), λ >0, (2.1)

whereα^∗(λ):=sup_r≥0(rλ−α(r))is the semi-Legendre transformation;

(iii) let(ξ_k)_k≥1be a sequence of i.i.d.r.v.valued inEof common lawμ,for everyf :E→RwithfLip≤1, P

1 n

n k=1

f (ξ_k)−μ(f ) > r

≤e^−nα(r) ∀r >0, n≥1. (2.2)

In such a caseαis called aW₁H-deviation functionofμ.

The equivalence of (i) and (ii) is a generalization of Bobkov–Götze’s criterion [3] for quadraticα. And their new characterization (iii) gives a very strong probabilistic meaning to theW₁H-inequality (α-W1H).

2.2. Main result

The main results of this paper are the following counterparts of (1.7) and (1.11) in the pure jumps case.

Theorem 2.2. Assume(C)and the dissipative condition(1.2)withK >0.Suppose that there is someU-measurable functionσ_∞(u)onUsuch that|σ (x, u)| ≤σ_∞(u), m-a.e.for everyx∈R^d and

∃λ >0: β(λ):=

U

e^λσ∞(u)−λσ_∞(u)−1

m(du) <+∞. (2.3)

Let Px,[0,T] be the law ofX_[0,T](x):=(X_t(x))_t∈[0,T], the solution of the SDE(1.1) withX₀=x.The following properties hold true:

(1) (X_t)admits a unique invariant probability measureμ,and for anyp∈ [1,2], Wp(νPt, μ)≤e^−KtWp(ν, μ) ∀t >0, ν∈M1

R^d

. (2.4)

(2) For eachT >0,P_T(x,dy)satisfies the followingW₁Htransportation inequality α_T

W1

ν, P_T(x,dy)

≤H

ν|P_T(x,dy)

∀ν∈M1

R^d

, (2.5)

where

α_T(r):=sup

λ≥0

rλ−

_T

0

β e^−Ktλ

dt

≥ 1

Kβ^∗(Kr), r≥0, T ∈ [0,+∞]

(β^∗(r):=sup_λ≥0(rλ−β(λ))).In particular for the invariant measureμ, 1

Kβ^∗

KW1(ν, μ)

≤α_∞

W1(ν, μ)

≤H (ν|μ) ∀ν∈M1

R^d

. (2.6)

(3) For eachT >0,Px,[0,T]satisfies on the spaceD([0, T],R^d)of right-continuous left-limitR^d-valued functions on [0, T],

α^P_T W_1,d

L1(Q,Px,[0,T])

≤H (Q|Px,[0,T]) ∀Q∈M₁ D

[0, T],R^d

, (2.7)

whered_L1(γ₁, γ₂):=_T

0 |γ₁(t)−γ₂(t)|dtis theL¹-metric,and α^P_T(r):=sup

λ≥0

λr−

_T

0

β

1−e^−Kt λ/K

dt

≥T β^∗ Kr

T

. (2.8)

(4) If the jumps ofX_t are bounded in size by some constantM >0,i.e., |σ (x, u)| ≤σ_∞(u)≤M, m-a.e.for allx, thenβ^∗(r)≥_2M^r log(1+^Mr_ϑ2)and in particular

KW_1,d

L1(Q,Px,[0,T])

2M log

1+KMW_1,d

L1(Q,Px,[0,T]) T ϑ²

≤H (Q|Px,[0,T]) ∀Q∈M₁ D

[0, T],R^d

, (2.9)

whereϑ²=

σ_∞²(u)dm(u).

Remark 2.3. In[5]the transportation inequality(1.11)for diffusions is proved by means of the Girsanov transforma- tion.We have not succeeded in adapting this last approach to the jumps case.Our proof here,based on the Malliavin calculus on the Poisson space,is completely different(but simple,too).

Remark 2.4. The W₁H inequalities(2.5)–(2.7)are sharp.Indeed for Example1.1,we have K=1, β(λ)=e^λ− λ−1,and

Ee^{λ(XT(x)−EXT(x))}=Eexp

λ _T

0

e^{−(T−t )}d(Nt−t )

=exp T

0

β λe^−t

dt

.

Similarly its invariant measureμbeing the law of_∞

0 e^−td(Nt−t )verifies

e^λxdμ(x)=exp(_∞

0 β(e^−tλ)dt ).Then (2.5)and(2.6)are optimal by Lemma2.1.Furthermore,

Ee^{λ(0TXt(x)dt−E0TXt(x)dt )}=Eexp

λ T

0

1−e^−t dN˜_t

=exp T

0

β

1−e^−t λ

dt

,

which shows that(2.7)is optimal,too.

Remark 2.5. The exponential integrability condition(2.3)on the jumps size,in the case of SDE(1.3)driven by Lévy process with Lévy measuremonR^d∗,reads as

∃λ >0,

R^d∗

e^λB∞|u|−λB_∞|u| −1

m(du) <+∞, B_∞:= sup

x∈R^d

|supz|=1

B(x)z<∞.

This condition is indispensable for theW₁H transportation inequalities in this theorem.Indeed in the special case dXt= −X_tdt +dLt,ifP_t(x,·)(resp.μ)satisfies(α-W1H)for some non-zero deviation functionα,then there is someδ >0 such thatEe^δ|Xt|<+∞(resp.

e^δ|x|dμ(x) <+∞),by Lemma2.1(ii) (and a monotone convergence argument).Either of those two conditions is equivalent to(2.3)by simple explicit calculus.

However without the exponential integrability condition (2.3) some convex concentration inequalities for Lip- schitzian functionals still hold,see Remark5.2,where the natural interpretation of this theorem in comparison form is also presented.

Remark 2.6. In the symmetric case,that is,P_t is symmetric onL²(μ),the exponential decay(2.4)ofP_t toμin the Wasserstein metricW_pimplies the Poincaré inequality([18],Lemma5.4):KVarμ(f )≤

Γ (f, f )dμ,∀f ∈C_b²(R^d).

2.3. Applications to concentration of empirical measure

We now explain parts (3) and (4) of Theorem2.2by the following:

Corollary 2.7. In the framework of Theorem2.2,letAbe a(non-empty)family of real Lipschitzian functionsf onR^d withfLip≤1,andZT :=sup_f∈A(_T¹_T

0 f (Xs(x))ds−μ(f )).We have for allr, T >0,

logP(Z_T >EZ_T +r)≤ −α_T^P(T r)≤ −T β^∗(Kr). (2.10) In particular in the situation of part(4)of Theorem2.2(bounded jumps case),

P(ZT >EZT +r)≤exp

−T Kr 2M log

1+KMr ϑ²

∀r, T >0. (2.11)

The same inequalities hold forZT =W1(LT, μ),whereLT :=_T¹T

0 δX_s(x)dsis the empirical measure.

Proof. We show at firstZT is measurable. Indeed we may assume thatf (0)=0 for allf∈A. Then for any closed ballB(0, R)¯ centered at 0 of radiusR >0,{f_B(0,R)¯ ;f∈A}is compact inCb(B(0, R))¯ (by Arzela–Ascoli). ThenZT

is measurable on the event sup_s≤t|X_s(x)| ≤R. It remains to letR→ +∞.

ConsiderF (γ ):=sup_f∈A|_T¹_T

0 (f (γ_t)dt−μ(f )|,FLip≤1/T w.r.t. thed_L1-metric. ThenZ_T =F (X_[0,T](x)) satisfies (2.10) and (2.11) by parts (3) and (4) of Theorem2.2and Lemma2.1.

Finally whenAis the whole family of allf withfLip≤1, thenZ_T =W₁(L_T, μ)by Kantorovitch–Rubinstein’s

identity.

Remark 2.8. The Poisson behavior in the concentration inequality(2.11)in the bounded jumps case is well known for the sequence of bounded i.i.d.r.v. (Bennett’s inequality)or for the stochastic integral_T

0

Uf (t, u)dN (dt,˜ du)with boundedf ∈L²(dtdm).

For statistical applications of the previous explicit inequality we should estimate the asymptotic biasEW₁(L_T, μ).

WhenT → ∞,sinceL_T →μ,a.s.weakly and

|x|²dLT(x)→μ(|x|²),a.s.,we haveW₁(L_T, μ)→0,a.s. [14], then inL¹(P)by dominated convergence.But transportation inequalitiesW₁Hdo not directly give information about the rate of that convergence(even not the much stronger log-Sobolev inequality,see Ledoux[11]).When the central limit theorem for√

T (LT(f )−μ(f ))holds uniformly over{f; fLip≤1},we haveEW1(LT, μ)=O(T^−1/2).

There is another widely used method to approach μ (which is unknown in general): instead of one sample X_[0,T](x), we producenindependent copiesN¹(dt,du), . . . , Nⁿ(dt,du)of the Poisson point processesN, and con- sider the solutionX^k_[0,T](xk) (k=1, . . . , n)of the SDE (1.1) driven byN^k (instead ofN). Then we approachμby one of

Lˆ_n,T :=1 n

n k=1

δ_Xk

T(x_k), L˜_n,T :=1 n

n k=1

1 T

T 0

δ_Xk t(x)dt.

By the tensorization of (α-W1H) in [7], that is,if αis a W1H-deviation function forμk ∈M1(E, d) for allk= 1, . . . , n, thennα(r/n)is aW₁H-deviation function for_n

k=1μ_k w.r.t. the metricd_l1(x, y):=_n

k=1d(x_k, y_k) (x= (x₁, . . . , x_n), y=(y₁, . . . , y_n)∈Eⁿ), and using the facts:

• W₁(Lˆ_n,T, μ)is a functional of(X¹_T(x₁), . . . , X_Tⁿ(x_n))whose Lipschitzian coefficient w.r.t. the metricd_l1 is≤1/n;

• W₁(L˜_n,T, μ)is a functional of(X_[¹_0,T](x₁), . . . , X_[ⁿ_0,T](x_n))whose Lipschitzian coefficient w.r.t. the sum-L¹metric _n

k=1d_L1(γk,γ˜k)(forγ ,γ˜∈(D([0, T];R^d))ⁿ)is≤1/(nT );

we get by parts (2) and (3) of Theorem2.2(and Lemma2.1):

Corollary 2.9. Under the assumptions of Theorem2.2,we have P

W₁(Lˆ_n,T, μ) >EW₁(Lˆ_n,T, μ)+r

≤exp

−n

Kβ^∗(Kr)

, r, T >0, n≥1, and similarly

P

W₁(L˜_n,T, μ) >EW₁(L˜_n,T, μ)+r

≤exp

−nα_T^P(T r)

, r, T >0, n≥1.

Remark 2.10. Application of transportation inequalitiesW₁H to concentration of empirical measure was explored amply by Bolley,Guillin and Villani[4]for i.i.d.sequences and for an interacting system of particles issued of granular media.

2.4. Generalization to uniform metrics

TheL¹-metricd_L1 in Theorem2.2may be too weak for some issues: for instance, Theorem2.2cannot be applied to concentration of functionals sup_t∈[0,T]|f (X_t)|, (1/n)_n−1

k=0f (X_tk)wheret_k∈ [kT /n, (k+1)T /n]may be ran- dom (chosen in practice according to the sample pathX_{[kT /n,(k+1)T /n]}, this Riemannian sum is more practical than the theoretic empirical mean LT(f )), etc. To cover those functionals, consider the uniform metric d_∞(γ1, γ2)= sup_t∈[0,T]|γ₁(t)−γ₂(t)|and the stronger metric (as in [5]):

d_∞,n(γ1, γ2):=

n−1

k=0

sup

t∈[kT /n,(k+1)T /n]

γ1(t)−γ2(t), γ1, γ2∈D

[0, T];R^d .

Theorem 2.11. Assume(C), (1.2) and (2.3). Assume that x →σ (x,·)∈L²(U, m) is globally Lipschitzian,that is, σLip:=sup_x=y^{σ (x,·)−σ (y,}_|x−y^·_|^)L2(U,m) <+∞. Then there is some constant C ≥0 depending only on K⁻:=

max{−K,0}andσLip[Cis given explicitly in Lemma3.3(3.5)]such that:

(1) for eachT >0,Px,[0,T]satisfies w.r.t.the uniform metricd_∞onD([0, T];R^d) T β^∗

W1,d_∞(Q,Px,[0,T])

√2Te^CT

≤H (Q|Px,[0,T]) ∀Q∈M₁ D

[0, T];R^d

; (2.12)

(2) ifK >0,thenPx,[0,T]satisfies w.r.t.the metricd_∞,nonD([0, T];R^d), T β^∗

W_1,d∞,n(Q,Px,[0,T]) T c_{T ,n}

≤H (Q|Px,[0,T]) ∀Q∈M₁ D

[0, T];R^d

(2.13)

for eachT >0, n≥1,wherec_{T ,n}:=²₁^√₋^{2 e}_e−KT /n^{CT /n}.

Remark 2.12. In the case thatσ (x, u)=σ (u)is independent ofx, (2.12)and(2.13)hold withC=0and without the factor√

2 (see Remark5.1).

Remark 2.13. The inequality(2.12)w.r.t.d_∞is bad for largeT,but sharp in order for smallT.Indeed in the bounded jumps case,forT =εsmall, (2.12)together with part(4)of Theorem2.2,implies that forZε:=sup_0≤t≤ε|Xt(x)−x| andr >0fixed,

P(Z_ε>EZ_ε+r)≤exp

−

1+o(1) r 2√

2Mlog

1+ Mr

√2εϑ²

,

whereo(1)is an infinitesimal,asε→0+.This is sharp in order as seen for Example1.1.

Remark 2.14. To illustrate (2.13), consider for a Lipschitzian observable f:R^d → R, the Riemannian sum F (γ )=(1/n)_n−1

k=0f (γ_tk),wheret_k =t_k(γ )∈ [kT /n, (k+1)T /n] is chosen so that|γ₁(t_k(γ₁))−γ₂(t_k(γ₂))| ≤ Dsup_{t∈[kT /n,(k+1)T /n]}|γ1(t)−γ2(t)|.It is easy to see that thed_∞,n-Lipschitzian coefficient of F is≤ fLipD/n.

Thus we obtain by part(2)of Theorem2.11thatF (X_[0,T](x))=(1/n)n−1

k=0f (Xt_k(x))satisfies the following con- centration inequality:for allr, T >0, n≥1,

P F

X_[0,T](x)

−EF

X_[0,T](x)

> r

≤exp

−T β^∗

n(1−e^{−KT /n})r 2√

2Te^{CT /n}fLipD

, (2.14)

which,asngoes to infinity,gives(2.10)with some slightly worse constant.

2.5. An application to transportation-information inequality Let

J (ν|μ):=sup

−LV

V dν; 1≤V ∈C_b² R^d

, ifνμ; +∞otherwise (2.15)

be the (modified) Donsker–Varadhan information, which is the rate function in the large deviations ofPμ(L_t∈ ·).

By [17], Theorem B.1, for any boundedf:R^d→RwithfLip≤1 andr >0, we have forμ-a.s.x∈R^d, lim inf

T→∞

1 T logP

1 T

_T

0

f X_s(x)

ds−μ(f ) > r

≥ −inf

J (ν|μ);ν(f )−μ(f ) > r

(which is true without the Lipschitzian property off). But the left-hand side is not greater than−g(r):= −β^∗(Kr) by applying (2.10) toA= {f}. Now for anyν∈M₁(R^d)different fromμandε >0, taking some boundedf with fLip≤1 such that (possible by Kantorovitch–Rubinstein’s identity)

ν(f )−μ(f ) > W₁(ν, μ)−ε >0,

and lettingr=W1(ν, μ)−ε, we haveJ (ν|μ)≥g(r)=g(W1(ν, μ)−ε). Lettingε→0, we get:

Corollary 2.15. In the framework of Theorem2.2, β^∗

KW₁(ν, μ)

≤J (ν|μ) ∀ν∈M₁ R^d

.

The argument above is borrowed from Guillin et al. [8], where the transportation-information inequalities are studied in different aspects.

3. Preparations

3.1. Malliavin calculus on the Poisson space

ThePoisson space (Ω,F,P)overR⁺×U with the intensity measure dt×m(du)(wheremis a positiveσ-finite measure on(U,U)) is given by

• Ω:= {ω=

iδ_(ti,ui)(at most countable);(t_i, u_i)∈R⁺×U};

• F=σ (ω→ω(B)|B∈B(R⁺)⊗U);Ft=σ (ω→ω(B)|B∈B([0, t])⊗U);

• ∀B∈B(R⁺)⊗U,∀k∈N:P(ω: ω(B)=k)=e^{−(dt×m)(B)[(dt×m)(B)}_k! ^]k;

• ∀B1, . . . , Bn∈B(R⁺)⊗Udisjoint,ω(B1), . . . , ω(Bn)areP-independent.

UnderP,N (ω,dt,du):=ω(dt,du)is exactly the Poisson point process onR⁺×Uwith intensity measure dt m(du).

For a realP-a.s. well-defined measurable functionF onΩ,Dt,uF (ω):=F (ω+δ(t,u))−F (ω)is well defined up toP(dω)dt m(du)-equivalence and thisdifference operatorplays in the Malliavin calculus on the Poisson space, the role of the Malliavin gradient on the Wiener space. We recall the following martingale representation [13,16]:

Lemma 3.1. For anyF∈L²(Ω,F,P),E

R⁺

U[E(D_t,uF|Ft)]²dt m(du) <+∞and F =EF+

_∞

0

U

f (t, u)N (dt,˜ du),

wheref (t,·)is theFt-predictableP×dt×m(du)version ofE(D_t,uF|Ft).

3.2. Klein,Ma and Privault’s convex concentration inequalities on the Poisson space The following lemma is one key for the results of this paper.

Lemma 3.2. Let F:Ω →R be a bounded measurable function. If there is a deterministic measurable function h(t, u)∈L²(dt m(du))on[0, T] ×Usuch that

E(D_t,uF|Ft)≤h(t, u),P(dω)×dt×m(du)-a.e. (3.1)

then for everyC²-convex functionφ:R→Rso thatφis convex, Eφ(F−EF )≤Eφ

_∞

0

U

h(t, u)N (dt,˜ du)

. (3.2)

In particular

Ee^F−EF ≤exp

e^h−h−1

dm(u)dt

. (3.3)

WhenE(D_t,uF|Ft)in condition (3.1) is replaced byD_t,uF, this lemma is proved by the author [16] using the L¹-log-Sobolev inequality therein.

Proof of Lemma 3.2. We shall use the forward–backward martingale method in Klein, Ma and Privault [10].

Without loss of generality we assume thatEF =0. Let f (t, u, ω)be an Ft-predictableP(dω)dt m(du)-version of E(D_t,uF|Ft). On the product space(Ω²,F²,P²), define

M_t ω, ω

:=

_t

0

U

f (t, u, ω)

ω(dt,du)−dt m(du)

,

ω, ω

∈Ω²,

which is a forward martingale w.r.t. the (increasing) filtrationF¯t:=Ft⊗FonΩ², and M_t^∗

ω, ω :=

_∞

t

U

h(t, u)

ω(dt,du)−dt m(du) ,

which is a backward martingale w.r.t. the (decreasing) filtration F¯_t^∗:=F ⊗F_t^∗ on Ω², where F_t^∗ :=σ (ω → ω(B);B ∈B([t,+∞))⊗U). Observe that M_t is F¯t^∗-adapted, M_t^∗ is F¯t-adapted (the starting condition for [10], Theorem 3.3). Now our condition (3.1) implies condition (3.6) in [10], Theorem 3.3, so [10], Theorem 3.3, says that Eφ(Mt+M_t^∗)is non-increasing int. SinceMt+M_t^∗→F−EF inL²ast→ +∞by Lemma3.1, we apply Fatou’s lemma (applicable forφ(Mt+M_t^∗)≥φ(0)(Mt+M_t^∗)) to get

Eφ(F−EF )≤ lim

t→∞Eφ

M_t+M_t^∗

≤Eφ M₀^∗

,

which is (3.2). Takingφ(x)=e^xin (3.2) gives (3.3).

3.3. Some estimates under the dissipativity We return to our SDE.

Lemma 3.3. Assume(C) and (1.2).For two different initial points x, y∈R^d, the solutions X_t(x), X_t(y)of the SDE(1.1)satisfy

EX_t(x)−X_t(y)²≤e^−2Kt|x−y|², t >0. (3.4)

If furthermoreσLip<+∞(see Theorem2.11for this notation),then there is some universal constantC₁>0such that forC:=2K⁻+(2C₁²+1)σ²_Lip,

E sup

0≤s≤h

Xt+s(x)−Xt+s(y)²≤2e^−2Kt+2Ch|x−y|², t, h >0. (3.5) Indeed C₁ is the best universal constant in the L¹ Burkholder–Davies–Gundy inequality Esup_s≤t|M_t| ≤ C1E√

[M]t for local martingaleMt withM0=0.

Proof. WritingXˆ_t:=X_t(x)−X_t(y),bˆ_t:=b(X_t(x))−b(X_t(y)),σˆ_t(u):=σ (X_t(x), u)−σ (X_t(y), u), we have by integration by parts and our dissipative condition (1.2),

d| ˆX_t|²=2 ˆX_t−,bˆ_t−dt+2

U

Xˆ_t−,σˆ_t−(u)N (dt,˜ du)+

U

σˆ_t−(u)²N (dt,du)

=2Xˆt(x),bˆt

dt+

U

σˆt(u)²m(du)dt+

U

2Xˆt−,σˆt−(u)

+σˆt−(u)²N (dt,˜ du)

≤ −2KXˆ_t(x)²dt+

U

2Xˆ_t−,σˆ_t−(u)

+σˆ_t−(u)²N (dt,˜ du), (3.6) which implies (3.4) by a localization procedure and Gronwall’s inequality.

For (3.5) letZ_t:=sup_s≤t| ˆX_s|², M_t:=_t

0

U ˆX_t−,σˆ_t−(u) ˜N (dt,du). By Burkholder–Davies–Gundy’s inequality, there is some best universal constantC1such that

Esup

s≤t

M_t≤C₁E

[M]t=C₁E t 0

U

Xˆ_s−,σˆ_s−(u)2

N (ds,du)

≤C₁E

Z_{t t}

0

U

σˆ_s−(u)²N (ds,du)≤C₁

EZ_tE _t

0

U

σˆ_s−(u)²m(du)ds

≤ C₁ 2

aEZ_t+σ²_Lip

a E

t 0

| ˆX_s|²ds

,

wherea >0 is arbitrary. Now using the first equality in (3.6) and ˆX_t,bˆ_t ≤ −2K| ˆX_t|², EZ_t≤ |x−y|²+2K⁻

_t

0

| ˆX_s|²ds+2E sup

0≤s≤t

|M_t| +E _t

0

U

σˆ_s−(u)²N (ds,du)

≤ |x−y|²+aC1EZt+ 2K⁻+(C1/a+1)σ²Lip t

0 E| ˆXs|²ds,

lettinga=1/(2C1)andC=2K⁻+(2C²₁+1)σ²_Lipwe obtain by Gronwall’s inequality EZ_h≤2 exp

2 2K⁻+

2C₁²+1 σ²_Lip

h

|x−y|²=2e^2Ch|x−y|², h >0, which is (3.5) fort=0. Now for (3.5) witht >0, it is enough to notice

E E

sup

0≤s≤h

X_t+s(x)−X_t+s(y)² Ft

≤2e^2ChEX_t(x)−X_t(y)²

and then to apply (3.4).

Remark 3.4. In the additive noise case,that is,σ (x, u)=σ (u)is independent ofx∈R^d,we will have d(Xt(x)− X_t(y))= [b(X_t(x))−b(X_t(y))]dt,which gives

dX_t(x)−X_t(y)²=2

X_t(x)−X_t(y), b X_t(x)

−b X_t(y)

dt≤ −2KX_t(x)−X_t(y)²dt.

WhenceP-a.s.,

X_t(x)−X_t(y)≤e^−Kt|x−y| ∀t >0. (3.7)

4. Proof of Theorem2.2

4.1. Part(1)

Though this part should be well known to specialists, yet we give its proof for the convenience of the reader. Recall thatM₁^p(R^d):= {ν∈M₁(R^d);

|x|^pdν <+∞}equipped with the Wasserstein metricW_p is complete [14]. Below letp∈ [1,2].

For any two initial probability measuresν1, ν2∈M₁^p(R^d), let(X0, Y0)be a couple ofR^d-valued random variables of lawν1, ν2respectively, independent of the Poisson point processN (dt,du)such thatE|X0−Y0|^p=Wp(ν1, ν2)^p. LetX_t (resp.Y_t) be the solution of the SDE (1.1) with initial conditionX₀(resp.Y₀).(X_t, Y_t)constitutes a coupling ofν₁P_tandν₂P_t. By Lemma3.3,

E

|Xt−Yt|^p|X0, Y0

≤ E

|Xt−Yt|²|X0, Y0

p/2

≤e^−pKt|X0−Y0|^p,