CHRISTIAN L ÉONARD Arnaud Guillin

Christian Léonard
email: christian.leonard@polytechnique.fr

AND Feng-Yu Wang
email: f.y.wang@swansea.ac.uk

Liming Wu

Yu Wang Feng

Transportation-information inequalities for Markov processes (II

Keywords: Wasserstein distance, entropy, Fisher information, transport-information inequality, deviation inequality 60E15, 60K35, 60G60

published or not. The documents may come

Introduction

Let (X , d) be a complete and separable metric space (say Polish) and µ a given probability measure on (X , B) where B is the Borel σ-field. Let (X t ) t≥0 be a µ-symmetric ergodic conservative Markov process valued in X , with transition semigroup (P t ) (which is symmetric on L 2 (µ)), and Dirichlet form (E(•, •), D(E)) where D(E) is the domain of E in L 2 (µ) := L 2 (X , B, µ). Here the ergodicity means simply : for g ∈ D(E), E(g, g) = 0 iff g = c.

For 1 ≤ p < +∞ fixed and for any probability measure ν on X (written as ν ∈ M 1 (X )), consider (i): L p -Wasserstein distance between ν and µ:

W p (ν, µ) := inf π∈C(ν,µ) E 2 d p (x, y)π(dx, dy) (1.1)
where C(ν, µ) are the set of all couplings of (ν, µ), i.e., probability measures π on E × E such that π(A × E) = ν(A) and π(X × A) = µ(A) for all A ∈ B.

I(ν|µ)

:= E( √ f , √ f ). if ν = f µ, √ f ∈ D(E), +∞ otherwise. (1.3) Fisher
The usual transport inequalities W p H, introduced and studied by K. Marton [START_REF] Marton | Bounding d-distance by informational divergence: a way to prove measure concentration[END_REF] and M. Talagrand [START_REF] Talagrand | Transportation cost for gaussian and other product measures[END_REF] mean that W p (ν, µ) 2 ≤ 2CH(ν|µ), ∀ν ∈ M 1 (X ).

(W p H(C)) Its study is very active: see Bobkov-Götze [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF], Otto-Villani [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF], Bobkov-Gentil-Ledoux [START_REF] Bobkov | Hypercontractivity of Hamilton-Jacobi equations[END_REF], Djellout-Guillin-Wu [START_REF] Djellout | Transportation cost-information inequalities for random dynamical systems and diffusions[END_REF] and references therein. Furthermore Gozlan-Léonard [START_REF] Gozlan | A large deviation approach to some transportation cost inequalities[END_REF] consider the following generalized transportation cost from ν to µ: T V (ν, µ) := sup{ν(u)µ(v); (u, v) ∈ V} (µ(u) := E udµ) where V is some given family of (u, v) ∈ (bB) 2 so that (A1) u ≤ v for all (u, v) ∈ V (or equivalently T V (ν, ν) ≤ 0 for all ν ∈ M 1 (X )); (A2) For all ν 1 , ν 2 ∈ M 1 (X ), there exists (u, v) ∈ V such that u dν 1v dν 2 ≥ 0 (or equivalently T V (ν 1 , ν 2 ) ≥ 0 for all ν 1 , ν 2 ∈ M 1 (X )). And they introduced the following generalization of W p H: for some convex, non-decreasing and left continuous function α on R + , α(T V (ν, µ)) ≤ 2CH(ν|µ), ∀ν ∈ M 1 (X ) (α-T V H(C)) and they established its equivalence with some concentration inequality of the underlying measure µ and of the i.i.d. sequences of common law µ.

Recall that T V (ν, µ) = W p (ν, µ) 2 iff V = V(p, d), the family of all couples (u, v) of real bounded measurable functions on X such that u(x)v(y) ≤ d p (x, y), ∀x, y ∈ E.

(1.4) A31 Guillin-Leonard-Wu-Yao [START_REF] Guillin | Transport-information inequalities for Markov processes (I)[END_REF] propose a new transport-information inequality, adapted to Markov processes (and in particular to consider deviation inequalities for integral functionals of Markov processes) W p (ν, µ) 2 ≤ 4C 2 I(ν|µ), ∀ν ∈ M 1 (X ) (W p I(C)) or the more general α (T V (ν, µ)) ≤ I(ν|µ), ∀ν ∈ M 1 (X ).

(α-T V I) Using large deviations techniques they prove the following characterization:

λ max (L + λu) := sup g∈D(E):µ(g 2 )=1 λ ug 2 dµ -E(g, g) ≤ λµ(v) + α * (λ) (1.5) L2bound
where L is the generator of (P t ) on L 2 (X , µ) and r) , ∀t, r > 0.

α * (λ) = sup r≥0 {λr -α(r)}, ∀λ ≥ 0 is the semi-Legendre transformation of α. (c): For any initial measure ν = f µ with f ∈ L 2 (µ) and for all (u, v) ∈ V P ν 1 t t 0 u(X s )ds ≥ µ(v) + r ≤ f 2 e -tα(
(1.6) thmA11a

Remarks 1.2. The meaning of the deviation inequality characterization (1.6) of α-W p I is clear in the ergodic behavior of the Markov process (X t ), as well as (1.5) in the study of the Schrödinger operator L + u. That is one more reason why α-T V I inequality is useful.

Remarks 1.3. If V is some family of (u, u) ∈ (bB) 2 , (1.6) becomes a deviation inequality of the empirical (time) mean from its space mean µ(u) for the observable u so that (u, u) ∈ V. Three criteria for W 1 I(C) are established in [START_REF] Guillin | Transport-information inequalities for Markov processes (I)[END_REF]: spectral gap in L 2 (µ); spectral gap in the space of Lipschitz functions and a very general Lyapunov function criterion if V = {(u, u); |u| ≤ φ} where φ > 0 is some fixed weight funtion. And it is also shown that on a Riemannian manifold X equipped with the Riemannian metric d, the log-Sobolev inequality

Notice that if V = {(u, u); u ∈ bB, u Lip ≤ 1} then T V (ν, µ) = W 1 (ν, µ),
H(ν|µ) ≤ 2CI(ν|µ), ∀ν ∈ M 1 (X ).
(HI(C))

implies W 2 I(C), which in turn implies the Poincaré inequality

V ar µ (g) ≤ c P E(g, g), ∀g ∈ L 2 (µ) D(E) (P (C))
where V ar µ (g) = µ(g 2 )µ(g) 2 is the variance. Furthermore W 2 I(C) =⇒ HI(C ′ ) once if the Ricci-Bakry-Emery curvature of µ is bounded from below.

We organize this paper around the four questions below:

(i): Investigate the relations between W p I with W p H. That is the objective of §2.

(ii): Prove that the spectral gap in the space of Lipschitz functions implies a Cheeger type's isoperimetric inequality, which is stronger than W 1 I. That is the purpose of §3. We will also establish deviation inequalities under natural quantities such as the variance of the test function, refining [START_REF] Guillin | Transport-information inequalities for Markov processes (I)[END_REF]. (iii): In §4 we study relations between (α-W 2 I) and the β-log-Sobolev inequality:

β • µ(g 2 log g 2 ) ≤ E(g, g), µ(g 2 ) = 1, g ∈ D(E), (1.7) Phi
where β is a positive increasing function. This inequality was connected in [START_REF] Wang | Transportation cost inequalities for the log-Sobolev inequality with powers[END_REF] to the well developed F -Sobolev inequality introduced in [START_REF] Wang | Functional inequalities for empty essential spectrum[END_REF], so that known criteria for the later can be applied directly to (1.7). (iv): Finally we present in §5 applications of Φ-Sobolev inequality

g 2 Φ ≤ C 1 E(g, g) + C 2 µ(g 2 )
in transportation-information inequalities α-T V I and then in the concentration phenomena of 1 t t 0 u(X s )ds under integrability conditions on u.

W p I implies W p H on a Riemannian manifold

Recall (cf. Villani [START_REF] Villani | Topics in Optimal Transportation[END_REF]) the well known Kantorovitch's dual characterization :

W p p (ν, µ) = sup (u,v)∈V(p,d) udν -vdµ (2.1) A32
where V(p, d) is given in (1.4), and Kantorovitch-Robinstein's identity

W 1 (ν, µ) = sup u Lip ≤1 ud(ν -µ). (2.2) A33
Throughout this section X is a connected complete Riemannian manifold equipped with the Riemannian metric d, and µ = e -V dx/Z (Z being the normalization constant assumed to be finite) with V ∈ C 1 (X ), and (E, D(E)) is the closure of

E(g, g) := E |∇g| 2 dµ(x), g ∈ C ∞ 0 (X )
where ∇ is the gradient on X , and C ∞ 0 (X ) is the space of infinitely differentiable functions on X with compact support. In such case our Fisher information of f µ with 0 < f ∈ C 1 (X ) w.r.t. µ becomes

I(f µ|µ) = 1 4 |∇f | 2 f dµ = 1 4 |∇ log f | 2 dµ. 2.1. W 1 I(C) =⇒ W 1 H(C). thm21 Theorem 2.1. Assume that µ satisfies W 1 I(C). Then W 1 (ν, µ) 2 ≤ 2CH(ν|µ), ∀ν ∈ M 1 (X )
i.e., µ satisfies W 1 H(C).

Proof. By Bobkov-Götze's criterion [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF] for W 1 H(C), it is enough to show that for any bounded g ∈ C 1 (X ) with |∇g| ≤ 1 and λ ≥ 0, e λ(g-µ(g)) dµ ≤ e λ 2 C 2 /2 .

(

2.3) thm21b

To this end we may assume that µ(g) = 0. Consider Z(λ) = e λg dµ, µ λ := e λg Z(λ) µ.

We have by Kantorovitch's identity (2.2)

d dλ log Z(λ) = µ λ (g) ≤ W 1 (µ λ , µ) but by W 1 I(C), W 1 (µ λ , µ) ≤ 2C I(µ λ |µ) = Cλ |∇g| 2 dµ λ ≤ Cλ. Thus log Z(λ) ≤ λ 0 Ctdt = Cλ 2 2 the desired control (2.3).
The implication "W 1 I(C) =⇒ W 1 H(C)" is strict, as shown by the following simple counter-example ( [START_REF] Djellout | Transportation cost-information inequalities for random dynamical systems and diffusions[END_REF]).

Example 2.2. Let X = [-2, -1] [1, 2] and µ(dx) = (1 [-2,-1] + 1 [1,2] )dx/2. The Dirichlet form (E, D(E)) is given by E(f, f ) = f ′2 dµ(x), ∀f ∈ D(E) = H 1 (X )
where H 1 (X ) is the space of those functions f ∈ L 2 (µ) so that f ′ ∈ L 2 (µ) (in the distribution sense). It corresponds to the reflecting Brownian Motion in X , which is not ergodic. But W 1 I(C) implies always the ergodicity. Thus µ does not satisfy W 1 I(C). However µ satisfies W 1 H(C) by the Gaussian integrability criterion in [START_REF] Djellout | Transportation cost-information inequalities for random dynamical systems and diffusions[END_REF].

The argument above can be extended to more general transportation information inequality α-W 1 I:

prop21 Proposition 2.3. Let α : R + → [0, +∞] be a left-continuous non-decreasing convex func- tion with α(0) = 0. Assume that µ satisfies α-W 1 I. Then µ satisfies α(W 1 (ν, µ)) ≤ H(ν|µ), ∀ν ∈ M 1 (X ) (α -W 1 H) prop21a
where α(r) = 2 r 0 α(s)ds. In particular for any Lipschitzian function g with g Lip ≤ 1, r) , ∀r > 0.

µ(g > µ(g) + r) ≤ e -α(
Proof. By Gozlan-Léonard's criterion [START_REF] Gozlan | A large deviation approach to some transportation cost inequalities[END_REF] for α-W 1 H, it is enough to show that for any bounded g ∈ C 1 (X ) with |∇g| ≤ 1 and λ ≥ 0, e λ(g-µ(g)) dµ ≤ e α * (λ) , ∀λ ≥ 0 (2.4) prop21c

which implies the last concentration inequality in this Proposition by Chebychev's inequality. To show (2.4) we may assume that µ(g) = 0. Let Z(λ) and µ λ be as in the previous proof of Theorem 2.1, we have

d dλ log Z(λ) = µ λ (g) ≤ W 1 (µ λ , µ).
But by the assumed α-W 1 I,

W 1 (µ λ , µ) ≤ α -1 (I(µ λ |µ)) = α -1 λ 2 4 |∇g| 2 dµ λ ≤ α -1 (λ 2 /4) where α -1 (t) := inf{t ≥ 0; α(r) > t}, t ≥ 0. Thus log Z(λ) ≤ λ 0 α -1 (t 2 /4)dt =: h(λ). Now by Fenchel-Legendre theorem, h = (h * ) * , but h * (r) = sup λ≥0 (λr -h(λ)) = 2 r 0 α(s)ds,
which completes the proof of the desired control (2.4).

W

2 I(C) =⇒ W 2 H(C). thm22 Theorem 2.4. Assume that µ satisfies W 2 I(C). Then W 2 (ν, µ) 2 ≤ 2CH(ν|µ), ∀ν ∈ M 1 (X ) i.e., µ satisfies W 2 H(C).
Proof. We shall use the method of Hamilton-Jacobi equation due to Bobkov-Gentil-Ledoux [START_REF] Bobkov | Hypercontractivity of Hamilton-Jacobi equations[END_REF]. Consider the inf-convolution

Q t g(x) := inf y∈E (g(y) + 1 2t d 2 (x, y))
which is viscosity solution of the Hamilton-Jacobi equation

∂ t Q t g + 1 2 |∇Q t g| 2 = 0. (2.

5) HJequ

By Bobkov-Götze's criterion [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF] for

W 2 H(C), it is enough to show that for any g ∈ C 1 b (X ), e Q 1 g/C dµ ≤ e µ(g)/C . (2.6) thm22b
To this end we may and will assume that µ(g) = 0. Let λ = λ(t) = κt where κ > 0 will be determined later and consider

Z(t) = e λQtg dµ, µ t := e λQtg Z(t) µ.
We have

d dt log Z(t) = 1 Z(t) λ ′ (t)Q t f + λ(t)∂ t Q t g e λQtg dµ = κ Q t gdµ t - λ 2 |∇Q t g| 2 dµ t = κ Q t gdµ t - 2 λ I(µ t |µ).
But by Kantorovitch's identity (2.1),

Q t gdµ t ≤ 1 2t W 2 2 (µ t , µ) (2.7) thm22c
and the assumed

W 2 I(C) gives W 2 2 (µ t , µ) ≤ 4C 2 I(µ t |µ). Thus for every t > 0, d dt log Z(t) ≤ 2κC 2 t - 2 κt I(µ t |µ)
Putting κ = 1/C, we obtain d dt log Z(t) ≤ 0 for all t > 0, which implies by the continuity of log

Z(t) on R + that e Q 1 g/C dµ = Z(1) ≤ Z(0) = 1 the desired (2.6).
Remarks 2.5. The proof above is adapted from that of Bobkov-Gentil-Ledoux [START_REF] Bobkov | Hypercontractivity of Hamilton-Jacobi equations[END_REF] for the implication HI(C) =⇒ W 2 H(C), originally established by Otto-Villani [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF].

Remarks 2.6. We have thus established in this section

HI(C) ⇒ W 2 I(C) ⇒ W 2 H(C).
It was also established in [START_REF] Guillin | Transport-information inequalities for Markov processes (I)[END_REF] that under a lower bound of the Ricci-Bakry-Emery curvature of µ that W 2 I implies back to HI, and with additional conditions on this lower bound that W 2 H implies back HI. It is then a natural question to know if the condition on the lower bound of the Ricci-Bakry-Emery curvature is also necessary to get the reverse implication. A partial answer was provided in [START_REF] Cattiaux | On quadratic transportation cost inequalities[END_REF] where an example of a real probability measure, with infinite lower bounded curvature, was shown to verify W 2 H but not HI. Inspired by this example, we furnish here an example where W 1 I holds (using Lyapunov conditions of [13, Section 5]) but not HI. Let then consider dµ(x) = e -V (x) dx, where V is symmetric C 2 (at least) and given for large x by

V (x) = x 4 + 4x 3 sin 2 (x) + x β .
Consider also the natural reversible process associated to this measure given by generator Lf = f ′′ -V ′ f ′ . Using W (x) = e ax 4 , by easy calculus, one sees that LW ≤ -cx 4 W + b (for some positive b and c) if β > 2. This Lyapunov condition also implies a Poincaré inequality (see [START_REF] Barthe | A simple proof or the Poincaré inequality for a large class of probability measure including the logconcave case[END_REF] for example), so that using a slight modification of [13, Lem. 5.7], we get that W 1 I holds and also W 2 H by [START_REF] Cattiaux | On quadratic transportation cost inequalities[END_REF]. Remark now that if β < 3 then V /V ′2 is not bounded, which is a known necessary condition for HI to hold (see [START_REF] Cattiaux | On quadratic transportation cost inequalities[END_REF]). Unfortunatly, we are not up to now able to prove that W 2 I holds.

W 1 I and the isoperimetric inequality of Cheeger's type by means of the spectral gap in C Lip

In this section we return to the general Polish space case (X , d). We assume that µ charges all non-empty open subsets of X .

Let C Lip be the space of all real functions g on X which are Lipschitz-continuous, i.e., g Lip := sup x =y

|f (x)-f (y)| d(x,y)
< +∞. We assume that there is an algebra A ⊂ C Lip D 2 (L) (here D 2 (L) is the domain of the generator L in L 2 (µ) associated with (E, D(E))), which is a form core for (E, D(E)). Hence the carré-du-champs operator

Γ(f, g) := 1 2 (L(f g) -f Lg -gLf ) , ∀f, g ∈ A admits a unique continuous extension Γ : D(E) × D(E) → L 1 (X , µ).
Throughout this section we assume that Γ is a differentiation, that is, for all (h k ) 1≤k≤n ⊂ A, g ∈ A and

F ∈ C 1 b (R n ), Γ(F (h 1 , • • • , h n ), g) = n i=1 ∂ i F (h 1 , • • • , h n )Γ(h i , g) (this can be extended to D(E)). thm31 Theorem 3.1. Assume that d 2 (x, x 0 )dµ(x) < +∞ for some (or all) x 0 ∈ E and Γ is a differentiation. Suppose that there is a form core D ⊂ C Lip D 2 (L) of (E, D(E)) such that 1 ∈ D and W 1 (ν, µ) = sup g∈D: g Lip ≤1 { gd(ν -µ)}, ∀ν with I(ν|µ) < +∞ (3.1) thm31a and
Γ(g, g) ≤ σ g Lip , µ -a.s., ∀g ∈ C Lip D 2 (L) (3.2) thm31b
and for some constant C > 0 and for any g ∈ D with µ(g) = 0, there is

G ∈ C Lip D 2 (L) so that -LG = g, G Lip ≤ C g Lip . (3.3) thm31c
Then the Poincaré inequality holds with c P ≤ C, and the following isoperimetric inequality of Cheeger's type

W 1 (f µ, µ) ≤ σC Γ(f, f )dµ, 0 ≤ f ∈ D(E), µ(f ) = 1 (3.4) thm31d
holds true. In particular,

W 1 (ν, µ) 2 ≤ 4(σC) 2 I(ν|µ), ∀ν ∈ M 1 (X ). (3.5) thm31e
Furthermore for any observable g with g Lip = 1,

gd(ν -µ) ≤ 2 I(ν|µ) V (g) 2 + 2(σC) 2 c P I(ν|µ) (3.6) thm31f
and for any t, r, δ > 0,

P β 1 t t 0 g(X s )ds > µ(g) + r ≤ dβ dµ 2 exp   -t r 2 (1 + δ)V (g) + [(1 + δ)V (g)] 2 + 8c P (σC) 4 δV (g) r 2   (3.7) thm31g
where V (g) := lim t→∞

1 t V ar Pµ t 0 g(X s )ds = 2
∞ 0 gµ(g), P t g µ dt is the asymptotic variance of g.

Proof.

Under the Lipschitzian spectral gap condition (3.3), it is noted in [START_REF] Guillin | Transport-information inequalities for Markov processes (I)[END_REF] that the Poincaré inequality holds with c P ≤ C.

For both (3.4) and (3.5) we may assume that ν = f µ with f ∈ D(E), f ≥ ε > 0. For any g ∈ D with g Lip ≤ 1 and µ(g) = 0, letting G := (-L) -1 g be the unique solution of the Poisson equation with µ(G) = 0, we have

gdν -gdµ = g, f µ = E(G, f ) = Γ(G, f )dµ ≤ Γ(G, G) ∞ Γ(f, f )dµ.
Taking the supremum over all such g and observing Γ(G, G) ∞ ≤ σ G Lip ≤ σC we obtain (3.4). Furthermore by Cauchy-Schwarz and the fact that Γ is a differentiation, we have

Γ(f, f )dµ ≤ Γ(f, f ) f dµ f dµ = 2 I(ν|µ)
where (3.5) follows from (3.4). For (3.6) writing f = h 2 , we have

gdν -gdµ = Γ(G, f )dµ = 2 hΓ(G, h)dµ ≤ 2 Γ(h, h)dµ • Γ(G, G)h 2 dµ.
Using the inequality in [13, Theorem 3.1]

Γ(G, G)h 2 dµ -Γ(G, G)dµ ≤ Γ(G, G) ∞ h 2 µ -µ T V ≤ (σC) 2 4c P I(ν|µ)
and noting that

V (g) = 2 (-L) -1 g, g µ = 2E(G, G) = 2 Γ(G, G)dµ, we obtain gdν -gdµ ≤ 2 I(ν|µ) V (g) 2 + 2(σC) 2 c P I(ν|µ)
which is (3.6). Using 2I

= N (0, σ 2 ). For f ∈ C ∞ b (R), from the explicit solution X t = e -σ -2 t X 0 + t 0 e σ -2 s √
2dB s , we see that (P t f ) ′ = e -σ -2 t P t f ′ .

Hence c

G ′ (x) = - x 0 g(t)dt, x ∈ [0, D].
It is now easy to see that c Lip,P = sup g Lip =1 G ′ ∞ is attained with g(x) = x -D/2 and then c Lip,P = D 2 /8. Thus by Theorem 3.1, the optimal constant C W 1 I for this process satisfies C W 1 I ≤ c Lip = D 2 /8. In comparison recall that the best Poincaré constant

c P = D 2 /π 2 . Since W 1 I(C) =⇒ W 1 H(C)
and the best constant of W 1 H(C) for the uniform law µ on [0, D] is D 2 /12, so we obtain

D 2 12 ≤ c W 1 I ≤ D 2 8 .
We do not know the exact value of c W 1 I for this simple example.

Example 3.6. Let X be a compact connected Riemannian manifold of dimension n with empty or convex boundary. Assume that the Ricci curvature is nonnegative and its diameter is D. Consider the Brownian Motion (with reflection in the presence of the boundary) generated by the Laplacian operator ∆.

In [START_REF] Wu | Lipschitzian norm of Poisson operators on Riemannian manifolds[END_REF] it is shown that c Lip,P = (-∆) -1 Lip ≤ D 2 /8 (the latest quantity is exactly c Lip,P for the reflected Brownian Motion on the interval [0, D]). Thus by Theorem 3.1,

W 1 I(C) holds with C = D 2 /8.
See [START_REF] Wu | Lipschitzian norm of Poisson operators on Riemannian manifolds[END_REF] for more examples for which c Lip,P is estimated.

Example 3.7. (One-dimensional diffusions) Now let us consider the one-dimensional diffusion with values in the interval (x 0 , y 0 ) generated by

Lf = a(x)f ′′ + b(x)f ′ , f ∈ C ∞ 0 (x 0 , y 0 )
where a, b are continuous such that a(x) > 0 for all x ∈ (x 0 , y 0 ). Let ((X t ) 0≤t<τ , P x ) be the martingale solution associated with L and initial position x, where τ is the explosion time. With a fixec c ∈ (x 0 , y 0 ),

s ′ (x) := exp - x c b(t) a(t) dt , m ′ (x) := 1 a(x) exp x c b(t) a(t) dt
are respectively the derivatives of Feller's scale and speed functions. Assume that

Z := y 0 x 0 m ′ (x) dx < +∞ (3.8) D1 and let µ(dx) = m ′ (x)dx/Z. It is well known that (L, C ∞ 0 (x 0 , y 0 )) is symmetric on L 2 (µ). Assume also that y 0 c s ′ (x) dx x c m ′ (x) dx = c x 0 s ′ (x) c x m ′ (x) dx = +∞ (3.9) D2
which, in the Feller's classification, means that x 0 , y 0 are no accessible or equivalently τ = ∞, P x -a.s. In this case by the L 1 -uniqueness in [START_REF] Wu | Uniqueness of Nelson's diffusions[END_REF], the Dirichlet form

D(E) = f ∈ AC(x 0 , y 0 ) L 2 (µ); y 0 x 0 (f ′ ) 2 dµ < +∞ , E(f, f ) = y 0 x 0 (f ′ ) 2 dµ, f ∈ D(E)
is associated with (X t ), where AC(x 0 , y 0 ) is the space of absolutely continuous functions on (x 0 , y 0 ). Fix some ρ ∈ C 1 (x 0 , y 0 ) such that ρ ∈ L 2 (µ) and ρ ′ (x) > 0 everywhere, consider the metric

d ρ (x, y) = |ρ(x) -ρ(y)|. A function f on (x 0 , y 0 ) is Lipschitz with respect to d ρ (written as f ∈ C Lip(ρ) ) if and only if f ∈ AC(x 0 , y 0 ) and f Lip(ρ) = sup x 0 <x<y<y 0 |f (y) -f (x)| ρ(y) -ρ(x) = f ′ ρ ′ ∞ .
The argument below is borrowed from [START_REF] Djellout | Spectral gap of one dimensional diffusions in Lipschitzian norm and application to log-Sobolev inequalities for Gibbs measures[END_REF]. Assume that

C(ρ) := sup x∈(x 0 ,y 0 ) 1 ρ ′ (x) y 0 x [ρ(t) -µ(ρ)]m ′ (t) dt < +∞. (3.10) C-rho
For every g ∈ C Lip(ρ) with µ(g) = 0, then f (x) =

x c dy

y 0 y g(t)m ′ (t) dt -A (in C 2 ) solves -(af ′′ + bf ′ ) = g. (3.11) cor44a It is obvious that f Lip(ρ) = sup x∈(x 0 ,y 0 ) 1 ρ ′ (x) y 0 x g(t)m ′ (t) dt.
An elementary exercise (as done in [START_REF] Djellout | Spectral gap of one dimensional diffusions in Lipschitzian norm and application to log-Sobolev inequalities for Gibbs measures[END_REF]) shows that the last quantity is always not greater than C(ρ) g Lip(ρ) . Thus f ∈ L 2 (µ) (for ρ ∈ L 2 (µ)). By Ito's formula, f ∈ D 2 (L). With the constant A so that µ(f ) = 0, f given above is the unique solution in L 2 (µ) with zero mean of (3.11) by the ergodicity of (X t ). We see also that C(ρ) is the best constant by taking g = ρµ(ρ). In other words condition (3.3) is verified with the best constant C = c Lip,P = C(ρ). Hence from Theorem 3.1, we get cor44 Corollary 3.8. Let a, b : (x 0 , y 0 ) → R be continuous such that a(x) > 0 for all x and conditions (3.8) (3.9) be satisfied. Assume (3.10) and σ := sup x∈(x 0 ,y 0 ) a(x)ρ ′ (x) < +∞.

Then µ satisfies W 1 I(σC(ρ)) on ((x 0 , y 0 ), d ρ ). In particular for Remarks 3.9. The quantity C(ρ) in (3.10) is not innocent: Chen-Wang's variational formula for the spectral gap λ 1 says that ( [START_REF] Chen | Eigenvalues, inequalities, and ergodic theory[END_REF][START_REF] Wang | Functional inequalities, Markov Semigroup and Spectral Theory[END_REF]):

ρ a (x) =
λ 1 = sup ρ 1 C(ρ) .

Functional inequalities and W 2 I inequalities

Throughout this section we consider the framework of Section 2, i.e. X is a connected complete Riemannian manifold M with µ(dx) := e -V (x) dx/Z for some V ∈ C(M ) with Z := M e -V (x) dx < ∞. Recall that in [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF][START_REF] Guillin | Transport-information inequalities for Markov processes (I)[END_REF] was proven the fact that a logarithmic Sobolev inequality implies W 2 I, and that (using HWI inequalities) under a lower bounded curvature, the converse was also true. We extend here this assertion for α -W 2 I inequalities.

T4.1 Theorem 4.1. (1) Let β ∈ C([0, ∞)) be increasing with β(0) = 0 such that γ(r) := 1 2 r 0 ds β(s) < ∞, r > 0.
Then the following β-log-Sobolev inequality

β • µ(g 2 log g 2 ) ≤ µ(|∇g| 2 ), g ∈ C 1 b (M ), µ(g 2 ) = 1, (4.1) P1 implies α(W 2 (ν, µ)) ≤ I(ν|µ), ν ∈ M 1 (X ) (4.2) AA for α(s) := β • γ -1 (s), s ≥ 0.
(2) Assume that Ric + Hess V ≥ -K for some K ≥ 0. Then (4.2) implies (4.1) for

β(r) := inf s > 0 : 2 √ 2sα -1 (s) + K(α -1 (s)) 2 ≥ r , r ≥ 0.
Proof. (1) According to [24, Theorem 2.2], (4.1) implies

W 2 (f µ, µ) ≤ γ • µ(f log f ), f ≥ 0, µ(f ) = 1. (4.
3) TP Then (4.2) follows from (4.3) and (4.1). For readers' convenience, we include below a brief proof of (4.3), inspired by the seminal work [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF] pushed further in [START_REF] Wang | Probability distance inequalities on Riemannian manifolds and path spaces[END_REF].

Since a continuous function can be uniformly approximated by smooth ones, we may and do assume that V is smooth. Let P t be the diffusion semigroup generated by L := ∆ -∇V.∇. Then P t is symmetric in L 2 (µ). For fixed f > 0 with µ(f ) = 1, let µ t = (P t f )µ, t > 0. According to [22, page 176] for p = 2 (see also [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF] under a curvature condition), we have

d + dt -W 2 (µ, µ t ) := lim sup s↓0 W 2 (µ, µ t ) -W 2 (µ, µ t+s ) s ≤ 2µ ∇ P t f 2 1/2 . (4.4) W2.1 Let γ(r) = 1 2 r 0 ds β(s) , r > 0.
It suffices to prove for the case that γ(r) < ∞ for r > 0. By (4.1) we have

d dt γ • µ(P t f log P t f ) = 4γ ′ • µ(P t f log P t f )µ ∇ P t f 2 = 2µ ∇ √ P t f 2 β • µ(P t f log P t f ) ≥ 2µ ∇ P t f 2 1/2 .
Combining this with (4.4) we obtain

d + dt -W 2 (µ, µ t ) ≤ d dt γ • µ(P t f log P t f ),
which implies (4.3) by noting that

P t f → µ(f ) = 1 as t → ∞.
(2) By the HWI inequality (see [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF][START_REF] Bobkov | Hypercontractivity of Hamilton-Jacobi equations[END_REF]), we have

µ(g 2 log g 2 ) ≤ 2(e 2Kt -1) K µ(|∇g| 2 ) + Ke 2Kt e 2Kt -1 W 2 (g 2 µ, µ) 2 , µ(g 2 ) = 1, t > 0.
Combining this with (4.2) we obtain

µ(g 2 log g 2 ) ≤ inf t>0 2(e 2Kt -1) K µ(|∇g| 2 ) + Ke 2Kt e 2Kt -1 [α -1 µ(|∇g| 2 ) ] 2 .
Taking t > 0 such that

e 2Kt = 1 + K α -1 (µ(|∇g| 2 )) 2µ(|∇g| 2 ) , we obtain µ(g 2 log g 2 ) ≤ 2 2µ(|∇g| 2 )α -1 (µ(|∇g| 2 )) + K[α -1 (µ(|∇g| 2
))] 2 . This completes the proof.

Let us give a natural family of examples, namely when β is a power function. -δ) . Then the first assertion follows from Theorem 4.1 [START_REF] Barthe | A simple proof or the Poincaré inequality for a large class of probability measure including the logconcave case[END_REF].

C4.2 Corollary 4.2. For any δ ∈ [1, 2), µ(g 2 log g 2 ) δ ≤ Cµ(|∇g| 2 ), g ∈ C 1 b (X ), µ(g 2 ) = 1 implies W 2 (ν, µ) 2 ≤ C 2/δ (2 -δ) 2 I(ν|µ) (2-δ)/δ . Inversely if Ric + Hess V is bounded below, then W 2 (ν, µ) 2 ≤ CI(ν|µ) (2-δ)/δ implies µ(g 2 log g 2 ) δ ≤ C ′ µ(|∇g| 2 ), g ∈ C 1 b (X ), µ(g 2 ) = 1 for some C ′ > 0. Proof. For β(r) := r δ /C we have γ(r) = √ C 2-δ r (2-δ)/2 so that β • γ -1 (s) = 1 C 2 -δ √ C s 2δ/(2-δ) = (2 -δ) 2δ/(2-δ) C 2/(2-δ) s 2δ/(2
Next, for α(r) = r 2δ/(2-δ) C -δ/(2-δ) , we have

α -1 (s) = √ Cs (2-δ)/2δ . Since 2 -δ ≤ 1, Theorem 4.1(2) implies µ(g 2 log g 2 ) ≤ 2 √ 2Cµ(|∇g| 2 ) 1/δ + KCµ(|∇g| 2 ) (2-δ)/δ , µ(g 2 ) = 1. (4.5) W* Since 2 -δ ≤ 1, this implies µ(g 2 log g 2 ) ≤ C ′ µ(|∇g| 2 ) 1/δ , µ(g 2 ) = 1, µ(|∇g| 2 ) ≥ 1 (4.6) W*2
for some C ′ > 0. Moreover, since δ ≥ 1, (4.5) implies the defective log-Sobolev inequality µ(g 2 log g 2 ) ≤ C 1 µ(|∇g| 2 ) + C 2 , µ(g 2 ) = 1 for some C 1 , C 2 > 0, which in particular implies that the spectrum of L := ∆ + ∇V is discrete (see e.g. [START_REF] Wang | Functional inequalities for empty essential spectrum[END_REF][START_REF] Wu | Uniformly integrable operators and large deviations for Markov processes[END_REF]), and hence the Poincaré inequality holds since λ 0 = 0 is the simple eigenvalue due to the connection of the manifold. Thus, the strict log-Sobolev inequality µ(g 2 log g 2 ) ≤ C ′ µ(|∇g| 2 ), µ(g 2 ) = 1 for some constant C ′ > 0. The proof is then completed by combining this with (4.6).

Example 4.3. Let Ric be bounded below, and ρ o the Riemannian distance function to a fixed point o ∈ E. Let V ∈ C(X ) such that Vaρ θ o is bounded for some a > 0 and θ ≥ 2. Then (4.2) holds for α(r) = Cr 2(θ-1) for some C > 0, i.e. CW 2 (ν, µ) 2(θ-1) ≤ I(ν|µ), ν ∈ M 1 (X ).

(4.7) W*3 The power 2(θ -1) is sharp, i.e. the above inequality does not hold if this power is replaced by any larger number, as seen from Proposition 2.3. Indeed, by [21, Corollaries 2.5 and 3.3], we have µ(g 2 log 2(θ-1)/θ (g 2 + 1)) ≤ C 1 µ(|∇g| 2 ) + C 2 , µ(g 2 ) = 1 holds for some C 1 , C 2 > 0. By Jensen's inequality we obtain µ(g 2 log g 2 ) 2(θ-1)/θ ≤ µ(g 2 log(g 2 + 1))

2(θ-1)/θ ≤ C 1 µ(|∇g| 2 ) + C 2 , µ(g 2 ) = 1.
Combining this with the log-Sobolev inequality as in the proof of Corollary 4.2, we obtain µ(g 2 log g 2 ) 2(θ-1)/θ ≤ µ(g 2 log(g 2 + 1)) 2(θ-1)/θ ≤ C ′ µ(|∇g| 2 ), µ(g 2 ) = 1 for some constant C ′ > 0. According to Corollary 4.2, this implies (4.7).

Φ-Sobolev inequality and concentration inequality for unbounded observables under integrability condition

Let Φ : R + → [0, +∞] be a convex, increasing and left continuous function with Φ(0) = 0, such that lim r→+∞ Φ(r) r = +∞.

(5.1) 51

Consider the Orlicz space L Φ (µ) of those measurable functions g on X so that its gauge norm

N Φ (g) := inf{c > 0; Φ(|g|/c)dµ ≤ 1}
is finite, where the convention inf ∅ := +∞ is used. The Orlicz norm of g is defined by

g Φ := sup{ gu dµ; N Ψ (u) ≤ 1}
where

Ψ(r) := sup λ≥0 (λr -Φ(λ)), r ≥ 0 (5.2)
is the convex conjugation of Φ. The so called (defective) Φ-Sobolev inequality says that for some two nonnegative constants

C 1 , C 2 ≥ 0 g 2 Φ ≤ C 1 E(g, g) + C 2 µ(g 2 ), ∀g ∈ D(E), µ(g 2 ) = 1. ( 5
f -1 Φ ≤ C ′ 1 I(f µ|µ) 2 + C ′ 2 I(f µ|µ) (5.5) thm51a where C ′ 1 = (C 1 + 2C 2 C P )C 1 , C ′ 2 = (C 1 + 2C 2 C P ) • 4C 2 ;
or equivalently for any observable u ∈ L Ψ (µ) (Ψ being the convex conjugation of Φ given above) so that N Ψ (u) ≤ 1 and for all t, r > 0,

P ν 1 t t 0 u(X s )ds > µ(u) + r ≤ dν dµ 2 exp -t • 4C ′ 1 r 2 + (C ′ 2 ) 2 -C ′ 2 2C ′ 1 .
(5.6) thm51b (b): for any µ-probability density f ,

sup u∈bB:N Ψ (u 2 )≤1 (f -1)udµ ≤ 2(C 1 + 4C 2 C P )I(f µ|µ) (5.7) thm51c
or equivalently for any u ∈ L 1 (µ) such that u 2 ∈ L Ψ (µ),

P ν 1 t t 0 u(X s )ds > µ(u) + r ≤ dν dµ 2 exp -t r 2 2(C 1 + 4C 2 C P ) u 2 Ψ
, ∀t, r > 0.

(5.8) thm51d (c): More generally for any p ∈ [1, +∞), there is a constant κ > 0 depending only of p, C 1 , C 2 , C P such that for any µ-probability density f ,

α p sup u∈bB:N Ψ (u 2 )≤1 (f -1)udµ ≤ I(f µ|µ) (5.9) thm51e
where α p (r) = (1 + r 2 /κ) p/2 -1; or equivalently for any u ∈ L 1 (µ) such that N Ψ (|u| p ) ≤ 1,

P ν 1 t t 0 u(X s )ds > µ(u) + r ≤ dν dµ 2 exp -t[1 + r 2 /κ) p/2 -1] , ∀t, r > 0. (5.10) thm51f
As there are numerous practical criteria for the Φ-Sobolev inequality (see e.g. [START_REF] Chen | Eigenvalues, inequalities, and ergodic theory[END_REF][START_REF] Ledoux | The Concentration of Measure Phenomenon[END_REF][START_REF] Wang | Functional inequalities, Markov Semigroup and Spectral Theory[END_REF]), this theorem is very useful and gives different concentration behaviors for 1 t t 0 u(X s )ds, according to the integrability condition |u| p ∈ L Ψ (µ) where p ∈ [1, +∞).

This result generalizes the sharp concentration inequality under the log-Sobolev inequality in Wu [START_REF] Wu | A deviation inequality for non-reversible Markov processes[END_REF]. For applications of Φ-Sobolev inequalities in large deviations see Wu and Yao [START_REF] Wu | Large deviations for Markov processes via Φ-Sobolev inequalities[END_REF].

Remarks 5.2. As the l.h.s. of (5.5), (5.7) and (5.9) are the transportation cost T V (f µ, µ), with V = {(u, u); u ∈ bB, N Ψ (|u| p ) ≤ 1}, p = 1, 2, p ≥ 1 respectively, so they are the transportation-information inequality. In this point of view, the equivalence between (5.5) and (5.6) in part (a), that between (5.7) and (5.8) in part (b) and that between (5.9) and (5.10) in part (c) are all immediate from Theorem 1.1 (the passage from bounded u to general u in the concentration inequalities (5.6), (5.8) and (5.10) can be realized easily by dominated convergence).

Remarks 5.3. The concentration inequalities (5.6), (5.8) and (5.10) are all sharp in order. Indeed consider the Ornstein-Uhlenbeck process on R generated by Lf = f ′′xf ′ : the Φ-Sobolev inequality (X t ) holds with Φ(r) = (1 + r) log(1 + r) and µ = N (0, 1). Consider u(x) := |x| 2/p where p ≥ 1. Then u p ∈ L Ψ (µ), and 1 t t 0 u(X s )ds = 1 t t 0 |X s | p ds possess exactly the concentration behaviors exhibited by the r.h.s. of (5.10) for large deviation value r, and for small deviation value r of order 1/ √ t if t is large enough (by the central limit theorem).

Proof of Theorem 5.1. As explained in the previous remarks, (5.6) (resp. (5.8); (5.10)) is equivalent to (5.5)(resp. (5.7), (5.9)), all by Theorem 1.1. It is not surprising that the proof relies on the ideas first used in [START_REF] Bolley | Weighted Csiszar-Kullback-Pinsker inequalities and applications to transportation inequalities Ann[END_REF], establishing criterions for W 1 H under integrability criteria. Note also that the reader may easily adapt the proof to use conditions on F -Sobolev inequalities (equivalent to some Orlicz-Poincaré inequality) and integrability on u (rather than Orlicz norm of u). Plugging those two estimates into the previous inequality we get (5.7). Substituting those estimates into the first inequality we get

  (ii): Relative entropy or Kullback's information of ν w.r.t. µ H(ν|µ) := dν dµ log dν dµ dµ, if ν ≪ µ; +∞, otherwise. (1.2) def-entropy-a (iii): The Fisher information of ν w.r.t. µ: Date: First Version: March 2008.

  thm-GLWY Theorem 1.1. ([START_REF] Guillin | Transport-information inequalities for Markov processes (I)[END_REF]) Let ((X t ) t≥0 , P µ ) be the µ-symmetric and ergodic Markov associated with the Dirichlet form (E, D(E)), α : R + → [0, +∞] a left-continuous non-decreasing convex function with α(0) = 0, and V as above.The following properties are equivalent: (a): µ satisfies the transport-information inequality (α-T V I).(b): For all (u, v) ∈ V and all λ ≥ 0

  Lip,P = (-L) -1 Lip = σ 2 . Therefore µ satisfies W 1 I(C) with C = c Lip = σ 2 by Theorem 3.1. Furthermore C = c Lip = σ 2 is also the best constant in W 1 I(C). Indeed W 1 I(C) =⇒ W 1 H(C) and the best constant in W 1 H(C) of µ is C = σ 2 . In other words Theorem 3.1 produces the exact best constant C in W 1 I(C) for this example. Example 3.5. (Reflected Brownian Motion) Consider the reflected Brownian Motion X D t on the interval X = [0, D] (D > 0) equipped with the usual Euclidean metric, whose generator is given by Lf = f ′′ with Neumann boundary condition at 0, D. The unique invariant measure µ is the uniform law on [0, D]. For every g ∈ C 2 b ([0, D]) with D 0 g(x)dx = 0, the solution G of the Poisson equation -LG = g satisfies

  (d ρa is the metric associated with the carré-du-champs operator of the diffusion), if C(ρ a ) < +∞, then µ satisfies W 1 I(C(ρ a )) on ((x 0 , y 0 ), d ρa ).

  .3) 52 Under the assumption of the Poincaré inequality with the best constant C P , (5.3) can be transformed into the following tight version (gµ(g)) 2 Φ ≤ (C 1 + C 2 C P )E(g, g), ∀g ∈ D(E) (5.4) 53 called sometimes Orlicz-Poincaré inequality. thm51 Theorem 5.1. Assume the Φ-Sobolev inequality (5.3) and the Poincaré inequality with constant C P . Then (a): for any µ-probability density f ,

2 Φ

 2 (a) For (5.5) we may assume that I(f µ|µ) is finite, i.e., √ f ∈ D(E) (and thenI(f µ|µ) = E( √ f , √ f )). For any u ∈ L Ψ (µ) with N Ψ (u) ≤ 1, we have by Cauchy-Schwartz |(f -1)u|dµ = | f -1|( f + 1)|u|dµ ≤ ( f -1) 2 |u|dµ ( f + 1) 2 |u|dµ ≤ ( f -1) 2 Φ ( f + 1)But by the assumed Φ-Sobolev inequality (5.3),( f -1) 2 Φ ≤ C 1 E( f , f ) + C 2 ( f -1) 2 dµ and ( √ f -1) 2 dµ = 2(1µ( √ f )) ≤ 2V ar µ ( √ f ) ≤ 2C P E( √ f , √ f ); moreover ( f + 1) 2 Φ ≤ C 1 E( f , f ) + C 2 ( f + 1) 2 dµand ( √ f + 1) 2 dµ ≤ 4. Thus we get|(f -1)u|dµ ≤ (C 1 + 2C 2 C P )E( f , f )(C 1 E( f , f ) + 4C 2 )where (5.5) follows by recalling I(f µ|µ) For any u so that N Ψ (u 2 ) ≤ 1 we use now differently Cauchy-Schwartz inequality to get:|(f -1)u|dµ ≤ ( f -1) 2 dµ ( f + 1) 2 u 2 dµBut as noticed in the proof of (a), ( f -1) 2 dµ ≤ 2V ar µ ( f ) ≤ 2 min{C P I(f µ|µ), 1} and ( f + 1) 2 u 2 dµ ≤ ( f + 1) 2 Φ ≤ C 1 I(f µ|µ) + 4C 2 .

  (c). Letting q := p/(p -1) we have by Hölder's inequality,|(f -1)u|dµ ≤ (µ(|f -1|)) 1/q |f -1||u| p dµ1/p Note that µ(|f -1|) ≤ 2 and by [13, Theorem 3.3], (µ(|f -1|)) 2 ≤ 4V ar µ ( f ) ≤ 4C P I where I := I(f µ|µ). On the other hand by part (a), |f -1||u| p dµ ≤ f -1 Φ ≤ C ′ 1 I 2 + C ′ 2 I.

  and W 1 I(C) is equivalent to the Gaussian deviation inequality (1.6) with α(r) = r 2 /(4C 2 ) for the Lipschitzian observable u with Lipschitzian coefficient u Lip ≤ 1, which generalizes the well known Hoeffding's inequality in the i.i.d. case.

  3/2 ≤ εI + I 2 /ε in(3.6), we obtain (3.7) by Theorem 1.1.Remarks 3.2. The W 1 I(σC) inequality (3.5) is due to Guillin and al.[START_REF] Guillin | Transport-information inequalities for Markov processes (I)[END_REF], but the method therein is based on the Lyons-Meyer-Zheng forward-backward martingale decomposition. The argument here is simpler and direct, and yields the stronger Cheeger type's isoperimetric inequality(3.4).Remarks 3.3. Letting δ be close to 0, we see that (3.7) is sharp for small r by the central limit theorem.Set C Lip,0 = {g ∈ C Lip,0 ; µ(g) = 0}. Under the Lipschitzian spectral gap condition (3.3), the Poisson operator (-L) -1 : C Lip,0 → C Lip,0 is a well defined bounded linear operator w.r.t. the Lipschitzian norm, and the best constant C in (3.3) is the Lipschitzian norm (-L)-1 Lip and will be denoted by c Lip,P (the index P is referred to Poincaré). We now present four examples for illustrating usefulness of Theorem 3.1.

	Example 3.4. (Ornstein-Uhlenbeck process) Consider the Ornstein-Uhlenbeck pro-cess dX t = √ 2dB

tσ -2 X t dt on X = R where σ > 0 and B t is the standard Brownian motion on R. Its unique invariant measure is µ

The last term is less than κ[(1 + I) 2/p -1] for some constant κ > 0. That yields to (5.9).

Let us finally relate previous inequalities to usual α -W I inequalities.

cor51 Corollary 5.4. Assume the Φ-Sobolev inequality (5.3) and the Poincaré inequality. Assume that d p (x, x 0 ) ∈ L Ψ (µ) for some p ≥ 1 where Ψ is the convex conjugation of Φ. Then there are positive constants C ′ 1 , C ′ 2 and κ such that for all ν ∈ M 1 (X ),

and when p ≥ 2,

Proof. Recall the following fact ([19, Proposition 7.10]),

Then this corollary follows directly from Theorem 5.1.


Julien Devillez

Tadashi Kawai

Denis Audo
email: audo@nigpas.ac.cn

Astacus multicavatus Bell, 1863 is a marine lobster (Decapoda: Erymoidea: Erymidae), not a freshwater crayfish

Keywords: Astacoidea, Cretaceous, fossil crayfishes, Speeton Clay

. It so appears that Eryma multicavatum (Bell, 1863) comb. nov. is not a freshwater crayfish but a marine representative of Erymidae.

Two co-occurring isolated chelae, also from the Speeton Clay, may correspond to larger, older specimens of E. multicavatum comb. nov.

INTRODUCTION

Crayfishes (Decapoda, Astacidea), comprising Astacoidea Latreille, 1802 and Parastacoidea Huxley, 1879, form a diverse group of freshwater lobster-like decapod crustaceans with ~669 modern species (several additional freshwater crayfishes have been described every year, for several years, so this figure is probably an underestimate) and 13 extinct fossil species [START_REF] Feldmann | New Eocene crayfish from the McAbee beds in British Columbia: first record of Parastacoidea in the Northern Hemisphere[END_REF][START_REF] Crandall | An updated classification of the freshwater crayfishes (Decapoda: Astacidea) of the world, with a complete species list[END_REF]. The ancestor of freshwater crayfishes derived from marine lobsters that were initially thought to have lived during the Permian or Triassic, both from palaeontological evidence [START_REF] Amati | A new family of Triassic lobsters (Decapoda: Astacidea) from British Columbia and its phylogenetic context[END_REF]Schram & Dixon, 2003;[START_REF] Porter | Model-based multi-locus estimation of decapod phylogeny and divergence times[END_REF] and molecular analyses [START_REF] Crandall | Global diversity of crayfish (Astacidae, Cambaridae, and Parastacidae-Decapoda) in freshwater[END_REF][START_REF] Toon | Gondwanan radiation of the Southern Hemisphere crayfishes (Decapoda: Parastacidae): evidence from fossils and molecules[END_REF][START_REF] Bracken-Grissom | The emergence of lobsters: Phylogenetic relationships, morphological evolution and divergence time comparisons of an ancient group (Decapoda: Achelata, Astacidea, Glypheidea, Polychelida)[END_REF]. The discovery of Late Permian burrows [START_REF] Hasiotis | Ichnology of Triassic and Holocene cambarid crayfish of North America: an overview of burrowing behaviour and morphology as reflected by their burrow morphologies in the geological record[END_REF] confirms this origin is at least Permian, and one recent molecular analysis [START_REF] Wolfe | A Phylogenomic framework[END_REF] points to an even older origin. This evidence provides an evolutionary scenario for dispersal distribution of freshwater crayfishes [START_REF] Ďuriš | Evolution and historical biogeography of crayfishes[END_REF]. A single marine ancestor species of the freshwater crayfishes invaded freshwater during the Permian or earlier; at that time our current five continents were aggregated as the supercontinent Pangea [START_REF] Dercourt | Atlas téthys palaeoenvironmental maps: explanatory notes[END_REF]. Despite this long history, freshwater crayfishes are extremely rare in the fossil record [START_REF] Feldmann | New Eocene crayfish from the McAbee beds in British Columbia: first record of Parastacoidea in the Northern Hemisphere[END_REF]. The number of extinct fossil species is less than 2% that of extant species [START_REF] Crandall | An updated classification of the freshwater crayfishes (Decapoda: Astacidea) of the world, with a complete species list[END_REF]. This is particularly unfortunate since the distribution of modern crayfishes is considered as a textbook example of vicariance, diversification events linked to the apparition of barriers to dispersal [START_REF] Ďuriš | Evolution and historical biogeography of crayfishes[END_REF][START_REF] Pârvulescu | Introducing a new Austropotamobius crayfish species (Crustacea, Decapoda, Astacidae): A Miocene endemism of the Apuseni Mountains, Romania[END_REF][START_REF] Scholtz | Freshwater crayfish evolution[END_REF].

With such a scarce fossil record, and the stakes to interpret correctly the rare occurrences that are known, it is important to examine correctly what is already known. We review Astacus multicavatus Bell, 1863 from the Hauterivian of Speeton Clay. This putative crayfish occurrence [START_REF] Feldmann | New Eocene crayfish from the McAbee beds in British Columbia: first record of Parastacoidea in the Northern Hemisphere[END_REF] had drawn our attention for several reasons: first, it is assigned to Astacus Fabricius, 1775, a genus that initially embraced the majority of European species before many of these were moved to other genera (i.e. the earliest subdivision of Astacus by Huxley, 1879, leading to the recognition of Parastacus Huxley, 1879, or more recently [START_REF] Bott | Die Flußkrebse Europas (Decapoda, Astacidae)[END_REF] who recognized subgenera within Astacus leading to the erection of Pacifastacus [START_REF] Bott | Die Flußkrebse Europas (Decapoda, Astacidae)[END_REF] including fossil species [START_REF] O'flynn | Systematic revision and palaeobiology of Emplastron edwardsi (Van Straelen, 1928) gen. et comb. nov. (Crustacea, Decapoda, Astacidae) entombed within travertine, from Sézanne, France[END_REF]; and also because this species was described from the Speeton Clay, a marine formation [START_REF] Lott | The stratigraphy of the Lower Cretaceous Speeton Clay Formation in a cored borehole off the coast of north-east England[END_REF]. This species is important because it was used as a calibration fossil to represent Astacus in a divergence time analysis [START_REF] Bracken-Grissom | The emergence of lobsters: Phylogenetic relationships, morphological evolution and divergence time comparisons of an ancient group (Decapoda: Achelata, Astacidea, Glypheidea, Polychelida)[END_REF]. Our observations show that this species is not a freshwater crayfish, but belongs to Erymidae, an extinct group of marine lobster-like crustaceans (Devillez et al., 2019).

MATERIAL AND METHODS

Sampling

Our study is based upon 14 specimens, including the syntypes of Astacus multicavatus [START_REF] Bell | Monograph of the fossil malacostracous crustacea of Great Britain. Part IIof those found in the Greensand and Gault: i[END_REF]Eryma sulcatum Harbort, 1905, the neotype, casts of the lectotype and paralectotypes, and additional specimens. The specimens are housed in Museum für Naturkunde, Berlin, Germany (MFN), Natural History Museum, London, UK (NHMUK), and Sedgwick Museum of Earth Sciences, Cambridge, UK (SM) (see Table 1).All photographs were taken in natural light, without cross-polarisation. The systematics follows that proposed by Devillez et al. (2019).

RESULTS

Systematic palaeontology

Malacostraca Latreille, 1802

Decapoda Latreille, 1802 Pleocyemata [START_REF] Burkenroad | The evolution of the Eucarida (Crustacea, Eumalacostraca) in relation to the fossil record[END_REF]Reptantia Bouvier, 1917 Astacidea Latreille, 1802

Erymoidea Van Straelen, 1925 Erymidae Van Straelen, 1925 Eryminae Van Straelen, 1925 Eryma [START_REF] Meyer | Neue Gattungen fossiler krebse aus Gebilden von Bunten Sandstein bis in die Kreide[END_REF] Eryma multicavatum (Bell, 1863) comb. nov. (Figs. 1-3) Astacus multicavatus [START_REF] Bell | Monograph of the fossil malacostracous crustacea of Great Britain. Part IIof those found in the Greensand and Gault: i[END_REF], 31, pl. 9 figs. 7, 8. -Glaessner, 1929: 61. -Schweitzer et al., 2010: 32. -Feldmann et al., 2011: table 1. -Bracken-Grisson, 2014: 465. -Crandall & De Grave, 2017: 621. Astacus? multicavatus -Woodward, 1877: 9.

Eryma sulcata [START_REF] Harbort | Die Fauna der Schaumburg-Lippe'schen Kreidemulde. Abhandlungen der Königlich Preussischen Geologischen Landesanstalt und Bergakademie[END_REF]nov. syn.: 15, pl. 1, fig. 11, pl. 11, fig. 4. -Glaessner, 1929: 159. -Woods, 1930: 80, pl. 22, figs. 5-7. -Van Straelen, 1936: 9. -Förster, 1966: 124, fig. 23, pl. 17 figs. 2, 4. -Taylor, 1979: 34. -Feldmann & Titus, 2006: 64. Eryma cf. sulcata -Aguirre- Urreta & Ramos, 1981: 610, pl. 1, fig. a. Eryma sulcatum -Schweitzer et al., 2010: 25. -Karasawa et al., 2013: 102 -Devillez et al., 2016 : 521, figs. 4I-4L;2017: 793;[START_REF] Pârvulescu | Introducing a new Austropotamobius crayfish species (Crustacea, Decapoda, Astacidae): A Miocene endemism of the Apuseni Mountains, Romania[END_REF]: 368, fig. 7F, G. -Devillez & Charbonnier, 2017: 3, figs. 3I-L. Astacodes falcifer (pro parte) - Bell, 1863: pl. 9, figs. 7, 8. Lectotype: NHMUK.In.61411 (Figs. 1A-D), selected herein from the two syntypes.

Paralectotype: NHMUK.In.61410 (Figs. 1E-H).

Type material of Eryma sulcatum: neotype SM.B11437; casts of lost original material lectotype NHMUK.In.27305 and paralectotypes NHMUK.In.27307, NHMUK.In.27308, NHMUK.In.27309, NHMUK.In.27310.

Examined material: see full list in Table 1.

Type locality and formation: Speeton, North Yorkshire, England, UK: Hauterivian of the Speeton Clay Formation.

Occurrence: Hauterivian of Speeton and Hilston (England, UK) and Stadthagen (Lower Saxony, Germany).

Diagnosis: fFusiform intercalated plate delimited by grooves; deep cervical groove, intercepting dorsal margin at ca 66° angle; short gastro-orbital groove originating as slight median inflexion of cervical groove; deep branchiocardiac groove, strongly inclined, subparallel to postcervical groove; massive P1 propodus, relatively thick, with rounded inner and outer margins; narrow dactylar bulge; fingers, almost straight, with short teeth on occlusal margin; branchial region with fine tubercles (heterogeneous ornamentation); tuberculate intercalated plate.

Emended description:Cephalothorax: Subcylindrical carapace (cephalothoracic shield); dorsal margin of cephalic region slightly down-curved; short rostrum with small lateral spines; fusiform intercalated plate delimited by grooves; post-orbital area narrowing dorsally; deep cervical groove, intercepting dorsal margin at ca 66° angle, joined to antennal groove; short gastro-orbital groove originating as slight median inflexion of cervical groove; postcervical groove slightly convex forward, not joined to dorsal margin, joined medially to branchiocardiac groove; deep branchiocardiac groove, strongly inclined, subparallel to postcervical groove, not joined to dorsal margin, joined to hepatic groove; concavo-convex hepatic groove; slightly inflated ω bulge, delimited ventrally by narrow, shallow depression extending between antennal and hepatic grooves; slightly inflated χ bulge; inferior groove curved backward.

Pleon and telson: Pleonal somites with subrectangular dorsal part of terga and sub-triangular tergopleura; longitudinal bulge above tergopleura basis; tergopleura with post tergopleural spine; ventral margin of tergopleura pointing backward; subrectangular telson with longitudinal carina.

Eyes and cephalic appendages: Small stalked eyes with numerous small ommatidia.

Thoracic appendages: Chelate P1 (thoracopod 4); massive P1 propodus, relatively thick with rounded inner and outer margins, longer than wide, inner and outer margins slightly curved in outline; inflated, narrow dactylar bulge; thin fingers, almost straight, of almost same width, with short teeth on occlusal margin; dactylus with longitudinal carina; short, rounded P1 carpus; P1 merus with a strong process on distal extremity of external margin; slender P2-P5 (thoracopods 5-8).

Pleonal appendages: Wide, rounded uropods with median longitudinal carina.

Ornamentation: Carapace and pleonal somites densely covered by small tubercles, pits with irregular outline; branchial region with finer tubercles; tuberculate intercalated plate; postorbital area slightly granular; cephalic region with oblique orbital row of tubercles ending by strong orbital spine (antennal row absent); telson and uropods ornamented with small rounded punctuations (cupula); P1 palm densely covered by small prominent tubercles preceded by pits; dactylar bulge with 2 spines, pointed forward; fingers covered by large rounded pits; P1 carpus with same ornamentation than propodus, with strong spine at distal extremity of external margin; P1 merus with same ornamentation of P1 carpus, spines on dorsal margin; P2-P5 with small punctation.

Remarks: Astacus multicavatus Bell, 1863 is described based upon two syntypes, NHMUK.In.61410 and NHMUK.In.61411. We herein designate NHMUK.In.61411 as the lectotype of A. multicavatus since it preserves more diagnostic characters. Consequently, the second syntype NHMUK.In.61410 is herein considered the paralectotype.

Eryma sulcatum [START_REF] Bell | Monograph of the fossil malacostracous crustacea of Great Britain. Part IIof those found in the Greensand and Gault: i[END_REF] is assigned to Eryma Meyer, 1840 based upon its fusiform intercalated plate (notably lacking in crayfishes), deep cervical groove joined to median line, short gastro-orbital groove, postcervical and branchiocardiac grooves subparallel (branchiocardiac diverging from postcervical in crayfishes), the sinuous (also termed concavo-convex) hepatic groove joined to inferior groove and inflated ω area, orbital row of tubercles with a large distal spine, large antennal spine, compressed P1 (thoracopod 4) propodus with narrow inner and outer margins, narrow P1 dactylar bulge; P1 fingers (dactylus and index) longer than the palm (propodus not including the index) and index wider than dactylus (diagnostic characters of Eryma following [START_REF] Devillez | Review of the Early Cretaceous erymid lobsters (Crustacea: Decapoda) from the Western Tethys[END_REF]Devillez et al. ( , 2017)).

Eryma sulcatum [START_REF] Harbort | Die Fauna der Schaumburg-Lippe'schen Kreidemulde. Abhandlungen der Königlich Preussischen Geologischen Landesanstalt und Bergakademie[END_REF] is considered herein as synonymous with Astacus multicavatus Bell, 1863 due to the absence of observable differences between the two species, and common origin and age of their type material, namely the lectotype and paralectotype of A. multicavatus and the neotype of E. sulcatum (designated by [START_REF] Devillez | Review of the Early Cretaceous erymid lobsters (Crustacea: Decapoda) from the Western Tethys[END_REF] Together with other specimens of Eryma multicavatum comb. nov. from England, two isolated P1 chelae were collected at Speeton in the Speeton Clay Formation (NHMUK.52081; Fig. 4). These chelae have a compressed and a slightly trapezoidal propodus, bearing long fingers progressively narrowing to their distal extremity. These features are diagnostic of Eryma. The propodus with slightly curved margins and the ornamentation made of numerous small tubercles and small depressions, are characteristics shared with E. multicavatum. The shape of the fingers, however, is different. In the two specimens, the index is strongly curved and wider than the straight one of E. multicavatum. These two P1 chelae are around twice as large as those of the type material of E. multicavatum. As such, these isolated chelae could be from older, more mature specimens of E. multicavatum, and the morphological differences could be the result of an allometric growth that modified their general shape. The isolated P1 chela from Hilston (MFN.MB.A.2853; Fig. 3D) is otherwise closer to the size of the unidentified chelae (NHMUK.52081; Fig. 4). The morphology of this specimen does not resemble closely to that of the unidentified P1 chelae, but it exhibits a morphology very similar to the type material of E. multicavatum. In conclusion, considering our previous remarks, and because no other species of Eryma has been reported in the same formation, we consider that these two isolated P1 chelae have affinities with E. multicavatum but cannot be related to this species. We therefore identify them as Eryma aff. E. multicavatum. The difference between these two types of claws is similar to the'form I' (straight fingers) and 'form II' (curved fingers) discussed by [START_REF] Hyžný | Intraspecific variation and taphonomy of a new erymid lobster (Crustacea: Decapoda) from the Middle Jurassic of Belmont (Beaujolais, France)[END_REF], [START_REF] Devillez | Review of the Early Cretaceous erymid lobsters (Crustacea: Decapoda) from the Western Tethys[END_REF], and Devillez & Charbonnier (2017). If more complete specimens of Eryma with the curved claws are found in the Speeton Clay, it may help to better understand the nature of this difference in morphology.

As the origin of crayfishes and the time at which they diverged from their marine relatives is still an ongoing question [START_REF] Ďuriš | Evolution and historical biogeography of crayfishes[END_REF], our work clears up one possible misunderstanding about the evolutionary history of crayfish: Eryma multicavatum is not a late case of marine crayfish but simply a member of the extinct lobster-like Erymidae.

Our study has also a broader importance to better understand the evolution of decapods as a whole. Astacus multicavatus, interpreted as a crayfish (Astacus), was certainly used as a calibration point in Bracken-Grisson et al. (2014), it may therefore have affected adversely the estimate for the age of clades presented in this particular study.

  due to the loss of the original syntypes). It is noteworthy that Devillez et al. (2016) had already considered the two syntypes (lectotype and paralectotype herein) of A. multicavatus as specimens of E. sulcatum without realizing their status. In application to the principle of priority (ICZN, 1999: article 23), the name A. multicavatum Bell, 1863 has the priority over E. sulcata Harbort, 1905. DISCUSSION Because of the preservation of eyes, carapace (shield), pleon, telson, pereiopods (thoracopods 4-8), and uropods, Eryma multicavatum comb. nov. is the most complete species of Eryma from Lower Cretaceous deposits. It is also one of the most complete erymoids, together with Enoploclytia seitzi Glaessner, 1932 (carapace, pleon and first pereiopod) and Palaeastacus uranusiensis Devillez & Charbonnier, 2019 (carapace, pleon, telson, first and second pereiopods and uropods).

  evolutionary timeline, and genomic resources for comparative studies of Decapod crustaceans. Proceedings of the Royal Society B, 286: 1901 [doi:10.1098/rspb.2019.0079]. Woods, H. 1925-1931. A monograph of the fossil macrurous crustacean of England. The Palaeontographical Society, London.Woodward, H. 1877. Catalogue of British fossil Crustacea, with their synonyms and the range in time of each genus and order.Taylor and Francis, London.

FIGURE CAPTIONS Figure 1 Figure1.

 1 FIGURE CAPTIONS

Figure 2 .

 2 Figure 2. Neotype SM B11437 of Eryma sulcatum Harbort,1905 from the Hauterivian of

Figure 3 .

 3 Figure 3. Additional specimens of Eryma multicavatum(Bell, 1863) comb. nov. specimen

Figure 4

 4 Figure 4

Figure 4 .

 4 Figure 4. Isolated P1 chelae NHMUK 52081 of Eryma aff. multicavatum from the

ACKNOWLEDGMENTS

We are indebted to Claire Mellish (Natural History Museum, London), Liz Harper and Matt Riley (Sedgwyck Museum, Cambridge, England, UK), Christian Neumann and Andreas Abele (Museum für Naturkunde, Berlin, Germany) for providing access to the specimens, and to Kevin Webb (Natural History Museum, London) for the photographs of E. multicavatum.

We acknowledge our two reviewers Guenter Schweigert and Sylvain Charbonnier as well as the assistant editor Rodney M. Feldmann for their valuable comments on an earlier version of the manuscript.


PharmD Elise Pape

MD Delphine Michel

PharmD Julien Scala-Bertola
email: j.scala-bertola@chu-nancy.fr

PharmD Thomas Schiestel

Alexandre Harlé

PharmD Stéphane Bouchet

Audrey Contet

Cécile Pochon

Nicolas Gambier

MD  Audrey Contet

Effect of esomeprazole on the oral absorption of dasatinib in a patient with Philadelphia-positive acute lymphoblastic leukemia

Keywords: Dasatinib, Esomeprazole, Drug-drug interaction, Pharmacokinetics, Leukemia

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ABSTRACT

Dasatinib is of particular interest in chronic myelogenous leukemia or in philadelphia chromosome-positive acute lymphoblastic leukemia resistant to imatinib. However, its efficacy is reported to be decreased when used simultaneously with gastric acid pH modulators. We report herein the first case reporting a drug-drug interaction between dasatinib and esomeprazole in a patient with Philadelphia-positive acute lymphoblastic leukemia leading to a decrease in dasatinib oral absorption. An extensive review of drug-drug interactions between tyrosine kinase inhibitors used in leukemia and proton pump inhibitors was performed in order to provide some strategies to avoid or to limit this drug-drug interaction.

We describe herein the first reporting case of a drug-drug interaction between dasatinib and esomeprazole in a patient with Philadelphia-positive acute lymphoblastic leukemia leading to a decrease in dasatinib oral absorption.

The patient was a 16-year-old man (1.76 m, 61.5 kg) who benefited from allogeneic stem cell transplantation after a myeloablative conditioning regimen based on 12-Gy total body irradiation and etoposide (60 mg/kg). This case is presented in the context of progression of imatinib-resistant chronic myelogenous leukemia (CML) in acute lymphoblastic leukemia (ALL) with B-cell lineage. One month after stem cell transplantation, Bcr/Abl transcripts were undetectable and chimerism (on total leucocytes) was assessed at 98 % donor cells in blood. Two months later, the patient experienced a molecular relapse with 0.06 % and 0.03 % Bcr/Abl transcripts in blood and bone marrow, respectively. The degree of chimerism was assessed at 99 % donor cells in bone marrow. Immunosuppressive treatment was quickly diminished and dasatinib was introduced at a daily dose of 100 mg but was thereafter decreased to 70 mg per day because of a severe leukopenia (white blood cells at 690/mm 3 ). At this dose, plasma trough concentration of dasatinib was 5.1 ng.mL -1 (Normal value <2.5 ng.mL -1 [START_REF] Miura | Therapeutic drug monitoring of imatinib, nilotinib, and dasatinib for patients with chronic myeloid leukemia[END_REF]) suggesting a suspected role of dasatinib in the occurrence of leukopenia. Despite the decrease in dasatinib dosing, a severe pancytopenia in a context of staphylococcus bacteraemia was observed and dasatinib was stopped. One week later, dasatinib was restarted at a daily dose of 50 mg then increased to 60 mg. Four months after transplantation, as the immunosuppressive drugs were withdrawn, the degree of chimerism reached 100 % donor cells in blood, and Bcr/Abl transcripts were still detectable but not quantifiable. Despite a prolonged mild pancytopenia, the patient was treated with 50 mg per day of dasatinib to limit molecular relapse. In this context, a study of plasma concentrations of dasatinib was performed as previously described by Bouchet et al. using liquid chromatography-tandem mass spectrometry (limit of quantification of 0.1 ng.mL -1 ) [START_REF] Bouchet | Simultaneous determination of nine tyrosine kinase inhibitors by 96well solid-phase extraction and ultra performance LC/MS-MS[END_REF]. This revealed a maximal concentration (C max ) of 23.1 ng.mL -1 (Normal value >50.0 ng.mL -1 [START_REF] Miura | Therapeutic drug monitoring of imatinib, nilotinib, and dasatinib for patients with chronic myeloid leukemia[END_REF]) 4 hours after the administration of dasatinib (Figure 1). The estimated area under the curve between 0 and 6.5 hours (AUC 0-6.5h ) using the trapezoidal rule was 89.6 ng.h.mL -1 . The terminal elimination half-life (t 1/2 ) was estimated to be 4.9 h. From these results, we concluded that there was a decrease in dasatinib absorption, probably related to a drug-drug interaction influencing dasatinib pharmacokinetics. Drugs concomitantly taken with dasatinib by the patient included prednisolone (15 mg q.d.), valaciclovir (500 mg b.i.d.), cefuroxime (250 mg b.i.d.), ursodeoxycholic acid (200 mg b.i.d.), itraconazole (200 mg b.i.d.), human immunoglobulins (25 g monthly), pentamidine (300 mg monthly) and esomeprazole (40 mg q.d.). As itraconazole was simultaneously used with dasatinib, an increase could have been expected due to an inhibition of CYP3A4 by itraconazole but it was not observed. This result could be explained by a limited absorption of itraconazole due to the effect of esomeprazole on gastric acidity [START_REF] Li | Pharmacokinetic drug interaction profile of omeprazole with adverse consequences and clinical risk management[END_REF]. Thus, esomeprazole was considered as the main suspected drug in the decrease of dasatinib absorption and its dosing was first decreased from 40 mg to 20 mg once daily but finally stopped. A second drug concentration-time profile of dasatinib was established four days after esomeprazole discontinuation (Figure 1). This revealed an increase of C max and the estimated AUC 0-6.5h from 23.1 to 52.0 ng.mL -1 and from 89.6 to 130.6 ng.h.mL -1 , respectively. The increase of these two pharmacokinetics parameters with an unmodified estimated t 1/2 (4.7 versus 4.9 hours) and a 1 hour-earlier T max (3 versus 4 hours) after esomeprazole discontinuation confirmed the influence of esomeprazole on the oral absorption of dasatinib.

The influence of esomeprazole on the pharmacokinetics of dasatinib resulting in a 56 % decrease of C max and a 32 % decrease of AUC 0-6.5h was in the same order of magnitude as the study led by Bristol-Myers Squibb. This study found a decrease in C max and AUC 0→∞ of 41 % and 42 %, respectively when dasatinib was concomitantly used with omeprazole [START_REF] Squibb | The Effect of Omeprazole on the Pharmacokinetics of Dasatinib (BMS-354825) in Healthy Subjects[END_REF]. Besides omeprazole, the efficacy of dasatinib was also described to be encumbered by a decrease of its absorption when used and administered simultaneously with gastric acid pH modulators such as lansoprazole [START_REF] Takahashi | Influence of H2-receptor antagonists and proton pump inhibitors on dasatinib pharmacokinetics in Japanese leukemia patients[END_REF], famotidine or aluminium-magnesium hydroxides [START_REF] Eley | Phase I study of the effect of gastric acid pH modulators on the bioavailability of oral dasatinib in healthy subjects[END_REF]. In the case of CML or ALL with no resistance to imatinib, imatinib remains an excellent first choice and presents the advantage of being relatively spared by the influence of PPI (pKa value of 7.7).

In alternative first-line therapies such as dasatinib or nilotinib with an absorption encumbered by a concomitant use of PPI, PPI could be replaced by aluminium-magnesium hydroxides or H2-receptor antagonists administered 2 hours before or 2 hours after the administration of TKI, respectively [START_REF] Eley | Phase I study of the effect of gastric acid pH modulators on the bioavailability of oral dasatinib in healthy subjects[END_REF]. In case of a concomitant use of PPI with TKI, the current available pharmacokinetics and pharmacodynamics studies seem to describe dexlansoprazole, esomeprazole and rabeprazole as the three most potent PPIs to modify the pH of the gastric medium (intragastric pH value above 4 for significantly longer periods of time (>12-16h)) [START_REF] Shin | Pharmacokinetics and pharmacodynamics of the proton pump inhibitors[END_REF][START_REF] Kirchheiner | Relative potency of proton-pump inhibitors-comparison of effects on intragastric pH[END_REF]. Furthermore, considering the study of Röhss et al., the administration of PPI in a oncedaily regimen during a 24h-period revealed for all the tested PPIs (esomeprazole, lansoprazole, omeprazole, pantoprazole, rabeprazole) the two lowest values of intragastric pH measured around 16 hours and 24 hours after PPI intake, respectively [START_REF] Röhss | Esomeprazole 40 mg provides more effective intragastric acid control than lansoprazole 30 mg, omeprazole 20 mg, pantoprazole 40 mg and rabeprazole 20 mg in patients with gastro-oesophageal reflux symptoms[END_REF]. In consequence, the administration of PPI in a once-daily dose regimen 2 or 3 hours after the TKI intake (as advised for H2 receptor agonist [START_REF] Eley | Phase I study of the effect of gastric acid pH modulators on the bioavailability of oral dasatinib in healthy subjects[END_REF]) could probably constitute an alternative option to explore limiting the decrease in TKI absorption due to a concomitant use of PPI therapy.

Finally, dasatinib was stopped 3 weeks after esomeprazole discontinuation because of a pancytopenia and severe acute hepatitis suspected to be attributable to the treatment toxicity, but was attributed to hepatic grade 3 chronic graft versus host disease (GVHD) after liver biopsy. Nine months after transplantation, our patient is still alive and he has not received any TKI since, with no relapse of his leukemia at this time. Immunosuppressive treatment for hepatic GVHD is still on-going.

Figure 1 :

 1 Figure 1: Dasatinib blood concentrations in the presence and the absence of Esomeprazole after an administration of a 50 mg daily dose.