P Cattiaux

I Gentil

A Guillin

Weak logarithmic Sobolev inequalities and entropic convergence

Keywords:

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.

Introduction

Since the beginning of the nineties, functional inequalities (Poincaré, logarithmic (or F-) Sobolev, Beckner's like, transportation) turned to be a powerful tool for studying various problems in Probability theory and in Statistics: uniform ergodic theory, concentration of measure, empirical processes, statistical mechanics, particle systems for non linear p.d.e.'s, stochastic analysis on path spaces, rate of convergence of p.d.e.... Among such functional inequalities, Poincaré inequality and its generalizations (weak and super Poincaré) deserved particular interest, as they are the most efficient tool for the study of isoperimetry, concentration of measure and L 2 long time behavior (see e.g. [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF][START_REF] Wang | Functional inequalities for empty essential spectrum[END_REF][START_REF] Wang | A generalization of Poincaré and log-Sobolev inequalities[END_REF][START_REF] Barthe | Concentration for independent random variables with heavy tails[END_REF][START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF]). However (except the usual Poincaré inequality) they are not easily tensorizable nor perturbation stable. That is why super-Poincaré inequalities have to be compared with (generalized) Beckner's inequalities or with additive ϕ-Sobolev inequalities (see [START_REF] Wang | A generalization of Poincaré and log-Sobolev inequalities[END_REF][START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF][START_REF] Barthe | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry[END_REF]). But for some aspects, generalized Poincaré inequalities are insufficient. Indeed L 2 controls are not well suited in various situations (statistical mechanics, non linear p.d.e), where entropic controls are more natural. It is thus interesting to look at generalizations of Gross logarithmic Sobolev inequality. In this paper we shall investigate weak logarithmic Sobolev inequalities (the "super" logarithmic Sobolev inequalities have already been investigated by Davies and Simon, or Röckner and Wang). In order to better understand the previous introduction and what can be expected, let us introduce some definitions and recall some known facts. In all the paper M denotes a Riemannian manifold and µ denotes an absolutely continuous probability measure with respect to the surface measure. We also assume that µ is symmetric for a diffusion semi-group P t associated to a non explosive diffusion process. Let H 1 (M, µ) be the closure of C ∞ b (M ) (the space of infinitely differentiable functions f on M with all |∇ n f |, n ≥ 0 bounded) w.r.t. the norm µ(|f | 2 + |∇f | 2 ).

Definition 1.1 We say that the measure µ satisfies a weak Poincaré inequality, WPI, if there exists a non-increasing function β W P : (0, +∞) → R + , such that for all s > 0 and any bounded function f ∈ H 1 (M, µ),

Var µ (f ) := f 2 dµ - f dµ 2 ≤ β W P (s) |∇f | 2 dµ + s Osc 2 (f ), (WPI)
where Osc(f ) = sup f -inf f .

Weak Poincaré inequalities have been introduced by Röckner and Wang in [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF]. If β W P is bounded, we recover the (classical) Poincaré inequality, while if β W P (s) → ∞ as s → 0 we obtain a weaker inequality. Actually, as shown in [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF] any Boltzman measure (dµ = e -V dx) on R n with a locally bounded potential V satisfies some WPI (the result extends to any manifold with Ricci curvature bounded from below by a possibly negative constant, according to Theorem 3.1 in [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF] and the local Poincaré inequality shown by Buser [START_REF] Buser | A note on the isoperimetric constant[END_REF] in this framework). WPI furnishes an isoperimetric inequality, hence (sub-exponential) concentration of measure (see [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF][START_REF] Barthe | Concentration for independent random variables with heavy tails[END_REF]). It also allows to describe non exponential decay of the L 2 norm of the semi group, i.e. WPI is linked to inequalities like ∀t 0, Var µ (P t f ) ≤ ξ(t) Osc 2 (f ), for some adapting function ξ (relations between β W P and ξ will be recalled later). Recall that a uniform decay of the Variance, is equivalent to its exponential decay which is equivalent to the usual Poincaré inequality. Let us note that a multiplicative form of weak Poincaré inequality (namely β(s) = s 2/p and choose s such that each term of the right hand side is of the same order) appears first in works of Liggett [START_REF] Liggett | L 2 rates of convergence for attractive reversible nearest particle systems: the critical case[END_REF] to prove an algebraic convergence in L 2 of some spin system dynamic.

If we replace the variance by the entropy the latter argument is still true. Indeed (at least for bounded below curvature) an uniform decay of Ent µ (P t h) is equivalent to its exponential decay which is equivalent to the logarithmic Sobolev inequality. In order to describe non exponential decays, it is thus natural to introduce the following definition:

Definition 1.2 We say that the measure µ satisfies a weak logarithmic Sobolev inequality, WLSI, if there exists a non-increasing function β W L : (0, +∞) → R + , such that for all s > 0 and any bounded function f ∈ H 1 (M, µ),

Ent µ f 2 := f 2 log f 2 f 2 dµ dµ ≤ β W L (s) |∇f | 2 dµ + s Osc 2 (f ) . (WLSI)
Remark that WPI is translation invariant. Hence it is enough to check it for non negative functions f and for such functions we get Var µ (f ) ≤ Ent µ f 2 . Hence WLSI is stronger than WPI (we shall prove a more interesting result), and we can expect that WLSI (with a non bounded β W L ) allows to describe all the sub-gaussian measures, in particular all super-exponential (and sub-gaussian measures) for which a strong form of Poincaré inequality holds. Remark that, as for weak Poincaré inequalities, multiplicative forms of the weak logarithmic Sobolev inequality appears first under the name of log-Nash inequality to study the decay of semigroup in the case of Gibbs measures, see Bertini-Zegarlinski [START_REF] Bertini | Coercive inequalities for Gibbs measures[END_REF][START_REF] Bertini | Coercive inequalities for Kawasaki dynamics. The product case. Markov Process[END_REF] or Zegarlinski [START_REF] Zegarlinski | Entropy bounds for Gibbs measures with non-Gaussian tails[END_REF].

Remark 1.3 One can easily check that Var µ (f ) ≤ 1 4 Osc 2 (f ) so that we may assume that β W P (s) = 0 as soon as s ≥ 1/4. In fact, one can also prove Ent µ f 2 ≤ 1 e Osc 2 (f ) and thus we can suppose that β W L (s) = 0 for s ≥ 1/e. Hence for WPI and WLSI what is important is the behaviour of β near 0.

In order to understand the picture and to compare all these inequalities we shall call upon another class of inequalities, namely measure-capacity inequalities introduced by Maz'ya [START_REF] Maz'ja | Springer Series in Soviet Mathematics[END_REF]. Then these inequalities are extensively used in this context [START_REF] Barthe | Sobolev inequalities for probability measures on the real line[END_REF][START_REF] Chen | Capacitary criteria for Poincaré-type inequalities[END_REF][START_REF] Barthe | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry[END_REF][START_REF] Barthe | Concentration for independent random variables with heavy tails[END_REF][START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF]. Given measurable sets A ⊂ Ω the capacity Cap µ (A, Ω), is defined as follow:

Cap µ (A, Ω) := inf |∇f | 2 dµ; 1 I A ≤ f ≤ 1 I Ω ,
where the infimum is taken over all function f ∈ H 1 (M, µ). By convention, if the set of function f ∈ H 1 (M, µ) such that 1 I A ≤ f ≤ 1 I Ω is empty then we note Cap µ (A, Ω) = +∞. We refer to Maz'ya [START_REF] Maz'ja | Springer Series in Soviet Mathematics[END_REF] and Grigor'yan [START_REF] Grigor ′ Yan | Isoperimetric inequalities and capacities on Riemannian manifolds[END_REF] for further discussion on capacities. The capacity defined by Maz'ya seems to be a little different but they are similar. If now A satisfies µ(A) ≤ 1/2 we note

Cap µ (A) := inf {Cap µ (A, Ω); A ⊂ Ω, µ(Ω) ≤ 1/2}. (1)
A measure-capacity inequality is an inequality of the form

µ(A) γ(µ(A)) ≤ Cap µ (A), (2)
for some function γ. They are in a sense universal, since they only involve the energy (Dirichlet form) and the measure. Furthermore, a remarkable feature is that most of known inequalities involving various functionals (variance, p-variance, F functions of F -Sobolev inequalities, entropy etc...) can be compared (in a non sharp form) with some measure-capacity inequalities. We shall thus start by characterizing WLSI via measure-capacity inequalities. Then we will study the one dimensional case, in the spirit of Muckenhoupt or Bobkov-Götze criteria for Poincaré or logarithmic Sobolev inequalities (see e.g. [ABC + 00] chapter 6). We shall then discuss in details the relationship between WLSI and the generalized Poincaré inequalities. Finally we shall discuss various properties and consequences of WLSI. In the final sections, we study in details the decay of entropy for large time. In particular we show that for a µ reversible gradient diffusion process, very mild conditions on the initial law are sufficient to ensure an entropic decay of type e -t β when µ satisfies interpolating inequalities between Poincaré and Gross introduced by Latala and Oleszkiewicz [START_REF] Lata | Between Sobolev and Poincaré. in geometric aspects of Functional Analysis[END_REF], those conditions preventing estimation via Poincaré inequalities. We also give the elements to compute this decay under general WLSI. The particular case of the double sided exponential measure is detailed.

Let us finally remark that the limitation to finite dimensional space is only instrumental and the main results would be readily extendable to infinite dimensional space with capacity defined to suitable Dirichlet forms (assuming for example the existence of a carré du champ operator).

2 Weak logarithmic Sobolev inequalities

Characterization via capacity-measure condition

We start this section by characterizing WLSI in terms of measure-capacity inequalities.

Theorem 2.1 Assume that the measure µ satisfies a WLSI with function β W L , then for every

A ⊂ M such that µ(A) ≤ 1/2, ∀s > 0, µ(A) log 1 + 1 2µ(A) -s β W L (s) ≤ Cap µ (A).
Proof ⊳ Let A ⊂ Ω with µ(Ω) ≤ 1/2 and let f be a locally Lipschitz function satisfying 1

I A ≤ f ≤ 1 I Ω .
The variational definition of the entropy implies

Ent µ f 2 f 2 gdµ,
for all g such that e g dµ ≤ 1. Apply this inequality with

g =      log 1 + 1 2µ(A) on A 0 on Ω\A -∞ on Ω c
which satisfies e g dµ ≤ 1. It yields Ent µ f 2 µ(A) log 1 + 1 2µ(A) . Therefore by the weak logarithmic Sobolev inequality and the definition of the capacity we obtain

µ(A) log 1 + 1 2µ(A) ≤ β W L (s)Cap µ (A, Ω) + s.
Taking the infimum over sets Ω with measure at most 1/2 and containing A we obtain

∀s > 0, µ(A) log 1 + 1 2µ(A) -s β W L (s) ≤ Cap µ (A).

⊲

Theorem 2.2 Let β : (0, +∞) → R + be non-increasing function such that for every

A ⊂ M with µ(A) ≤ 1/2 one has ∀s > 0, µ(A) log 1 + e 2 µ(A) -s β(s) ≤ Cap µ (A). (3)
Then the measure µ satisfies a WLSI with the function β W L (s) = 16β(3s/14), for s > 0.

Proof ⊳ Let a bounded function f ∈ H 1 (M, µ), we will prove that

∀s > 0, Ent µ f 2 ≤ 16β(s) |∇f | 2 dµ + 14s/3 Osc 2 (f ). ( 4
)
Let m be a median of f under µ and let Ω + = {f > m}, Ω -= {f < m}. Then, using the argument of Lemma 5 in [START_REF] Barthe | Sobolev inequalities for probability measures on the real line[END_REF], we obtain

Ent µ f 2 ≤ sup F 2 + hdµ; h 0, e h dµ ≤ e 2 + 1 + sup F 2 -hdµ; h 0, e h dµ ≤ e 2 + 1 , (5)
where

F + = (f -m)1 I Ω + and F -= (f -m)1 I Ω -.
We will study the first term in the right hand side, the second one will be treated by the same method.

There are two cases depending on the value of s. Let s 1 := 1 2 log 1 + 2e 2 , and assume that s ∈ (0, s 1 ). Let define c by

c = inf t 0, µ(F 2 + > t) log 1 + e 2 µ(F 2 + > t) ≤ s .
If c = 0 then one get that for some constant

C sup F 2 + hdµ; h 0, e h dµ ≤ e 2 + 1 ≤ s log(1 + e 2 ) F + 2 ∞
and the problem is solved on that case. If now c > 0, since µ is absolutely continuous with respect to the surface measure of the Riemannian manifold M , one can find Ω 0 such that

F 2 + > c ⊂ Ω 0 ⊂ F 2 + c and µ(Ω 0 ) log 1 + e 2 µ(Ω 0 ) = s. (6)
Note that the function x → x log(1 + e 2 /x) is increasing on (0, ∞), and realize a bijection between (0, 1/2] and (0, s 1 ]. Pick some ρ ∈ (0, 1) and introduce for any k > 0, Ω k = F 2 + cρ k . The sequence (Ω k ) k is increasing so that, for every function h 0,

F 2 + hdµ = Ω 0 F 2 + hdµ + k>0 Ω k \Ω k-1 F 2 + hdµ.
For the first term we get

Ω 0 F 2 + hdµ ≤ Osc 2 (f ) Ω 0 h dµ,
then Lemma 6 of [START_REF] Barthe | Sobolev inequalities for probability measures on the real line[END_REF] implies that sup

Ω 0 hdµ; h 0, e h dµ ≤ e 2 + 1 = µ(Ω 0 ) log 1 + e 2 µ(Ω 0 ) .
So that, using the definition of c (equality (6)) we get sup

Ω 0 F 2 + hdµ; h 0, e h dµ ≤ e 2 + 1 ≤ s Osc 2 (f ).
For the second term we have for all k > 0, due to the fact that

cρ k ≤ F 2 + ≤ cρ k-1 on Ω k \Ω k-1 , Ω k \Ω k-1 F 2 + hdµ ≤ cρ k-1 Ω k \Ω k-1
hdµ.

Then we obtain using again Lemma 6 of [START_REF] Barthe | Sobolev inequalities for probability measures on the real line[END_REF], for any k > 0,

sup

Ω k \Ω k-1 F 2 + hdµ; h 0, e h dµ ≤ e 2 + 1 ≤ cρ k-1 µ(Ω k \Ω k-1 ) log 1 + e 2 µ(Ω k \Ω k-1 )
.

Using now inequality (3) we get

cρ k-1 µ(Ω k \Ω k-1 ) log 1 + e 2 µ(Ω k \Ω k-1 ) ≤ cρ k-1 β(s)Cap µ (Ω k \Ω k-1 ) + scρ k-1 .
Let set for any k > 0,

g k = min 1, F + -cρ k+1 cρ k -cρ k+1 + , so that we have 1 I Ω k ≤ g k ≤ 1 I Ω k+1 with µ(Ω + ) ≤ 1/2. This implies, using the definition of Cap µ (Ω k \Ω k-1 ) (see (1)), cρ k-1 Cap µ (Ω k \Ω k-1 ) ≤ 1 ρ(1 - √ ρ) 2 Ω k+1 \Ω k |∇F + | 2 dµ.
Note that the constant c satisfies c ≤ F + 2 ∞ ≤ Osc 2 (f ). We can now finish the proof in the case s ∈ (0, s 1 ), sup F 2 + hdµ; h 0, e h dµ ≤ e 2 + 1 ≤ sup

Ω 0 F 2 + hdµ; h 0, e h dµ + k>0 sup Ω k+1 \Ω k F 2 + hdµ; h 0, e h dµ ≤ s Osc 2 (f ) + k>0 scρ k-1 + k>0 1 ρ(1 - √ ρ) 2 Ω k+1 \Ω k |∇F + | 2 dµ ≤ β(s) ρ(1 - √ ρ) 2 |∇F + | 2 dµ + s 2 -ρ 1 -ρ Osc 2 (f ).
Using inequality (5) and the previous inequality for F -we get

∀s ∈ (0, s 1 ), Ent µ f 2 ≤ β(s) ρ(1 - √ ρ) 2 |∇f | 2 dµ + 2s 2 -ρ 1 -ρ Osc 2 (f ), (7)
for all ρ ∈ (0, 1). Choosing ρ = 1/4 furnishes inequality (4) for any s ∈ (0, s 1 ). Assume now that s s 1 , then take c = 0 and we get

µ(Ω 0 ) log 1 + e 2 µ(Ω 0 ) ≤ s,
and the same argument used for s ∈ (0, s 1 ) implies

∀s s 1 , Ent µ f 2 ≤ 2sOsc 2 (f ). ( 8
)
Then inequality (8) and the previous result implies inequality (4) for any s > 0. Note that we do not obtain the optimal function β W L (s) for s large, but, as explained in remark 1.3, this is not important for the WLSI. ⊲ Remark 2.3 The following two inequalities hold µ(A)

2 log 1 + 1 2µ(A) β W L µ(A) 2 log 1 + 1 2µ(A) ≤ sup s>0    µ(A) log 1 + 1 2µ(A) -s β W L (s)    ≤ µ(A) log 1 + 1 2µ(A) β W L µ(A) log 1 + 1 2µ(A)
and

µ(A) 2 log 1 + e 2 µ(A) β W L µ(A) 2 log 1 + e 2 µ(A) ≤ sup s>0    µ(A) log 1 + e 2 µ(A) -s β W L (s)    ≤ µ(A) log 1 + e 2 µ(A) β W L µ(A) log 1 + e 2 µ(A) . (9)
Proofs of these inequalities are the same as in [BCR05, Theorem 1]. The lower bounds of these inequalities correspond to a specific choice, s = µ(A)

2 log 1 + 1 2µ(A)
for the first one and s = µ(A)

2 log 1 + e 2 µ(A)
for the second one. For the upper bound of the first inequality we use the fact that

sup s>0    µ(A) log 1 + 1 2µ(A) -s β W L (s)    ≤ sup 0<s<µ(A) log 1+ 1 2µ(A)    µ(A) log 1 + 1 2µ(A) β W L (s)    ,
and the non-increasing property of β gives the result. The method holds for the second inequality.

A Hardy like criterion on R

Proposition 2.4 Let µ be a probability measure on R. Assume that µ is absolutely continuous with respect to Lebesgue measure and denote by ρ µ its density. Let m be a median of µ and β W L : (0, ∞) → R + be non-increasing. Let C be the optimal constant such that for all f ∈ H 1 (R, µ),

∀s > 0, Ent µ f 2 ≤ Cβ W L (s) |∇f | 2 dµ + s Osc 2 (f ).
Then we get max(b

-, b + ) ≤ C ≤ max(B -, B + ), where b + := sup x>m µ([x,+∞)) 2 log 1 + 1 2µ([x,+∞)) β W L µ([x,+∞)) 2 log 1 + 1 2µ([x,+∞)) x m 1 ρ µ b -:= sup x<m µ((-∞,x]) 2 log 1 + 1 2µ((-∞,x]) β W L µ((-∞,x]) 2 log 1 + 1 2µ((-∞,x]) m x 1 ρ µ B + := sup x>m 16µ([x, +∞)) log 1 + e 2 µ([x,+∞)) β W L 14 3 µ([x, +∞)) log 1 + e 2 µ([x,+∞)) x m 1 ρ µ (10) B -:= sup x<m 16µ((-∞, x]) log 1 + e 2 µ((-∞,x]) β W L 14 3 µ((-∞, x]) log 1 + e 2 µ((-∞,x]) m x 1 ρ µ
Proof ⊳ The proof of the lower bound on C is exactly the same as in [BCR05, Theorem 3] using Theorem 2.1 and Remark 2.3. For the upper bound denote

F + = (f -f (m))1 I [m,+∞) and F -= (f -f (m))1 I (-∞,m] . Then Ent µ f 2 ≤ Ent µ F 2 + + Ent µ F 2 -.
We work separately with the two terms and explain the arguments for Ent µ F 2 + only. We follow the method of proof in [BCR05, Theorem 3]. Using equality (10) we get

∀x > m, 16µ([x, +∞)) log 1 + e 2 µ([x,+∞)) β W L 14 3 µ([x, +∞)) log 1 + e 2 µ([x,+∞)) x m 1 ρ µ ≤ B + .
This means that

∀x > m, 16µ([x, +∞)) log 1 + e 2 µ([x,+∞)) B + β W L 14 3 µ([x, +∞)) log 1 + e 2 µ([x,+∞)) ≤ Cap µ ([x, +∞), [m, +∞)). If A ⊂ [m, +∞) then Cap µ (A, [m, +∞)) = Cap µ ([inf A, +∞), [m, +∞)) (
B + β W L 14 3 µ(A) log 1 + e 2 µ(A) ≤ Cap µ (A, [m, +∞)).

Using now inequality (9) one has for all

A ⊂ [m, +∞), sup s>0    16 µ(A) log 1 + e 2 µ(A) -s B + β W L ( 14 3 s)    ≤ Cap µ (A, [m, +∞)),
and then by the same argument as in Theorem 2.2 one has

Ent µ F 2 + ≤ B + β W L (s) |∇F + | 2 dµ + sOsc(f ) 2 .
It follows that C ≤ B + . The same argument gives also C ≤ B -and the proposition is proved. ⊲ Corollary 2.5 Let Φ be a function on R such that dµ Φ (x) := e -Φ(x) dx, x ∈ R is a probability measure and let ε ∈ (0, 1).

Assume that there exists an interval I = (x 0 , x 1 ) containing a median m of µ such that |Φ| is bounded on I, and Φ is twice differentiable outside I with for any x ∈ I,

Φ ′ (x) = 0, |Φ ′′ (x)| Φ ′ (x) 2 ≤ 1 -ε and A ′ Φ(x) ≤ Φ(x) + log Φ ′ (x) ≤ AΦ(x), (11)
for some constants A, A ′ > 0.

Let β be a non-increasing function on (0, ∞). Assume that there exists c > 0 such that for all x ∈ I it holds

Φ(x) Φ ′ (x) 2 ≤ cβ Ae -Φ(x) Φ(x) |Φ ′ (x)| .
Then µ Φ satisfies a WLSI with function Cβ for some constant C > 0.

Proof ⊳ Corollary 2.4 of [BCR05] gives for x ≥ x 1 , µ([x, +∞)) ≤ e -Φ(x) εΦ ′ (x) ≤ 2 -ε ε µ([x, +∞)).
Then using Proposition 2.4 and inequality (11) we obtain the result. ⊲ Example 2.6 Let us give two examples:

• For α > 0, the measure dm α (t) = α(1 + |t|) -1-α dt/2, t ∈ R satisfies the WLSI with the function ∀s > 0, β W L (s) = C (log 1/s) 1+2/α s 2/α ,
for some constant C > 0.

• Let α ∈ (0, 2) and defined the probability measure dµ α (t) = Z α e -|t| α dt, t ∈ R, (Z α is a normalization constant). Then µ α satisfies the WLSI with the function

∀s > 0, β W L (s) = C(log 1/s) (2-α)/α ,
for some C > 0.

Contrary to the WPI, one can study the case α ∈ (1, 2]. In particular for α = 2 we get that β W L is bounded, i.e. we recover (with a non sharp constant) the classical logarithmic Sobolev inequality for the gaussian measure.

3 Weak Logarithmic Sobolev inequalities and generalized Poincaré inequalities

Link with weak Poincaré inequalities and classical Poincaré inequality

Barthe, Cattiaux and Roberto investigated in [START_REF] Barthe | Concentration for independent random variables with heavy tails[END_REF] the measure-capacity criterion for WPI.

Their results read as follows: WPI with a function β W P implies a measure-capacity inequality with γ(u) = 4β W P (u/4) (see inequality (2)) while a measure-capacity inequality with non-increasing function γ implies WPI with β W P = 12γ (we may assume that γ(u) = γ(1/2) for u 1/2). Comparing with Theorem 2.1 and Theorem 2.2, we can state: Proposition 3.1 Assume that a probability measure µ satisfies a WLSI with function β W L then µ satisfies a WPI with function β W P defined by

∀s > 0, β W P (s) = 24β W L s 2 log 1 + 1 2s log 1 + 1 2s . ( 12
)
Conversely, a WPI with function β W P implies a WLSI with function β W L , defined by,

       ∀s ∈ (0, s 0 ), β W L (s) = c ′ β W P c s log (1/s) log (1/s), ∀s s 0 , β W L (s) = c ′ β W P c s 0 log (1/s 0 ) log (1/s 0 ), (13)
for some universal constants c, c ′ , s 0 > 0.

Finally assume that µ satisfies a WLSI with function β W L , then it verifies a classical Poincaré inequality if and only if there exists c 1 , c 2 > 0 such that for s small enough,

β W L (s) ≤ c 1 log(c 2 /s).
Proof ⊳ For the first statement, first note that β W P is non-increasing. Then Theorem 2.1 and Remark 2.3 imply that µ(A)

2 log 1 + 1 2µ(A) β W L µ(A) 2 log 1 + 1 2µ(A) ≤ Cap µ (A).
This means that 12µ(A)

β W P (µ(A)) ≤ Cap µ (A),
where β W P is defined by ( 12), the result holds using Theorem 2.2 of [START_REF] Barthe | Concentration for independent random variables with heavy tails[END_REF].

To prove the second statement we use the same argument (replacing Theorem 2.1 by Theorem 2.2) and the fact that there exist constants A, A ′ , s 0 > 0 such that

∀s ∈ (0, s 0 ), A ′ s log (1/s) ≤ ϕ -1 (s) ≤ A s log (1/s) , (14)
where ϕ(s) = s log(1 + e 2 /s). Then µ satisfies a WLSI with function β W L defined by (13). Note that β W L is non-increasing. Finally, the last two results prove that

β W L (s) ≤ c 1 log(c 2 /s) for s enough is equivalent to classical Poincaré inequality. ⊲ Remark 3.2
• It is interesting to remark that when considering the usual derivation "Logarithmic Sobolev inequality implies Poincaré inequality" by means of test function 1 + ǫg and ǫ → 0, we get a worse result: a weak logarithmic Sobolev inequality with function β implies a weak Poincaré inequality with the same function β, whereas the result of the proposition 3.1 gives a better result.

• As a byproduct, we get that any Boltzman's measure (with a locally bounded potential) satisfies some WLSI if Ricci(M ) is bounded from below (see [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF]).

• Finally the above proof shows that we obtain the best function (up to multiplicative constants) for WPI or WLSI as soon as we have the best function for the other. In particular we recover the good functions for the examples 2.6.

Link with super Poincaré inequalities

Let us recall the definition of the super Poincaré inequality introduced by Wang in [START_REF] Wang | Functional inequalities for empty essential spectrum[END_REF].

Definition 3.3 We say that the measure µ satisfies a super Poincaré inequality, SPI, if there exists a non-increasing function β SP : [1, +∞) → R + , such that for all s 1 and any function

f ∈ H 1 (M, µ), f 2 dµ ≤ β SP (s) |∇f | 2 dµ + s |f |dµ 2 . (SPI)
Note that as for WLSI in Remark 1.3, for the SPI what is important is the behaviour of β near ∞.

As for Proposition 3.1 we can now relate WLSI and SPI.

Proposition 3.4 Suppose that µ satisfies a WLSI with function

β W L . Assume that β W L verifies that x → β W L log(x/2) 2x /log(x/2) is non-increasing on (2, ∞). Then µ satisfies a SPI with function β SP given by ∀t 2e, β SP (t) = 2 β W L log(t/2) 2t log(t/2) , (15)
and constant on [1, 2e).

Proof ⊳ If µ satisfes a WLSI then one obtains by Theorem 2.1 and Remark 2.3:

µ(A) 2 log 1 + 1 2µ(A) β W L µ(A) 2 log 1 + 1 2µ(A) ≤ Cap µ (A), (16)
for any A ⊂ M , with µ(A) ≤ 1/2. Finally the function t → t β W L log(t/2) 2t

/log(t/2) is clearly non decreasing for t 2e, then Corollary 6 of [START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF] gives the result. ⊲ Note that the last proposition is not entirely satisfying, we hope that WLSI is equivalent to SPI via a measure-capacity measure criterion.

Link with general Beckner inequalities

Definition 3.5 Let T : [0, 1] → R + , be a non-decreasing function, satisfying in addition x → T (x)/x is non-increasing on (0, 1]. We say that a measure µ satisfies a general Beckner inequality, GBI, with function

T if for all function f ∈ H 1 (M, µ), sup p∈(1,2) f 2 dµ - |f | p dµ 2 p T (2 -p) ≤ |∇f | 2 dµ. (GBI)
Note that our hypotheses imply that

∀x ∈ [0, 1], T (1)x ≤ T (x) ≤ T (1).
The two extremal cases correspond respectively to the Poincaré inequality (T is constant, T (x) = T (1)) and the logarithmic Sobolev inequality (T (x) = T (1)x). The intermediate cases T (x) = x a for 0 ≤ a ≤ 1, have been introduced and studied in [START_REF] Lata | Between Sobolev and Poincaré. in geometric aspects of Functional Analysis[END_REF], while a study of general T is partly done in [START_REF] Barthe | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry[END_REF]. Also note that (up to multiplicative constants) the interesting part of T is its behaviour near 0, that is we can always define T near the origin and then take it equal to a large enough constant. Recall finally that the usual Beckner inequality concerns T (x) = x and was introduced by Beckner to get quantitative information on an interpolation between Poincaré's inequality and logarithmic Sobolev inequality for the Gaussian measure, see [START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF].

In [START_REF] Barthe | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry[END_REF] Theorem 10 and Lemma 9, it is shown that (up to a multiplicative constant 3) GBI is equivalent to a measure-capacity. More precisely, the inequality measure-capacity (2) with the function

γ(u) = T 1 log 1 + 1 u , (17)
for u small enough implies a GBI with the function 20T . And GBI implies a measure-capacity inequality with the function 6γ defined on (17). We thus obtain:

Proposition 3.6 Assume that µ satisfies a WLSI with function β W L . Let ∀t ∈ (0, 1], T (t) = tβ W L 1 4te 1/t , (18)
and assume that T non-decreasing (0, t a ] for some t a ∈ (0, 1]. Then the measure µ satisfies a GBI with function 20T . Conversely assume that µ satisfies a GBI with function T and constant c, then µ satisfies a WLSI with function β W L given by

β W L (s) = C T C ′ 1 log(1/s) log(1/s), (19)
for s small enough and some constants C, C ′ .

Proof

Link with an other weak logarithmic Sobolev inequality

The next inequality is useful to control the decay in entropy of the semigroup. It will by used in Theorem 4.2.

Theorem 3.8 If µ satisfies a WLSI with function β W L , then µ satisfies for any function f ∈ H 1 (M, µ) and any u > 0 small enough,

Ent µ f 2 ≤ β SW L (u) |∇f | 2 dµ + √ 3u Var µ (f 2 ) 1 2 , (20)
with

β SW L (u) = 16β W L κ u 3 log 6 (1/u)
for some universal constant κ and u small enough.

Proof ⊳ According to Theorem 2.1 and Remark 2.3 we know that for every

A ⊂ M such that µ(A) ≤ 1/2, Cap µ (A) µ(A) 2 log 1 + 1 2µ(A) β W L µ(A) 2 log 1 + 1 2µ(A) µ(A) 2k log 1 + e 2 µ(A) β W L µ(A) log 1 + e 2 µ(A) for k = log(1 + 2e 2 )/ log(2) using k log(1 + y/2)
log(1 + e 2 y) for y 2 and that β W L is nonincreasing. Hence we are in the situation of Theorem 2.2 with β = 2k β W L . Note that we may assume that f is non-negative. Let Ω 0 ⊂ M , it will be fixed latter. Indeed the first quantity we have to control is Ω 0 F 2 + hdµ which is less than

Ω 0 h 2 dµ 1 2 F 4 + dµ 1 2
.

We thus have to bound

X 0 := sup{ Ω 0 h 2 dµ; h 0, e h dµ ≤ 1 + e 2 } = sup{ Ω 0 h 2 dµ; h 0, Ω 0 e h dµ ≤ e 2 + µ(Ω 0 )} ,
(see [START_REF] Barthe | Sobolev inequalities for probability measures on the real line[END_REF] Lemma 6 for the latter equality). But ϕ(x) = (1 + log 2 (x)) 1 I x e + 2 e x 1 I x<e is concave and non-decreasing on R + . It follows that

ϕ e 2 + µ(Ω 0 ) µ(Ω 0 ) Ω 0 ϕ(e h ) dµ µ(Ω 0 ) Ω 0 (1 + h 2 ) 1 I h 1 dµ µ(Ω 0 ) Ω 0 h 2 dµ µ(Ω 0 ) - Ω 0 h 2 1 I h<1 dµ µ(Ω 0 ) Ω 0 h 2 dµ µ(Ω 0 ) -1 , so that X 0 ≤ µ(Ω 0 ) 2 + log 2 1 + e 2 µ(Ω 0 ) := ψ(µ(Ω 0 )) .
Once again onlys small values of s are challenging, consider then s ≤ 1. We can mimic now the proof of Theorem 2.2 , briefly we define c by

c = inf t 0, ψ(µ(F 2 + > c)) ≤ s ,
and then we choose Ω 0 such that

F 2 + > c ⊂ Ω 0 ⊂ F 2 +
c and ψ(µ(Ω 0 )) = s a , for some a > 0. This choice being possible since ψ is increasing on [0, 1/2[, the maximal possible s being greater than 1. Then and obtain sup

F 2 + hdµ; h 0, e h dµ ≤ e 2 + 1 ≤ √ s a F 4 + dµ 1 2 + s c 1 -ρ + β W L (s) ρ(1 - √ ρ) 2 |∇F + | 2 dµ . ( 21
)
It remains to estimate c. Note that there exists an universal constant θ such that ψ -1 (x) θ x/ log 2 (1 + e 2 x ) . It follows using this two inequalities

θ s a log 2 (1 + e 2 s a ) ≤ µ(Ω 0 )
and by Markov inequality

µ(Ω 0 ) ≤ F 2 + dµ c ≤ F 4 + dµ 1 2 c ,
so that choosing a = 2/3 and ρ = 1/4 we finally obtain sup

F 2 + hdµ; h 0, e h dµ ≤ e 2 + 1 ≤ s 1 3 1 + 4 3θ log 2 (1 + e 2 s 2/3 ) F 4 + dµ 1 2 + 16β W L (s) |∇F + | 2 dµ . ( 22
)
The same inequality for F -and the elementary

√ a + √ b ≤ √ 2 √
a + b yield, since there exists an universal constant θ ′ such that the inverse function of s → √ 2s

1 3 1 + 4 3θ log 2 (1 + e 2 s 2/3 ) is greater than u → θ ′ u 3 /(log 6 (1/u)) for u > 0 small enough, Ent µ f 2 ≤ 16β W L θ ′ u 3 1 + log 6 (1 + e 2 u 2 ) |∇f | 2 dµ + u (f -m) 4 dµ 1 2 . ( 23
)
Since we have assumed that f is non-negative, a median of f 2 is m 2 , and

(f -m) 4 ≤ (f 2 -m 2 ) 2 . Finally, if M denotes the mean of f 2 , (f 2 -M ) -(m 2 -M ) 2 dµ = Var µ (f 2 ) + (m 2 -M ) 2 and since m 2 -M is a median of f 2 -M , provided m 2 -M 0 Var µ (f 2 ) (f 2 -M ) 2 1 I f 2 -M m 2 -M dµ 1 2 (m 2 -M ) 2 while if m 2 -M ≤ 0 Var µ (f 2 ) (f 2 -M ) 2 1 I f 2 -M ≤m 2 -M dµ 1 2 (m 2 -M ) 2 .
We thus finally obtain

(f -m) 4 dµ ≤ 3 Var µ (f 2 )
and the proof is completed. ⊲

One may of course derive other weak logarithmic Sobolev inequalities by this method, such inequalities as well as further applications will be treated elsewhere. We will apply this theorem on the section 4 for the decay to the equilibrium of the semigroup.

Convergence of the associated semigroup

In this section we shall study entropic convergence for the associated semi-group. Namely we assume that (P t ) t 0 is a "nice" diffusion µ symmetric semi-group. Here by "nice" we mean that (P t ) t 0 is the semi-group associated to a non-explosive diffusion process on some Polish space admitting a "carré du champ". For a precise framework we refer to [START_REF] Cattiaux | A pathwise approach of some classical inequalities[END_REF] Section 1.1. Roughly speaking, these assumptions allow us to give a rigorous meaning to all computations below. Let h be a bounded density of probability with respect to the measure µ. The two results of this section connect the decay of the entropy with the infinite norm of h. More precisely, using the WLSI we will compute the function C(t, h ∞ ) such that for all t > 0,

Ent µ (P t h) ≤ C(t, h ∞ ).
We will give here conditions under which C(t, h ∞ ) → 0 when t goes to ∞.

The first result connects the decay of the entropy with the oscillation of h, one gets:

Proposition 4.1 Let µ satisfies a WLSI with function β W L and let h 0, bounded with hdµ = 1.

Then for any ε > 0 and for t large enough we get:

Ent µ (P t h) ≤ (e -1 + ε) ξ ε (t) Osc 2 ( √ h) (24)
where ξ ε (t) is given by

ξ -1 ε (r) = - 1 2 β W L (r) log r ε ,
for r small enough. Conversely, if there exists a decreasing function ξ such that, for any bounded h 0, with hdµ = 1 we have ∀t > 0,

Ent µ (P t h) ≤ ξ(t) Osc 2 ( √ h),
then µ satisfies a WLSI with function β W L (t) = ψ -1 (t) where ψ(t) = 2 2 ξ(t). In particular if ξ(t) ≤ ce -αt , for some α > 0, the measure µ satisfies a Poincaré inequality.

Proof ⊳ We start with the direct part. Denote I(t) = Ent µ (P t h). Then I ′ (t) = -1 2 |∇P t h| 2 P t h dµ, thus the weak logarithmic Sobolev inequality yields

I ′ (t) ≤ - 2 β W L (r) I(t) + 2r β W L (r)
Osc 2 ( P t h).

Using Gronwall's lemma yields

Ent µ (P t h) ≤ inf r>0 r sup s∈[0,t]
Osc 2 ( P s h) + e -2t/β W L (r) Ent µ (h) .

We may now use Osc 2 ( √ P t h) ≤ Osc 2 ( √ h) and Ent µ (h) ≤ 1/e Osc 2 ( √ h) as we quoted in Remark 1.3 and finally choose r such that r = ε e -2t/β W L (r) (which is optimal up to constants) to get the result. Let us prove the second statement. Denote f = √ h. According to [START_REF] Cattiaux | A pathwise approach of some classical inequalities[END_REF] (2.5) with α 1 = -1 and

α 2 = 2 it holds Ent µ (h) ≤ t |∇f | 2 dµ + 2 log f P t h dµ . (25)
But

f P t h dµ = f (1 + (P t h -1)) dµ ≤ 1 + (f -f dµ) (P t h -1) dµ ≤ 1 + Osc(f ) |P t h -1| dµ ≤ 1 + Osc(f ) 2 Ent µ (P t h) ≤ 1 + 2ξ(t) Osc 2 (f ) ,
where we used successively f dµ ≤ 1 , Pinsker inequality and the hypothesis. It remains to use log(1 + a) ≤ a to get the first result. The particular case follows from Proposition 3.1. ⊲

The previous result is the exact analogue of Theorem 2.1 in [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF] for WPI. The converse statement (Theorem 2.3 in [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF]) is remarkable in the following sense: it implies in particular that any exponential decay (Var µ (P t f ) ≤ ce -αt Ψ(f -f dµ)) for any Ψ such that Ψ(af ) = a 2 Ψ(f ) (in particular Ψ(f ) = Osc 2 (f )) implies a (true) Poincaré inequality. This result is of course very much stronger than the usual one involving a L 2 bound. Its proof lies on the fact that t → log( (P t f ) 2 dµ) is convex. This convexity property (even without the log) fails in general for the relative entropy (Bakry-Emery renowned criterion was introduced for ensuring such a property). Actually a similar statement for the entropy is false.

Not that the previous result is only partly satisfactory for the convergence of the entropy. Indeed recall that for a density of probability h, the following holds

Var µ ( √ h) ≤ Ent µ (h) ≤ Var µ (h)
so that a weak Poincaré inequality implies for t > 0

Ent µ (P t h) ≤ ξ W P ε (t) (1 + ε)Osc 2 (h) ,
whereas our WLSI implies

Ent µ (P t h) ≤ ξ W LS ε (t) (e -1 + ε)Osc 2 ( √ h),
so that for small time, the WLSI furnishes better bounds than a weak Poincaré inequality (and justifies the use of LSI for this kind of evaluation), though the rate of convergence is not the expected one.

In order to correct this unsatisfactory point, at least when a Poincaré inequality holds, and always for bounded density h, we will make use of the other weak logarithmic Sobolev inequality stated in Theorem 3.8. Indeed, another way to control entropy decay was introduced in [CG06b, Theorem 1.13]. It was proved there that a Poincaré inequality (with constant C P ) is equivalent to a restricted logarithmic Sobolev inequality

Ent µ (h) ≤ C (1 + log( h ∞ )) |∇h| 2 h dµ
for all bounded density of probability h, where the constant C only depends on C P . It follows that

Ent µ (P t h) ≤ e - t C(1+log( h ∞)) Ent µ (h)
for such an h.

We shall describe below one result in this direction for WLSI, using Theorem 3.8 and Poincaré inequality.

Proposition 4.2 Let µ be a probability measure satisfying a WLSI with function β W L and the usual Poincaré inequality with constant C P . Let β SW L be the function defined in Theorem 3.8. Then for all f ∈ H 1 (E, µ),

Ent µ f 2 ≤ A(C P , f ∞ ) |∇f | 2 dµ where A(C P , f ∞ ) = inf u∈(0,s 0 ] β SW L (u) + u 3C P f 2 ∞ .
Here s 0 is any positive number such that β SW L is defined by the formula in Theorem 3.8 for s ≤ s 0 and then extended by β SW L (s) = β SW L (s 0 ) for s s 0 . As a consequence, for all t 0,

Ent µ (P t h) ≤ e -t / A C P , √ h ∞ Ent µ (h)
for any bounded density of probability h.

Proof

⊳ Due to homogeneity we may assume that |∇f | 2 dµ = 1 (if it is 0 the result is obvious). But since µ satisfies a Poincaré inequality

Var µ (f 2 ) ≤ 4C P f 2 |∇f | 2 dµ ≤ 4C P f 2 ∞ , so that Ent µ P t f 2 ≤ β SW L (u) + 2u f 2 ∞ √ 3C P . ⊲
Note now that the previous entropic decay is always better for small time. Indeed if for x ∈ (0, 1). Then one can find C(α), C ′ (α) > 0 such that for all bounded density of probability f ,

t ≤ C P A(C P , h 1 2 ∞ ) A(C P , h 1 2 ∞ ) -C P log Var µ (h) Ent µ (h)
Ent µ f 2 ≤ C(α) 1 + log (2/α)-1 ( f ∞ ) |∇f | 2 dµ .
As a consequence, for all t 0,

Ent µ (P t h) ≤ e -t / C ′ (α) (1+log (2/α)-1 ( h ∞ )) Ent µ (h) ,
for any bounded density of probability h.

It seems very unlikely that one can derive such a result from a direct use of Proposition 4.1. As noticed in [START_REF] Cattiaux | On quadratic transportation cost inequalities[END_REF], these restricted logarithmic Sobolev inequalities (restricted to the (P t stable) L ∞ balls) can be used to obtain modified (or restricted) transportation inequalities. We recall below a result taken from section 4.2 in [START_REF] Cattiaux | On quadratic transportation cost inequalities[END_REF]. If ν = hµ is a probability measure, it can be shown

W 2 2 (ν, µ) ≤ η(0) Ent µ (h) + +∞ 0 η ′ (t) Ent µ (P t h) dt , ( 26
)
where η is a non-decreasing positive function such that (1/η(t))dt = 1, and W 2 denotes the (quadratic) Wasserstein distance between ν and µ. We may take here

η(t) = 2 A(C P , h 1 2 ∞ ) e 1 2 t/A(C P , h 1 2 ∞ )
which yields

W 2 (ν, µ) ≤ D (1 + A 1 2 (C P , h 1 2 ∞ )) Ent µ (h) . (27)
In the Latala-Oleszkiewicz situation, we recover, up to the constants, Theorem 1.11 in [START_REF] Cattiaux | On quadratic transportation cost inequalities[END_REF]. Using Marton's trick, (27) allows us to obtain a concentration result (a little bit less explicit than the one obtained via GBI in Proposition 29 of [START_REF] Barthe | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry[END_REF]) namely there exist r 0 and σ such that if µ(A) 1/2 and A c r = {x, d(x, A) r} one has

r -r 0 ≤ σ A 1 2 (C P , (1/µ 1/2 (A c r ))) log(1/µ(A c r )) .
In the Latala-Oleszkiewicz situation, we recover up to the constants, the same concentration function as µ α , showing that our restricted logarithmic Sobolev inequality is (up to the constants) optimal. Note that another way to get the concentration result is to use the modified logarithmic Sobolev (and transportation) inequalities discussed in [START_REF] Gentil | Modified logarithmic sobolev inequalities and transportation inequalities[END_REF][START_REF] Gentil | Modified logarithmic sobolev inequalities in null curvature[END_REF].

Let us finally note that even if the results obtained by the WLSI are always efficient in the regime between Poincaré and Gross inequality, it relies on the crucial assumption that h is a bounded density. The goal of the next section is to get rid of this assumption.

Convergence to equilibrium for diffusion processes

In this section we shall discuss the rate of convergence to equilibrium for particular diffusion process, both in total variation and in entropy. The main difference between the previous section is that we do not assume that the initial law of the diffusion processes has a density of probability with respect to symmetric measure µ. The initial entropy is not necessarily finite.

For simplicity we only consider the case when M = R n and µ = e -2V dx. Hence our diffusion process is given by the stochastic differential equation

dX t = dB t -(∇V )(X t )dt , Law(X 0 ) = ν (28)
where B . is a standard Brownian motion. We assume that V is C 3 and that there exists some ψ such that ψ(x) → +∞ as |x| → +∞ and 1 2 ∆ ψ -∇V.∇ψ is bounded from above. This assumption ensures the existence of an unique non explosive strong solution for (28). If ν = δ x we will denote by X x t the associated process (cf e.g. [START_REF] Royer | Une initiation aux inégalités de Sobolev logarithmiques[END_REF]). A remarkable consequence of Girsanov theory (see [START_REF] Royer | Une initiation aux inégalités de Sobolev logarithmiques[END_REF] in our situation) is that with our assumptions, for all ν and all t > 0 the law of X t denoted by P t ν is absolutely continuous with respect to µ, its density will be denoted by h t . Of course if ν = hµ, P t ν = (P t h)µ and µ is a reversible measure. In particular P t ν = (P t-u h u )µ, and the rate of convergence of P t ν towards µ can be studied by using the semigroup properties only. In the sequel we shall make the abuse of notation P t ν = (P t-u h u ) i.e. we shall abusively identify the measure with its density. What we need to understand is thus the behavior of P t h, where h is a density of probability (in a sense it is P t f 2 rather than P t f which is interesting). Of particular interest is the case when

|∇V | 2 (x) -∆V (x) -C min > -∞ (29)
for a nonnegative C min since in this case one can show (see [Roy99, Theorem 3.2.7]) that Ent µ (P t δ x ) is finite for all t > 0. Actually the proof of Royer can be used in order to get the following more general and precise result Proposition 5.1 With the previous hypotheses

P t δ x log p + (P t δ x ) dµ ≤ 4 p-1 V p + (x) + C min t 2 p + n 2 log( 1 2πt ) p + e V (x)+p(log p-1)+ 1 2 C min t (30)
for all t ∈]0, 1/2π[ and p 1.

If in addition V + (y) ≤ D(V + (x) + |y -x| 2 + D ′ ) ( 31
)
for some D > 0, D ′ and all pair (x, y), then for all t ∈]0, 1/2D ∧ 1/2π[

P t δ x log p + (P t δ x ) dµ ≤ 4 p-1 (1 + D p ) (V + (x) + D ′ ) p + C min t 2 p + n 2 log( 1 2πt ) p . (32)
In particular, if e V + dν := M < +∞,

P t ν log p + (P t ν) dµ 1 p ≤ p C(ν, t 0 ) ( 33
)
for all t t 0 > 0, where C(ν, t 0 ) only depends on t 0 , M , C min (λ) and the dimension. If in addition (31) holds, it is enough to assume that e λ V + dν := M < +∞ for some λ0.

Proof ⊳ Let F = exp V (x) -V (W t ) - 1 2 t 0 |∇V | 2 -∆V (W s )ds ,
where W is a Brownian motion starting from x. Recall that F is a density of probability (with our hypotheses). If I(t) = P t δ x log p + (P t δ x ) dµ we may use the argument in [Roy99, Theorem 3.2.7] and the convexity of u → u p in order to get

I(t) ≤ E F 4 p-1 V p + (x) + (V (W t ) - 1 2t |W t -x| 2 ) p + + (C min t/2) p + n 2 log( 1 2πt ) p .
The first statement follows easily bounding (V (W t ) -1 2t |W t -x| 2 ) + by D(V (W t ) + D ′ ) + and u p e -u by p p e -p . The second one is immediate since (31) allows us to bound the same term by V + (x) for t small enough. The last statements are obtained by using two arguments. First u p ≤ p! e u (or u p ≤ p!(1/λ) p e λp ), so that for a given t the result follows from (p!) 1 p ≤ cp. The second one is standard, namely t → P t h log p + P t h dµ is non-increasing. ⊲

We shall come back to the condition (31) later on. Note however that such a condition is trivially verified for V (x) = |x| γ , 0 < γ ≤ 2.

Rate of convergence for the relative entropy

Theorem 5.2 Let dµ = e -2V dx be a probability measure which satisfies a WLSI with function β W L and let ξ be defined as in (24) of Proposition 4.1. Assume that (29) holds and let ν be a probability measure such that (33) holds.

Then for all 1 ε > 0 and all k > 0, there exist a constant C(ε, k) depending (in addition) on M , C min and the dimension only, and t ε > 0 such that

Ent µ (P kt ν) ≤ C(ε, k) log k(1-ε) (1/ξ(t)) ,
for all t > t ε .

Before proving the theorem we need a preliminary result. Recall first that for all non-negative functions f, g we have Ent µ (f + g) ≤ Ent µ (f ) + Ent µ (g). Then for h 0, applying this with f = P t (h1 I h≤K ) and g = P t (h1 I h>K ), and using the fact that entropy is decaying along the semigroup, we obtain that Ent µ (P t h) ≤ Ent µ (P t (h1 I h≤K )) + Ent µ (h1 I h>K ) ,

for all K > 0. The next Lemma explains how control the second term of the right hand side of (34) using the estimate of the Proposition (5.1).

Lemma 5.3 Let h be a density of probability with respect to µ. Assume that there exists c > 0 such that for all p > 1,

h log p + h dµ 1 p ≤ cp.
For K e 2 , if Ent µ (h) ≤ 1 2e log K then we get

Ent µ (h1 I h>K ) ≤ (ec + 2) Ent µ (h) log K log log K Ent µ (h) . ( 35
)
Proof ⊳ It is easily seen (see e.g. [CG06b, Lemma 3.4]) that if K e 2 , 1 I h>K hdµ ≤ 2 log K Ent µ (h) . ( 36
)
Hence

h log h 1 I h>K dµ ≤ h 1 I h>K dµ p-1 p h log p + (h) dµ 1 p (37) ≤ c p Ent µ (h) log K p-1 p ≤ ce Ent µ (h) log K log log K Ent µ (h) provided Ent µ (h) ≤ 1 e log K.
The last inequality is obtained by an optimization upon p (for which we need Ent µ (h) ≤ 1 e log K).

If Ent µ (h) ≤ 1 2e log K, - 1 I h>K hdµ log 1 I h>K hdµ ≤ - 2 log K Ent µ (h) log 2 log K Ent µ (h) ,
using (36), so that we have finished the proof. ⊲ Proof of Theorem 5.2 ⊳ Let h = P s ν. According to (34), Proposition 4.1 and Lemma 5.3, it holds for all t > s > 0,

Ent µ (P t ν) ≤ Kξ(t -s) + c s H log K log log K H ,
where H = Ent µ (h), provided K is large enough. Since H can be bounded from above by a quantity H 0 depending on M , C min and the dimension only, we may choose K > K 1 independent of H.

Choosing K = c H 0 ξ(t-s) 1 1+log + H ξ(t-s)
, we obtain

Ent µ (P t ν) ≤ C 1 + log + log + (1/ξ(t -s)) 1 + log + (1/ξ(t -s)) . ( 38
)
It follows that, for all 1 ε > 0 there exists some t ε such that for t t ε

Ent µ (P t ν) ≤ C log 1-ε (1/ξ(t)) . ( 39
)
Using again (34) and ( 35) (we may choose c = c s for all t s) we may write

Ent µ (P 2t ν) ≤ Kξ(t) + c Ent µ (P t ν) log K log log K Ent µ (P t ν) ≤ Kξ(t) + cc ′ log K log 1-2ε (1/ξ(t)) + c log log + K log K log 1-ε (1/ξ(t))
where we have used y log(1/y) ≤ c ′ y 1-ε for y ≤ 1/e. Hence choosing K = 1/ξ(t) log 2 (1/ξ(t)) we obtain a bound like

Ent µ (P 2t ν) ≤ C log 2-2ε (1/ξ(t)) ,
for t large enough. Note that C depend on ε. We may iterate the method and get the result. ⊲ Of course this result is not totally satisfactory, but it indicates that the decay of entropy is faster than any 1/ log k(1-ε) (1/ξ(t/k)).

Example 5.4 Let us study the two classical examples we already mentioned. To be rigorous |t| := √ 1 + t 2 in what follows (to ensure the required regularity), so that (29) is satisfied.

• For α > 0, the measure dm α (t) = Z α (1 + |t|) -1-α dt, t ∈ R satisfies the weak logarithmic Sobolev inequality with ∀s ∈ (0, 1),

β W L (s) = C (log 1/s) 1+2/α s 2/α ,
for some constant C > 0. Hence,

ξ(t) = c α t α/2 log 1+α (t)
for large t, and

Ent mα (P kt ν) ≤ C α,k,ε log k(1-ε) (t) .
Notice that, if roughly the rate of decay does not depend on α (it is faster than any log k (t)), the dependence on α of all constants shows that this regime is attained for smaller t when α increases.

• For α ∈ (0, 2), the measure dµ α (t) = Z α e -|t| α dt, t ∈ R, (Z α is a normalization constant) satisfies the weak logarithmic Sobolev inequality with

β W L (s) = C(log 1/s) (2-α)/α , C > 0.
Hence ξ(t) = c e -dt α/2 and for t large enough,

Ent µα (P kt ν) ≤ C α,k 1 + t (α/2)(k-ε) .
Of course this result is not satisfactory for α 2 where we know that the decay is exponential. See below for an improvement.

If we replace Proposition 4.1 or Proposition 4.2 we can greatly improve the previous results. Let us describe the latter situation.

Theorem 5.5 In the situation of Example 4.3 (i.e. the Latala-Oleszkiewicz situation) and Theorem 5.2, there exists s > 0 such that for all 1 ε > 0 one can find T ε in such a way that for t T ε ,

Ent µ (P t+s ν) ≤ e 1-t (1-ε)α 2-εα .
In particular for α = 2 relative entropy is exponentially decaying.

Proof ⊳ The beginning of the proof is similar to the one of Theorem 5.2 but replacing the estimate of Proposition 4.1 by the one of Example 4.3 (in particular we may take K = +∞ if α = 2). The first step yields

H t := Ent µ (P t+s ν) ≤ C(1 + log α 2-α + (t)) 1 + t α 2-α H log(1/H) .
Let us choose s in such a way that H ≤ 1/e, i.e. H log(1/H) ≤ 1. Then

H 2t ≤ C(1 + log α 2-α + (t)) 1 + t α 2-α H t log(1/H t ) ≤   C(1 + log α 2-α + (t)) 1 + t α 2-α   2 log(1 + t α 2-α ) ,
provided C 1 that we can assume. Iterating the procedure we get

H kt ≤   C(1 + log α 2-α + (t)) 1 + t α 2-α   k k-1 j=1 log (1 + t α 2-α ) j ≤   C (1 + log α 2-α + (t)) log(1 + t α 2-α ) 1 + t α 2-α   k (k -1)! log(1 + t α 2-α )
Now, we may find t ε such that for t t ε ,

C (1 + log α 2-α + (t)) log(1 + t α 2-α ) 1 + t α 2-α ≤ 1 t α 2-α (1-ε) ,
and log(1 + t α 2-α ) 1, so that

H kt ≤ k e t α 2-α (1-ε) k
as soon as k is large enough (for (k -1)! ≤ (k/e) k ). Choosing t = k (2-α)/α(1-ε) (hence k large enough for t to be greater than t ε ) we obtain that

H u ≤ e -k for u = k 2-εα (1-ε)α , i.e. H t ≤ e e -t
(1-ε)α 2-εα .

⊲

Of course the statement of the Theorem is not sharp (we have bounded some logarithm by some power) but it is tractable and shows that (up to some ε) the decay is similar to ξ. Of course we are able to derive a similar (but not very explicit) result with the general bound (A) in Proposition 4.2.

Comparison results and convergence in total variation distance

It is interesting to see what can be done by using the usual Poincaré inequality. Indeed recall that Ent µ (g) ≤ Var µ (g)/ gdµ for a nonnegative g. Using this with g = P t (h1 I h≤K ), using also (34) and Poincaré yield a decay

Ent µ (P t ν) ≤ C 1 + log + (t)
1 + t that is a slightly better result than the one we may obtain at the first step of the previous method (up to a log + (t) factor) in this situation (corresponding to α = 1). But iterating the procedure also yields a polynomial decay. Nevertheless if P s ν ∈ L 2 (µ) for some s, we obtain an exponential decay. It is thus particularly interesting to study stronger integrability condition. It turns out that Royer's method furnishes a much better result (in a sense) than the one shown in Proposition 5.1, namely Proposition 5.6 Under the hypotheses of Proposition 5.1, for all t > 0, all x ∈ R n ,

(P t δ x ) 2 dµ ≤ (2πt) -n 2 e C min t e 2V (x) .
This result can be shown exactly as Proposition 5.1 replacing the convex function u → u log p + (u) by u → u 2 (see e.g. [START_REF] Cattiaux | Deviation bounds for additive functionals of Markov processes[END_REF]). It shows that for an initial condition ν, a sufficient condition for

P t ν ∈ L 2 (µ) (for t > 0) is e 2V dν < +∞ .
This is of course a very strong assumption. In particular, it has been shown by P.A. Zitt ([Zit06]), that Theorem 5.2 can be used to show the absence of phase transitions in some infinite dimensional situations, while the control in the previous Proposition is not useful. We shall study an example in the next subsection, showing that actually, one can expect a still better integrability for P t δ x .

To finish this section we shall now discuss the weaker convergence in total variation distance. Denoting again h = P s ν, we thus have for K > 0

|P t h -1|dµ ≤ |P t (h ∧ K) -P t h|dµ + |P t (h ∧ K) -(h ∧ K)dµ|dµ + | (h ∧ K)dµ -1| ≤ |P t (h ∧ K) -(h ∧ K)dµ|dµ + 2 (h -K)1 I h K dµ (40)
where we have used the fact that P t is a contraction in L 1 . The second term in the right hand sum is going to 0 when K goes to +∞, while the first term can be controlled either by Var µ (P t (h ∧ K))

or by 2( (h ∧ K)dµ) Ent µ (P t (h ∧ K)) according respectively to Cauchy-Schwarz and to Pinsker inequality. In both cases, WPI or WLSI inequalities imply that P t ν goes to µ in total variation distance, for all initial ν.

If we want a rate of convergence, we immediately see that WPI will furnish a better rate than WLSI for the µ that do not satisfy Poincaré inequality. If µ satisfies a Poincaré inequality with constant C P then Var µ (P t (h ∧ K)) ≤ Ke -t/C P , so that the optimal K is given (up to a factor 2) by 2 (h -K)1

I h K dµ = K 1 2 e -t/2C P . In particular if (29) holds, 2 (h -K)1 I h K dµ ≤ 2C(p) log p (K)
for K > 1 and p 1, so that we obtain P t+s ν -µ T V ≤ κ(p)/t p for all s > 0, p 1, where κ depends on s, C min , p, M and the dimension. But if we directly use Theorem 5.5 and Pinsker we have the much better

P t+s ν -µ T V ≤ κ e -1 2 t (1-ε)α
2-εα at least for s large enough. In particular for α = 1 we obtain a faster decay. Once again, if P s ν ∞ is finite for some positive s then one should use the entropic convergence of Proposition 4.2 to get an exponential decay.

Example(s)

In the previous subsections, we have seen that finite entropy conditions are quite natural for the law of the diffusion at any positive time, but that starting from an initial Dirac mass, we immediately reach L 2 (µ). Before to study examples indicating that one can expect much better, we shall give a generic example showing that some natural measures ν never satisfy P s ν ∈ L 2 (µ), but satisfy the conditions in Proposition 5.1. Consider V such that for all λ > 0 , e -λV dx < +∞. Let dµ = e -2V dx and dν = e -(2-ε)V /Z ε dx so that dν/dµ

:= h = Z ε e εV / ∈ L 2 (µ) for 2 > ε > 1, but e 2-ε 2 V dν < +∞. Set G = e V = h 1 ε . If P s h ∈ L 2 (µ)
for some s > 0, then P s G ∈ L 2ε (µ). If (29) holds, it follows from [Cat05, Theorem 2.8] that µ satisfies a logarithmic Sobolev inequality. Thus if it is not the case, P s h / ∈ L 2 (µ) for all s 0, while if (31) is satisfied (for instance for V (y) = |y| α , 1 ≤ α < 2 see below) ν satisfies the conditions in Proposition 5.1. This example shows that the set of initial measures satisfying the conditions in the previous subsection but not the necessary conditions to simply apply Poincaré is non empty.

We shall go further, and for simplicity we shall only consider the measures µ α for α 1, and essentially discuss the case α = 1. First of all notice that if 1 ≤ α ≤ 2,

|y| α ≤ 2 α-1 (|x| α + |y -x| 2 + 1)
so that (31) is satisfied. Hence as soon as e λ|x| α ν(dx) < +∞ for some λ > 0, we may apply all the results of the previous subsection. We shall now give a precise description of h = P s δ x . This will allow us to give a similar sufficient condition for P s ν to belong to L 2 (µ).

We thus consider (in one dimension)

dX t = dB t -sign(X t )dt , X 0 = x , (41)
corresponding to α = 1. Elementary stochastic calculus (inspired by the first sections of [START_REF] Gradinaru | A singular large deviations phenomenon[END_REF]) furnishes

E[f (X t )] = E f (x + B t ) e -t 2 exp - t 0 sign(x + B s )dB s = e |x| e -t 2 E [f (x -W t ) exp (-|W t -x| + L x t )]
where W s = -B s is a new Brownian motion with local time at x denoted by L x s . Now as usual we introduce the hitting time of x of (W s ) denoted by T x , and the supremum S t = sup 0≤s≤t W s . We also assume here that x > 0. Then

E[f (X t )] = E[f (X t ) 1 I t≤Tx ] + E[f (X t ) 1 I t>Tx ] = e |x| e -t 2 E[f (x -W t ) 1 I St≤x e Wt-x ] + e -t 2 E[1 I St>x E[f (B ′ t-Tx ) exp -|B ′ t-Tx | + L ′ t-Tx ]]
where B ′ is a Brownian motion independent of W and L ′ its local time at 0. For the first term, we know that the joint law of (W t , S t ) is given by the density (w, s) → 1 I w≤s 2/πt 3 (2s -w) exp(-(2s -w) 2 /2t) so that (recall x > 0)

E[f (X t ) 1 I t≤Tx ] = f (u) 1 I u 0 2/πt e -t 2 e x e -u e -(x-u) 2 /2t -e -(x+u) 2 /2t du .
For the second term, we know that the law of T x is given by the density

T → x 1/2πT 3 e -x 2 /2T
and that (|B ′ s |, L ′ s ) has the same law as (S ′ s -B ′ s , S ′ s ) so that (noting that only the even part of f has to be considered)

E[f (X t ) 1 I t>Tx ] = e -t 2 1 I 0<T <t 1 I u>0 1 I v>u f (u) + f (-u) 2 g(T, u, v) dudvdT , with g(T, u, v) = 1/2πT 3 2/π(t -T ) 3 v e v e -2u e -v 2 /2(t-T ) e -x 2 /2T . But Q := t 0 +∞ u 1/2πT 3 2/π(t -T ) 3 v e v e -v 2 /2(t-T ) e -x 2 /2T dvdT is such that Q ≤ t 0 1/2πT 3 2/π(t -T ) e u e -u 2 /2(t-T ) + 2e t-T e -x 2 /2T dT ≤ t 0 1/2πT 3 2/π(t -T ) e t/2 + 2e t-T e -x 2 /2T dT ≤ C(t)
independently of x. The first inequality is obtained by performing an integration by parts in v, the second one by bounding e u e -u 2 /2(t-T ) and the final one by bounding separately t/2 0 and t t/2 . We thus see that

E[f (X t ) 1 I t>Tx ] = C ′ (t) f (u) e -2|u| g(u) du
where g is bounded.

Putting all this together we have obtained the following

(P t δ x )(u) = c(t) 1 I u 0 e x e u e -(x-u) 2 /2t -e -(x+u) 2 /2t + C ′ (t)g(u) (42)
for all x > 0. A similar result holds for x < 0, while P t δ 0 is bounded. Of course the previous (42) shows that for a fixed x, P t δ x is bounded. This result is not so surprising. Indeed for α = 2 (more precisely for the normalized gaussian measure i.e. the Ornstein-Uhlenbeck process) (P t δ x )(u) = c(t) e (1-e -t )x 2 /2(1-e -t ) e -(e -t/2 u-x) 2 /2(1-e -t ) is bounded too. One may adapt our proof and Proposition 4 in [START_REF] Gradinaru | A singular large deviations phenomenon[END_REF] in order to show that a similar result actually holds for all 1 ≤ α ≤ 2. But (42) allows us to look at more general P t ν. In particular we see that P t ν ∈ L 2 (µ) if and only if u>0 x>0

e x e -(u-x) 2 /2t ν(dx)

2 du < +∞ (43)
and a similar property is available on the negative real numbers. We then easily recover and complete the discussion at the beginning of this subsection, i.e. if dν = e -λ|x| dx/Z ,

P t ν / ∈ L 2 (µ) if λ ≤ 1, but belongs to L 2 (µ) if λ > 1.
Let us finally give some discussion concerning the obtainable rate of entropic convergence depending on the initial measure: i. if ν = δ x , then P t 0 δ x ∞ < ∞ and using respectively Proposition 4.1, Proposition 4.2 or Poincaré inequality, one gets

Ent µ (P t+t 0 ν) ≤ C min e -a √ t P t 0 δ x ∞ , e -bt/(1+log Pt 0 δx ∞) , e -ct P t 0 δ x ∞ ,
(note that it easily extends to the case where ν has compact support.)

ii. if ν does not satisfy (43) but for some positive λ, e λ|x| dν is finite then we can only use Theorem 5.5 to get that for all ε > 0, there exists T ε such that for all t ≥ T ε we have

Ent µ (P t ν) ≤ e 1-t 1-ε 2-ε .
6 Classical properties of WLSI

Tensorization

Let us begin by the following naive procedure of tensorization.

Proposition 6.1 Assume that µ satisfies a WLSI with function β and let n ≥ 1. Then the measure µ n satisfies a WLSI with function β s n , for s > 0.

Proof ⊳ By the sub-additivity property of the entropy we get

Ent µ n (f ) ≤ n i=1 Ent µ (f (x 1 , . . . , x i-1 , •, x i+1 , . . . , x n )) j =i dµ(x j ).
For each i we get for all (x 1 , . . . ,

x i-1 , x i+1 , . . . , x n ) ∈ M n-1 Ent µ (f (x 1 , . . . , x i-1 , •, x i+1 , . . . , x n )) ≤ β(s) |∇ i f | 2 (x 1 , . . . , y i , . . . , x n )dµ(y i ) + s Osc(f (x 1 , . . . , •, . . . , x n )) 2 , It yields ∀s > 0, Ent µ n (f ) ≤ β(s) |∇f | 2 dµ n + ns Osc 2 (f ). ⊲
The tensorization result above is of course the same as the one in [START_REF] Barthe | Concentration for independent random variables with heavy tails[END_REF] for weak Poincaré inequality. As explained in Section 5 of this paper, one cannot expect a better result beyond the exponential case. However as we have already seen, WLSI may take place between the exponential and the gaussian regime (when GBI holds), so that we obtain this corollary:

Corollary 6.2 If µ i (1 ≤ i ≤ n) satisfy a WLSI with the same function β W L satisfying the hypotheses in Proposition 3.6, then the tensor product ⊗ n i=1 µ i satisfies a WLSI with function

β n W L (u) = C β W L (C ′ u)
where C, C ′ are constants which don't depend on n.

Proof ⊳ It is enough to use both parts of Proposition 3.6 and the (exact) tensorization property of GBI. One can see [START_REF] Lata | Between Sobolev and Poincaré. in geometric aspects of Functional Analysis[END_REF] for the proof of the tensorization of GBI. ⊲ Among the most important consequences of functional inequalities, one find concentration of measure and isoperimetric profile. Unfortunately weak inequalities are not easily tractable to derive results in this direction (due to the Oscillation term). However results for WPI are contained in [RW01, BCR05] with a particular interest in dimension dependence in the latter. Actually we do not succeed in deriving similar estimates starting from WLSI, as Herbst's argument or Aida-Masuda-Shigekawa iteration argument are more intricate and we can only recover weak Poincaré non optimal concentration rate. The situation is still worse (from the WLSI point of view) when a SPI holds. In this case various (more or less explicit) results have been obtained. Let us mention on one hand [Wan00] Section 6, [GW02] Section 5 (using super Poincaré) and [START_REF] Wang | A generalization of Poincaré and log-Sobolev inequalities[END_REF] Corollary 2.4 (using GBI), on the other hand [BCR06a] Section 6 (using GBI) and Section 8 (using F -Sobolev inequalities) and [START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF] Theorem 12 for an improvement of [Wan00] Section 6. The previous result may be used in conjunction with the above mentioned results to get dimension free concentration (or isoperimetric) results, completing thus the transportation approach presented before.

Perturbation

Among the methods used to obtain functional inequalities, an efficient one is to perturb measures satisfying themselves some functional inequalities. The most known result in this direction was first obtained by Holley and Stroock who showed that a logarithmic Sobolev inequality is stable under a log-bounded perturbation. The same is true for a SPI (using the related GBI [Wan05, Proposition 2.5]), and actually one can replace the bounded assumption by a Lipschitz assumption (this was shown by Miclo for logarithmic Sobolev, and by Wang [Wan05, Proposition 2.6] for a SPI).

For the WPI, a similar result is shown in [RW01, Theorem 6.1]. Actually this result shows that one can consider non bounded perturbation, but with very strong integrability assumptions, the final result being far to be explicit. For WLSI we may state Proposition 6.3 Suppose that µ satisfies a WLSI with function β W L . Let ν V = e V µ/Z V , where Z V = e V dµ and assume that V is bounded on M . Then ν V satisfies a WLSI with function

β V W L (u) = e 2Osc(V ) β W L (ue -Osc(V ) ) .
We may replace WLSI by WPI replacing β W L by β W P , or by SPI with

β V SP (u) = e 2Osc(V ) β SP (ue -2Osc(V ) ). Proof ⊳ Recall that Ent ν V f 2 ≤ e Osc(V ) Ent µ f 2 . Applying WLSI for µ yields Ent ν V f 2 ≤ e Osc(V ) β W L (s) |∇f | 2 dµ + s Osc 2 (f ) ≤ e 2Osc(V ) β W L ue -Osc(V ) |∇f | 2 dν V + u Osc 2 (f ) ,
which is exactly the first statement. The second one is similar since Var ν V (f ) ≤ e Osc(V ) Var µ (f ).

For SPI the proof is immediate. ⊲

The second way to get perturbation results is to use a natural isometry between L 2 spaces. For notational convenience we assume now that ν V = e -2V µ. Then g → f := e -V g is an isometry between L 2 (ν V ) and L 2 (µ). It is thus immediate that on one hand

Ent ν V g 2 = Ent µ f 2 + 2 g 2 V dν V . (44)
On the other hand, an integration by parts yields

|∇f | 2 dµ = |∇g| 2 dν V + g 2 2LV -|∇V | 2 dν V , (45)
where L is the generator of P t reversible for µ.

Combining these two facts, yields perturbation results for logarithmic Sobolev inequalities (the idea goes back to Rosen [START_REF] Rosen | Sobolev inequalities for weight spaces and supercontractivity[END_REF], and was used in [START_REF] Carlen | Superadditivity of Fisher's information and Logarithmic Sobolev inequalities[END_REF][START_REF] Cattiaux | Hypercontractivity for perturbed diffusion semigroups[END_REF]). In order to see how to use it in our framework, we shall first introduce some notation.

Definition 6.4 Let G be a positive continuous function defined on R + . We shall say that a smooth V is (G, µ)-good, if V (x) → +∞ as |x| → +∞ and if there exists A 0 such that one has for any

x such that V (x) A, |∇V | 2 (x) -2LV (x) G(V (x)) .
Our first general result is a bounded (but not log-bounded) perturbation result.

Proposition 6.5 Let µ be a positive measure (not a necessarily probability measure) satisfying a WLSI with continuous function β W L . Let V be (G, µ)-good, such that ν V = e -2V µ is a probability measure.

Then for all u > 0 and b A the following inequality holds for any g ∈ H 1 (E, µ), (which is finite) and β V W P is the best function such that ν V satisfies WPI (if it does not take β V W P (u) = +∞ for small u). Proof ⊳ First according to Rothaus inequality, we may assume that gdν V = 0 up to 2Var ν V (g). Applying WLSI in (44) and (45) we get for all s > 0,

Ent ν V g 2 ≤ C(u
Ent ν V g 2 ≤ β W L (s) |∇g| 2 dν V + g 2 β W L (s) 2LV -|∇V | 2 + 2V dν V + sOsc 2 (ge -V ) . (48)
Note that if β W L is bounded, we may replace it by any β(s) β W L (0).

• On {V ≤ A}, the second integrand is bounded by (β W L (s)M (V ) + 2A) Var ν V (g), and can be controlled (together with the term 2Var ν V (g) coming from Rothaus inequality) with the WPI for the measure ν V .

• On {b V A}, we choose s = s b then the second integrand is non-positive.

• On {b ≤ V }, 2LV -|∇V | 2 is still non-positive, so that the second integrand is bounded by

{V b} 2V g 2 dν V ≤ {V b}
2V dν V Osc 2 (g) , since gdν V = 0. ⊲ For this proposition to be useful, we must choose u and b in such a way that D(u, b) → 0 as b → +∞.

If µ is a probability measure, e 2V dν V = 1 so that if b > 1/2,

{V b} 2V dν V ≤ Ent ν V (1 I V b ) = ν V (V b) log 1 ν V (V b) ≤ b e -2b
where we used Markov inequality and the fact that x log(1/x) is non decreasing on [0, 1/e] for the latter.

If µ is not bounded, we assume in addition that e -pV dµ = K(p) < +∞ for some p < 2, so that a similar argument (changing the constants) yields again If h(b) Cb we obtain that β V W L (s) behaves like a function greater than or equal to (up to some constants) log(1/s)β V W P (s/ log(1/s)) in the first case, β W L (s)β V W P (s/β W L (s)) in the second case, with β W L (s) larger than log(1/s) in the latter case. Hence the result is not better (even worse) than (13) in Proposition 3.1. If h(b)/b → 0 as b → +∞ we obtain the same results, but replacing log(1/s) by h(log(1/s)), provided β V W P is not bounded (otherwise β V W L (s) = Ch(log(1/s)) for some C). Hence if β W L (s) ≪ log(1/s) we obtain a better result that the one in Proposition 3.1, namely ν V satisfies WPI with a function β(s) h(log(1/s)) log(1/s) β V W P (cs) provided this function is non-increasing. But if there exists M such that β V W P (cs) ≤ M β V W P (s), we may thus choose β ≤ (1/2)β V W P , which leads to a contradiction since β V W P is assumed to be the best one. We have thus obtained (recall that we leave some constants away in the previous argument) Corollary 6.6 Let µ be a positive measure (not necessarily bounded) satisfying a WLSI with continuous function β W L . Let V be (G, µ)-good, such that ν V = e -2V µ is a probability measure. If µ is not bounded, we assume in addition that there exists p < 2 such that e -pV dµ < +∞. • β W L (s)/ log(1/s) → 0 as s → 0 (that is, if µ is bounded, µ satisfies some SPI which is stronger than the usual Poincaré inequality).

{V b} 2V dν V ≤ ν V (V b) (2/2 -p) log K(p) ν V (V b) ≤ (2K (

Assume in addition that

Then ν V satisfies a Poincaré inequality, and a WLSI with function β V W L (s) = ah(a ′ log(1/s)) for some constants a and a ′ .

In particular if G(z) cz for large z, ν V satisfies the usual logarithmic Sobolev inequality.

The previous result extends part of the results in [START_REF] Cattiaux | Hypercontractivity for perturbed diffusion semigroups[END_REF] since we do not assume that µ satisfies a logarithmic Sobolev inequality. It has to be noticed that the conditions in Corollary 6.6 are far to be optimal for ν V to satisfy Poincaré inequality. Indeed if µ = dx on the euclidean space, it is known that G(b) k > 0 for large b is sufficient (i.e. h asymptotically linear) (see [START_REF] Cattiaux | Hypercontractivity for perturbed diffusion semigroups[END_REF] for a reference). In the general manifold case with µ the riemannian measure, Wang ([Wan99] Theorem 1.1 and Remark 1) has obtained a beautiful sufficient condition, namely -Lρ(x) k > 0 for ρ(x) large, when ρ is the riemannian distance to some point o. In the flat case, this condition reads |∇V |(x) > k > 0 for |x| large. In the one dimensional case, it is easy to see that this condition is weaker than our G(b) k > 0 for large b. Wang's condition thus appears as the best general one, though it is not necessary as shown in one dimension by a potential V (x) = x + sin(x) for large x. But Wang's approach, based on Cheeger inequality and the control of local Poincaré inequality outside large balls, seems difficult to extend to more general functional inequalities (though it can be used in particular cases, see [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF] section 3 and [START_REF] Wang | Functional inequalities for empty essential spectrum[END_REF]).

Example 6.7 For 1 < α ≤ 2 and G(u) = u 2(1-1 α ) we recover (here dµ = dx) the same β W L as the one corresponding to the measure µ α studied at the end of section 2. This furnishes a new proof of some results in [START_REF] Barthe | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry[END_REF] section 7.2. For more general G the result is linked to the perturbation results in [START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF].

  , b)|∇g| 2 dν V + D(u, b) Osc 2 (g) , with C(u, b) = h(b) + (2 + 2A + M (V )h(b)) β V W P (u) ,(46)D(u, b) = s b e -2 inf V + (2 + 2A + M (V )h(b))) u + {V b} 2V dν V ,(47)where h(b):= sup {A≤z≤b} 2z G(z) , s b := inf {s > 0 , β W L (s) ≤ h(b)}, M (V ) := sup {V ≤A} (2LV -|∇V | 2 ),

  p)/(2 -p)) b e (p-2)b if b (1 + log(K(p))/(2 -p). In both cases, defining ε as the upper bound, one can find constants a and a ′ (depending on p if necessary) such that b = a log a ′ log(1/ε) ε , and the appropriate choice for u is then u = ε/h(b), provided β W L (ε) ≤ h(b). Conversely, if β W L (ε) h(b), s b is greater than ε (up to multiplicative constants) and the good choice is then u = s b /h(b).

•

  h(b) := sup {A≤z≤b} 2z G(z) is such that h(b)/b → 0 as b → +∞,

  ⊳ Assume that µ satisfies a WLSI with function β W L . Using Theorem 2.1 and Remark 2.3 one has inequality (16). Using the fact that Note that if the function T defined on (18) is non-decreasing near 0 then one can prove that β W L (s) ≤ c 1 log(c 2 /s) for s small enough and some constants c 1 , c 2 > 0. Then by Proposition 3.1, µ satisfies a Poincaré inequality. The last proposition can be applied only for measures satisfying a Poincaré inequality.

	∀x ∈ (0, 1],	log 1 +	1 2x	1 2	log 1 +	1 x	,
	one obtains that inequality (16) implies that the function T defined on (18) satisfies a measure-
	capacity inequality. The function x → T (x)/x is non-increasing and due to the fact that T is non-decreasing by hypothesis, then Theorem 10 and Lemma 9 of [BCR06a] prove that µ satisfies a
	GBI of function T .
	To prove the second statement we need also Theorem 10 and Lemma 9 of [BCR06a], Theorem 2.2
	and inequality (14). ⊲
	Example 3.7

  Example 4.3 Let α ∈ [1, 2] and dµ α (t) = Z α e -|t| α dt, t ∈ R where Z α is a normalization constant. Using Example 2.6 and Proposition 3.6 one obtains that µ α satisfies a GBI with T (x) = C x

	2α-2
	α

then the entropic decay obtained by Proposition 4.2 is better than the estimate with Poincaré inequality.