On $\mathcal H^1$ and entropic convergence for contractive PDMP

Explicit rate of convergence in variance (or more general entropies) is obtained for a class of Piecewise Deterministic Markov Processes such as the TCP process, relying on functional inequalities. A method to establish Poincar\'e (and more generaly Beckner) inequalities with respect to a diffusion-type energy for the invariant law of such hybrid processes is developped.


Introduction
This work is devoted to the study of convergence to equilibrium for a class of Piecewise Deterministic Markov Process (PDMP). These hybrid processes, satisfying a deterministic differential equation between random jumps, have received much attention recently : we refer to [5] and the references therein for an overview of the topic. Ergodicity and, then, speed of convergence to the steady state are particularily studied. As far as this last point is concerned, coupling methods have recently proved efficient in order to get explicit rate of convergence in Wasserstein distances for PDMP (see [17,7,10,26,13] for instance, among many others). On the other hand, another classical approach to quantify ergodicity, based on functional inequalities, is hardly used, since the usual methods do not directly apply. Our aim is to adapt them (see also [33] in this direction).
Let Ω be an open set of R d . The dynamics is defined thanks to a vector field b : Ω → R d , a jump rate λ : Ω → R + , and a transition kernel Q which will be seen either as a function from Ω to P (Ω) the set of probability measures on Ω, or as an operator on the space C (Ω) of all continuous functions on Ω. For x ∈ Ω and t > 0 let ϕ x (t) be the flow associated to b, namely the solution of Starting at point x, the process (X t ) t≥0 deterministically follows this flow up to its first jump time T x with law P (T x < s) = s 0 λ (ϕ x (u)) e − u 0 λ(ϕx(w))dw du = 1 − e − s 0 λ(ϕx(w))dw .
At time T x , the process jumps according to the law Q (ϕ x (T x )), and starts anew from its new position. The infinitesimal generator of the process is defined for smooth f ∈ C (Ω). We note the associated semi-group. The following assumptions hold throughout this work: • the flow is well-defined and it stabilizes Ω: if x ∈ Ω then ϕ x (t) ∈ Ω for all t > 0.
• the process is non-explosive: there can't be infinitely many jumps in a finite time interval, so that the process (and therefore the semi-group) is defined for all time.
• the functions λ and b are smooth; we write J b (x) = [∂ i b k (x)] 1≤i,k≤d the Jacobian matrix of b = (b k ) 1≤k≤d .
• The process admits a unique invariant law µ, and P t is ergodic in the sense it converges weakly to µ as t goes to infinity. Moreover The generator L is well-defined on the set A of all compactly supported smooth functions on Ω, and A is dense in L 2 (µ).
The test functions will always belong to A in order to keep the study at a formal level, all the forthcoming elementary definitions and calculations being licit in this framework. These strong assumptions allow us to focus only on the quantification of ergodicity. Note that the uniqueness of the invariant measure and the ergodicity of the process may often be proved by checking it is irreducible and admits a Lyapunov function (cf. [24]). Throughout this work the test functions will always belong to the set A, in order to keep the study at a formal level, all the forthcoming elementary definitions and calculations being licit in this framework.
We recall here some classical arguments (see [4] for a general introduction to functional inequalities and for the detailed proofs of the assertions in this paragraph). For f ∈ A, we write Γ (f ) = 1 2 L(f 2 ) − f Lf the carré du champ operator of L, Γ(f , g) the corresponding symetric bilinear operator obtained by polarization, and Writing ψ(s) = P s Γ (P t−s f ), from ∂ t P t f = LP t f = P t Lf one gets ψ ′ (s) = 2P s Γ 2 (P t−s f ) .
Hence, if the Bakry-Emery (or Γ 2 ) criterion Γ 2 > ρΓ holds for some ρ > 0, the Gronwall Lemma yields ψ(0) ≤ e −2ρt ψ(t), namely For instance for the Ornstein-Uhlenbeck process with generator this reads where |.| is the euclidian norm of R d . In fact, the sub-commutation (2) is equivalent to the Bakry-Emery criterion. Nevertheless the latter does not usually hold in our settings. That said, a simple adaptation of the Γ 2 argument will give, at least in the constant jump rate case, a gradient estimate similar to (3). In the following we denote by A * the usual transpose of a matrix A and thus by u * v the scalar product of two vectors.
Inequality (4) is a balance condition on the drift and the jumps, reminiscent of the condition on the curvature in [19]. More precisely, suppose |∇Qf (x)| 2 ≤ M (x)Q|∇f | 2 (x) for some function M on Ω. If M < 1, Q is a contraction of the Wasserstein distance (this will be detailed in Section 2); it means two particles that simultaneously jump can be coupled so that they get closer. More generaly M measures how two such particles can be coupled in order for them not to get too far away one from the other. On the other hand, J b measures how two trajectories of the deterministic flow tends to get closer or to drift appart. Indeed, We see that the condition u * J b (x)u < 0 for all (x, u) ∈ Ω × R d implies the flow contracts the space in the neighborhood of all points of Ω.
Note that by integrating Inequality (5) with respect to µ and writing In the non-constant jump rate case, under a condition similar to (4), we will prove there exists constants β > 0 and η ∈ R such that where E t is defined as Both W t and E t are usually called energy ; we may say W t is the classical (or diffusion-like) energy, while E t is the markovian one. They coincide in the case of the Ornstein-Uhlenbeck process. The markovian energy usually appears in particular when one is concerned with the variance of P t f with respect to µ, We say µ satisfies a Poincaré (or spectral gap) inequality with respect to Γ if there exist a constant c > 0 such that c V 0 , namely to an exponential decay in L 2 (µ). The same goes for entropy and Gross log Sobolev inequality, or general Φ-entropies (see [16] and Section 3 for some definitions), at least for diffusion processes.
For reversible processes (i.e. when L is symmetric in L 2 (µ)) there is a strong link between, on the one hand, Wasserstein distances and coupling and, on the other hand, variance (or entropy) and functional inequalities (see [6,15,29]); nevertheless PDMP are not reversible. Furthermore their invariant measures usually do not satisfy a Poincaré inequality for Γ, which is non-local, not easy to handle, satisfying no chain rule (nevertheless, see [14] for a case in which such an inequality does indeed hold).
However, they may satisfy a diffusion-like Poincaré inequality of the form in other words V t ≤ cW t . Such an inequality, which involves the classical energy rather than the markovian one, implies concentration properties for the measure µ (see [4]), but is a priori not directly linked to the convergence to equilibirum in general. Suppose such an inequality holds. Then, from inequality (6), if η > 0, This yields: Assume the Poincaré inequality (7) holds, and |∇Qf (x)| 2 ≤ M (x)Q|∇f | 2 (x) with M such that for µ-almost all x ∈ Ω and for all u ∈ R d , for some constants η, β > 0. Then .
is equivalent to the square of the usual Sobolev H 1 -norm of P t f − µf . Thus Theorem 2 provides a decay in H 1 (µ) rather than in L 2 (µ). In this sense, our method can be seen as an hypocoercive method of modified Lyapunov functional (see [37,23,9], etc.), although it is quite simple. In these settings, it is usual to assume a Poincaré inequality (7) holds. There are classical criteria on a function F on R d to decide wether the law e −F (x) dx satisfies such an inequality, and several ways to estimate the constant c. However, for PDMP, the invariant law is usually quite unknown. The second part of this work will thus be dedicated to the obtention of such inequalities, which are interesting by themselves as they provide concentration bounds for the measure µ.
The original motivation of the present work was the study of the so-called TCP process on Ω = R + , whose generator is for some δ ∈ (0, 1). It has been studied in [17], which inspired the main ideas of this work. In addition to the previous difficulties (no Poincaré inequality for Γ, non-constant rate of jump), there is an other one which is particular to this process : the jump vanishes at the origin. Nevertheless, as an illustration of the efficiency of our method, we will prove the following: Then if (P t ) t>0 is the semi-group associated to the generator (8), there exists c, r > 0 such that for all f ∈ A, Moreover it is possible to get explicit values for c and r such that this holds.
The paper is organized as follow. Slightly generalized versions of Theorems 1 and 2 are proved in Section 2. A general strategy to obtain some functional inequalities (including the Poincaré inequality) for PDMP by the study of their embedded chain is exposed in Section 3 and applied in several illustrative models in Section 4, where in particular Proposition 3 is proved. A perturbative results for Poincaré and log-Sobolev inequalities is stated and proved in an Appendix.

Exponential decay
We keep the notations and assumptions of the introduction. In particular we study the semi-group (P t ) t≥0 with generator L defined by (1).
When A is a linear operator on A and φ is a bilinear symmetric one, for f , g ∈ A we define .
With respect to f , Γ A,φ (f , f ) is quadratic, and linear with respect to A and φ. We will always note f → φ(f ) the quadratic form associated to a bilinear form f , g → φ(f , g) and similarly we will always note f , g → q(f , g) the symetric bilinear form associated by polarization to a quadratic form f → q(f ) on A. Let which interpolates between φ (P t f ) and P t (φf ). Then To prove Theorems 1 and 2 we should consider φ(f ) = |∇f | 2 . In fact it will be convenient for the applications to work with a weighted gradient φ a (f ) = a|∇f | 2 with a > 0 a scalar field on Ω.

Suppose there exists a function
, and let I be the identity operator on A. Then for all f ∈ A Proof. First we note that As far as the second point is concerned, We conclude by We can now state the following : Theorem 5. Assume λ is constant and there exist a function M on Ω and a constant η ∈ R such that, for all f ∈ A, φ a (Qf ) ≤ M Q (φ a (f )) and Then φ a (P t f ) ≤ e −ηt P t (φ a (f )) .
In particular with a = 1 we retrieve Theorem 1.
Remark that we did note use the ergodicity of the process here, and that η can be negative.
This commutation between the semigroup and the gradient leads to a contraction in Wasserstein distance. More precisely, define on Ω the distance associated to the weighted gradient D = √ a∇ by a (γ(s)) ds, γ : [0, 1] → Ω, smooth, γ(0) = x, γ(t) = y and the associated Wasserstein distance between two probability laws ν 1 , ν 2 having a finite p th moment (i.e. for which there exists a x 0 ∈ Ω with ν i [d p (., A function f will be called κ-Lipschitz with respect to D if ∀x, y ∈ Ω, This is equivalent for a smooth function to Df ∞ ≤ κ. At fixed x, the function y → d(x, y) is 1-Lipschitz with respect to D. Since the metric space (Ω, d) is locally diffeomorphic to (R d , |.|), thanks to Rademacher's theorem, this means a κ-Lipschitz function g is differentiable almost everywhere, with Dg ∞ ≤ κ. We have the Kantorovich-Rubinstein dual representation (see [38]) where we use the operator notation νf = f dν.
Recall that by duality a Markov semi-group acts on the right on probability laws by If P t were absolutely continuous with respect to the Lebesgue measure for t > 0 -which is not the case for a PDMP since for all time t there is a non-zero probability that the process hasn't jumped yet -the gradient estimate of Theorem 5 would yield, from [30, Theorem 2.2], a contraction of the W d,2 distance : Instead of trying to adapt Kuwada's result, since our work is more concerned about variance than Wasserstein distance, we will only state the weaker result : In the settings of Theorem 5, for all laws ν 1 , ν 2 with finite first moment, if P t ν 1 and P t ν 2 still have finite first moment, Proof. Theorem 5 yields the weaker gradient estimate This implies the W d,1 decay, thanks to the Kantorovich-Rubinstein dual representation Note that the invariant measure does not intervene neither in Theorem 5 nor in Corollary 6, so that its existence and uniqueness are not necessary. Besides, on a complete space, a contraction of the Wasserstein distance would imply ergodicity, from [18,Theorem 5.23].
We won't push the analysis further concerning the Wasserstein distance, but refer to the study in [7] of the TCP process where an exponential decay is first obtained for a distance equivalent to d(x, y) = |x − y| and then is transposed to d(x, y) = |x − y| p via moments estimates and Hölder inequality. For further considerations on gradient-semigroup commutation, one shall consult [12,3,30].
We now turn to the non-constant jump rate case. Let a be a non-negative scalar field on Ω. Throughout all the text we will say a probability measure ν satisfies a weighted Poincaré inequality with constant c and weight a if for all f ∈ A

Theorem 7.
Assume that µ satisfies the weighted Poincaré inequality (9) with constant c and weight a, that µ-amost everywhere λ > 0 and that there exist a function M and constants η, β > 0 such that Proof. Since µ is the invariant measure of the process, µLg = 0 for all g ∈ A. In particular if φ is a quadratic form on A, µ (Lφ(f )) = 0 and In particular From Lemma 4, Again from Lemma 4 and from Inequality (10), On the other hand, if the last inequality being a consequence of the Cauchy-Schwartz inequality for Q. At the end of the day, we get and, thanks to the weighted Poincaré inequality (9), which yields the first assertion. Then Note that η could depend on x, so that the weight that intervenes in the Poincaré inequality may be different from a. For instance for the TCP with linear rate on R + (Example 4.4), one could consider a(x) = x and η(x) = −κ − αx for some κ, α > 0. Then it would be sufficient to prove an inequality with weightã(x) = 1 + x, which is weaker than both the classical inequality with constant weight and the inequality with weight a.

Functional inequalities for PDMP
This section is devoted to the obtention of the Poincaré inequality (9) and of slightly more general functional inequalities for µ the invariant measure of the process (X t ) t≥0 with generator (1).

Confining operators
The variance is a way among others to quantify the distance to equilibrium. In this section we suppose that for all f ∈ A the so-called p-entropies , For p = 2 this is the Poincaré inequality, for p = 1 this is the Gross log Sobolev one. Since Ent p f is non increasing with p ∈ (1, 2] (see [32]; note that we took the definitions of [11]), B(p, c) implies B (q, c) whenever q ≥ p. On the other hand by Jensen inequality (1 − p)Ent p f is non decreasing with p ∈ [1,2]. In particular all Beckner's for p ∈ (1, 2] are equivalent up to some factor. For the global study of this inequalities and of more general Φ-entropies, we refer to [16] and [11]. For α = 0 this is still the Poincaré inequality, for α = 1 this is the log Sobolev one, and for α ∈ (0, 1) this is an interpolation between these two cases which implies the following concentration property: there exists a constant L > 0 such that for any borellian set A with In this section, for the sake of simplicity, we won't consider weighted inequalities such as the weighted Poincaré inequality (9). Everything would work the same, and, at least in dimension one, a weighted inequality can be seen as a non-weighted one through a change of variable (see an application in Section 4.4).
Remark that if µ satisfies B(p, c) for p ∈ [1, 2], then it satisfies a Poincaré inequality. In this case, providing the inequality (10) of Theorem 7 holds, W t decays exponentialy fast, and Let ψ : Ω → Ω be a smooth function with Jacobian matrix J ψ , and let |J ψ | be the euclidian operator norm of J ψ , namely It is clear that in this case when the law of a random variable Z satisfies B(p, c) then the law of ψ(Z) satisfies B(p, γ 2 c). In order to get Beckner's inequalities for the invariant law of a PDMP we will prove a generalization of this fact, based on an initial idea of Malrieu and Talay [36].
Let H be a Markov kernel on Ω.
We say that H is (c, γ, p)-confining if both the following conditions are satisfied : • sub-commutation: ∀f ∈ A, ∀x ∈ Ω, • Local Beckner's inequality: ∀f ∈ A, ∀x ∈ Ω, If γ < 1 we say H is (c, γ, p)-contractive. When there is no ambiguity for p, H will simply be called confining (or contractive) if there exist c, γ > 0 satisfying both conditions.
• The sub-commutation is always satisfied with γ = 0 if H(x) = ν is a constant kernel, namely is a probability on Ω, so that ν is confining iff it satisfies a Beckner's inequality.
• if N is a standard Gaussian vector on R d and (B t ) t≥0 a Brownian motion on R d then is (t, 1)-confined for the usual gradient and p = 1 (see [4]). If the Brownian motion is replaced by an elliptic diffusion, a sub-commutation is given by its Bakry-Emery curvature.
• Remark this definition could be extended to a Markov kernel H : Here is maybe our most important, although very simple result: 2. If ν ∈ P (Ω) satisfies B(p, c) then νH 2 satisfies B(p, c 2 + γ 2 c).
3. If H is (c, γ, p)-contractive and if the Markov chain generated by H is ergodic, meaning that H n converges weakly to some ν ∈ P (Ω) when n goes to infinity, then the invariant law ν satisfies Proof. Let p ∈ (1, 2] (the case p = 1 is similar and already treated in [17]). First, The second point is obtained from the first one by considering H 1 = ν. Concerning the third assertion, by induction from the first one we get for all n ∈ N The weak convergence of H n to ν concludes.
Example: Let (E k ) k≥0 be an i.i.d. sequence of standard exponential variables, and (X k ) k≥0 be the Markov chain on R + defined by X k+1 = X k +E k 2 . Its transition operator is Clearly On the other hand P (x), the law of x+E 2 , is the image by a 1 2 -Lipschitz transformation of the exponential law E(1), which satisfies a Poincaré inequality B(2, 4) (cf. Theorem [4, Theorem 6.2.2] for instance). Thus P is (2, 1 4 , 2)-contractive. On the other hand it is clear the chain is irreducible, it admits C = [0, 3] as a small set and V (x) = x + 1 as a Lyapunov function (since P V (x) ≤ 3 4 V (x) + 1 x<3 ) so that it is ergodic (see [24] for definitions and proof). According to Lemma 9, the invariant measure satisfies a Poincaré inequality B 2, 8 3 .

The embedded chain
Recall X = (X t ) t≥0 is a process on Ω with generator given by (1). Let (S k ) k≥0 be the jump times of X and let Z k = X S k . The Markov chain (Z k ) k≥0 is called the embedded chain associated to X.
we shall say that a function f is non-decreasing (resp. constant, concave, etc.) along the for all x), and define Then P = KQ is the transition operator for the chain Z.
Transfering properties from X to Z, or the converse, is far from obvious. In fact it is quite easy to find counter-examples for which one is ergodic and not the other (see examples 34.28 and 34.33 of [21]). In [20] this problem is solved with the definition of another embedded chain by adding observation points at constant rate. That being said, in the following we won't delve into this issue, and simply assume Z has a unique invariant law µ e (which can often be proved under conditions of irreducibility, aperiodicity and existence of a Lyapunov function). In this case we can express µ from µ e : Lemma 10 (Theorem 34.31 of [21], p.123).
In other words, µ = ν e K where In the following we will always assume the condition C < ∞ holds, so that ν e and K are well defined.
Here is our plan: from Lemma 9, we may establish a Beckner's inequality for µ e by proving the operator P is contractive. By perturbative results on functional inequalities (see [16] or Appendix) this may give an inequality for ν e . Finally, again from Lemma 9, we may transfer the inequality from ν e to µ by proving the operator K is confining.
The rest of this section will thus enlight some general facts which will later help us (mostly in dimension 1) prove K and K are confining. It is strongly inspired by the work of Chafaï, Malrieu and Paroux [17], in which a log-Sobolev inequality is proved for the invariant measure of the embedded chain of a particular PDMP, the TCP with linear rate (see example 4.4).
Until the end of this section we suppose λ > 0 almost everywhere. Then is invertible for all x ∈ Ω. Since we assumed the jump times are a.s. finite, necessarily, for all which yields both and, taking If X 0 = x and if T x is the next time of jump then is independant from X 0 , and has a standard exponential law. In other words T

Lemma 11.
If λ is non-decreasing along the flow, then for all x ∈ Ω and t > 0, the law of T ϕx(t) is the image of the law of T x by a 1-Lipschitz function.
The assumption that the jump rate is non-decreasing along the flow is natural in several applications where the role of the jump mechanism is to counteract a deterministic trend (growth/fragmentation models for cells [13], TCP dynamics [17], etc.). In this context, the more the system is driven away by the flow, the more it is likely to jump. From a mathematical point of view, thanks to Lemma 11, a Beckner's inequality for the law K(x) may be transfered to K (ϕ x (t)) for all t > 0.
In fact this is also true for K. Let T x be a random variable on R + with density

Lemma 12.
If λ is non-decreasing along the flow, then for all x ∈ Ω and t > 0, the law of T ϕx(t) is the image of the law of T x by a 1-Lipschitz function.
Proof. We will prove Lemma 11 applies here. Indeed the law of T ϕx(t) is the law of T x − t conditionnaly to the event T x > t, exactly as the law of T ϕx(t) is the law of T x −t conditionnaly to the event T x > t. We need to find a jump rate which define T x as the jump time of a Markov process.
Let e −V (s) ds be a positive probability density on R + , assume V is convex and let Differentiating this equality yields r(t)e − t 0 r(s)ds = e −V (t) .
We want to prove r is non-decreasing. From the convexity of V , As a consequence, In the case of T x , if λ is non-decreasing along the flow then V (t) = Λ x (t) − ln ∞ 0 e −Λx(w) dw is convex, so that the corresponding r is non-decreasing and Lemma 11 applies.

Proof. From the representation
we compute (recall f ∈ A is smooth and compactly supported) (17)) If λ is non-decreasing along the flow, λ (ϕ x (t)) ≥ λ(x) for all t ≥ 0.

Proof.
Kf ∞ 0 e −Λx(w) dw ds the cumulative function of T x is invertible. Let U be a uniform random variable on [0, 1]. Then On the first hand F ′ On the other hand from Equality (16)

Relation (18) yields
x (u). Thanks to Equation (15), Bringing the pieces together, we have proved When λ is non-decreasing along the flow, from (16), x → Λ x (t) is non-decreasing along the flow for all t ≥ 0, and h (ϕ x (t)) ≤ h(x).

The TCP with constant rate
A simple yet instructive example on R + is the TCP with constant rate of jump with generator where R is a random variable on [0, 1) and λ > 0 is constant. It is a simple growth/fragmentation model, or obtained by renormalizing a pure fragmentation model (cf. [27] for instance). In [34,31], ergodicity is proved and it is shown the moments of the invariant measure µ are all finite; so instead of the set of compactly supported smooth functions, A may be chosen as the set of smooth functions for which all derivatives grow at most polynomialy at infinity. Applying Theorem 5 with J b = 0, M = E R 2 and a = 1, we get Corrolary 6 then yields a contraction at rate λ 1 − E R 2 of the Wasserstein distance W 1 (ν 1 P t , ν 2 P t ). In fact by coupling two processes starting at different points to have the same jump times and the same factor R at each jump, one get that for any p ≥ 1, the W p distance decays at rate λp −1 (1 − E (R p )) (see [17]), and those rates are optimal (see [35]). In particular λ 1 − E R 2 is the rate of decay of W 2 2 .
As far as the jump operator Qf (x) = E (f (Rx)) is concerned, we have already used the sub-commutation However a local Poincaré inequality (14) for Q(x) would mean ∀f ∈ A, x > 0, with g x (r) = f (rx). This implies the law of R satisfies B 2, cx −2 for all x > 0, hence B (2, 0), which means R is deterministic. Indeed, when R is deterministic, the local inequality always holds: When R is random, what prevent to straightforwardly use our argument is the possibility of arbitrarily little concentrated jump, for instance with uniform law on (0, x) for any x. It's a shame because if, say, R is uniform on 0, 1 2 , it means when the process jumps it is at least divided by 2 but can be even much more contracted. In particular its invariant measure should be more concentrated near zero than the process with R = 1 2 a.s. for which, as we will see, the invariant measure satisfies a Poincaré inequality. This illustrates a limit of our procedure.
In fact in this example the spectrum of the generator in L 2 (µ) is explicit: there are polynomial eigenfunctions, and since the tail of µ is exponential, polynomials are dense in L 2 (µ) and these eigenfunctions are the only one in L 2 (µ) . The eigenvalues are l k = λ(E R k − 1) with k ∈ Z + . The convergence rate of the L 2 -norm obtained in Proposition 18 for a deterministic R appears to be 1 2 |l 2 | and not the spectral gap |l 1 |, and of course Nevertheless 1 2 |l 1 | ≤ |l 2 |/2 so we get the right rate up to a factor 1/2.
Note that, contrary to the Ornstein-Uhlenbeck case (see p.81 of [4]), a pointwise Poincaré inequality of the form cannot hold. Indeed P t (x) is the mix of a Dirac mass at x + t and of a smooth density with support included in [0, δx + t].
If one consider f ∈ A which is constant equal to 0 on [0, δx + t] and constant equal to 1 in the neighborhood of x + t then the left part of (19) is non-zero while the right part vanishes. More precisely if there have been n jumps during the time t, then X t ∈ I n := [δ n (x + t), δ n x + t], so that the support of P t (x) is included in n≥0 I n . To control the variance of P t f , one need to control the variations of f inside each interval I n , which is done with P t (f ′ ) 2 , but also the variations between two different intervals, which may be done with P t (Qf − f ) 2 .

Proposition 19.
If R = δ is deterministic then P t satisfies the local inequality: ∀x, t > 0, f ∈ A, and Hence for all β > 0, With ψ(s) = P s φ(P t−s f )(x) and β = 1 + s we roughly get From Proposition 15, this yields With φ 2 (f ) = f 2 we now consider the usual carré-du-champs operator so that It would be natural to expect an inequality of the form with a bounded t → c(t). We could prove such an inequality in the same way as Lemma 9 if we had a function γ such that and a time t for which γ(t) < 1. Note that in the present case the Bakry-Emery criterion is not satisfied, which implies that an inequality of the form (20) cannot hold with γ ′ (0) < 0 or, equivalently, with γ(t) = e −rt for some r > 0.

The storage model
Let U be a positive random variable, and consider the generator on R + This is, in a sense, the converse of the TCP: the jumps send the process away from 0 and the flow brings it back. Applying Theorem 5 with M = 1, a = 1 and J b = −1, we get Besides in this case it is easy to obtain a Wasserstein decay, as the distance s between two processes starting at different point and coupled to have the same jump times and the same U at each jump satisfies s ′ = −s, and such a decay implies (21) (see [30]; the converse is not clear, since P t is a mix of a Dirac mass and a smooth density).
To prove a Beckner's inequality, the same problem arises as in the previous example with a random R: here the law K(x), namely the law of e −T x with T an exponential random variable, can be as little concentrated as possible when x goes to infinity, so that K does not satisfy a local Beckner's inequality (14).

The TCP with increasing rate
Consider the generator on R + We have already studied the constant rate case. Before tackling the case of λ(x) = x, we consider in this section an intermediate difficulty, with the following assumptions: λ is nondecreasing, λ(0) = λ * > 0, and ln λ is a κ-Lipschitz function. Let β = 1−δ 2 2κ 2 , so that In other word, Inequality (10) holds with η = − λ * (1−δ 2 ) 2 and a = 1. To apply Theorem 7, we also need to prove a Poincaré inequality.

Lemma 20. The operators
Proof. The sub-commutation (13) is a direct consequence of Lemma 13 and 14, since the rate of jump is non-decreasing and b = 1. On the other hand K(x) (resp K(x)) is the law of x + T x (resp. x + T x ) which is from Lemma 11 the image by a 1-Lipschitz function of T 0 (resp. T 0 ). Thus we only need to prove the inequality holds for K(0) and K(0). For the case of K(0), denote by F (t) = 1 − e −Λ 0 (t) the cumulative function of T 0 . Then, if E is a standard exponential random variable, Since Λ −1 0 is a non-decreasing concave function with Λ −1 0 ′ (0) = 1 λ * , T 0 is a 1 λ * -Lipschitz transformation of E, whose law satisfies the Poincaré inequality B (2,4).

Corollary 21.
The invariant measure µ of the process satisfies a Poincaré inequality B(2, c) with Proof. It is clear the jump operator Q is (0, δ 2 , 2)-contractive, so that from Lemma 9, P = KQ is 4δ 2 λ 2 * , 1, 2 -contractive, and µ e the invariant measure of the embedded chain associated with the process satisfies a Poincaré inequality B 2, 4δ 2 It is a non-increasing function with h(0) ≤ ∞ 0 e −λ * s ds = 1 λ * . In order to prove the perturbation ν e of µ e , defined by ν e (f ) = 1 µe(h) µ e (f h), satisfies a Poincaré inequality, we will use Lemma 31, which requires an upper bound on the median m e of µ e . Note that it is possible to couple a process X with rate λ and a process Z with constant rate λ * so that, if they start at the same point, the first one will always stay below the second one: suppose such a coupling (X, Z) has been defined up to a jump time T k of X. Then both process increases linearily up to the next jump time T k+1 of X. At time T k+1 , X jumps, but Z jumps only with probability λ * λ(X T k +T k ) , else it does not move. In other words the jump part of the generator of Z is thought as Such a coupling proves m e is less than the median of the invariant law of the process with constant rate λ * . Let Z ∞ be a random variable with this invariant law, so that, if E is a standard exponential random variable, Hence from Markov's inequality, m e ≤ 2δ λ * (1−δ) , and Finally, from Lemma 31, ν e satisfies a Poincaré inequality with constant c ′ = 32 , and since K is confining, from Lemma 9, µ = ν e K satisfies such an inequality with constant Remark: if, again, λ(x) ≥ k(1 + x) q for some k > 0 and q ∈ [0, 1], these arguments prove the invariant measure satisfies a generalized Poincaré inequality I(α, c) for some c > 0 and α = 2q q+1 . Thus the invariant measure inherits the concentration properties of the law of the jump time T 0 : the logarithm of its density tail is (at most) of order −x q+1 . Let (P t ) t≥0 be the semi-group associated to the generator (23) and for f ∈ A let W t = µ ((P t f ) ′ ) 2 and V t = µ (P t f − µf ) 2 . We have proved Theorem 7 holds:
To cope with the rate of jump that vanishes at the origin, we will apply Theorem 7 with a weight a that behaves linearily near 0. More precisely, let

Lemma 23. Suppose µ satisfies the weighted Poincaré inequality
for some c > 0, and let Then for all β > η −1 , t > 0 and f ∈ A, Proof. Note that a is a concave function, so that Hence, from Lemma 4, for any β > 0, The function g(x) = 1 e x −1 + x goes to +∞ at 0 and +∞ and admits a unique positive critical point for which Hence for all x > 0, g(x) ≥ g ln 3+ Following the proof of Theorem 7, with V t = µ (P t f − µf ) 2 , this yields and, thanks to the weighted Poincaré inquality, if βη > 1, Finally, Remark that h(β) = βη−1 β+β 2 c goes to 0 when β either goes to η −1 or to +∞, and admits a unique positive critical point for which Corollary 24. Suppose µ satisfies the weighted inequalities, for all f ∈ A, for some c 1 , c 2 > 0, and let η be such as defined in Lemma 23. Then for all β > η −1 , t > 0 and f ∈ A, Proof. From Lemma 23 and the fact a ≤ 1, Thus, to prove Proposition 3, it only remains to prove a weighted log-Sobolev inequality holds. In order to simplify some upcoming computations, we consider an intermediate . Then, by concavity, a(x) = α 1−δ √ 2 x ≥ 1−δ √ 2 α(x). If we prove a log-Sobolev inequality (24) holds with weight α, it implies such an inequality with weight a. Let It is a concave, non-decreasing, one-to-one function. If Z is a random variable with law µ and Y = ψ(Z), then with g(y) = f ψ −1 (y) . As a consequence we will study the Markov process ψ(X) = (ψ(X t )) t≥0 , where X = (X t ) t≥0 has generator (23), and prove a classical non-weighted log-Sobolev for the invariant measure of this twisted process, which will imply the weighted log-Sobolev assumed in Corollary 24. We still denote by Q the jump kernel of X, so that is the jump kernel of ψ(X). Let K α and K α be such as defined in Section 3.2, but corresponding to the process ψ(X).

Lemma 25.
For all g ∈ A, Proof. By concavity, α (δx) ≥ δα(x) for all x ≥ 0. Thus On the other hand the vector field associated to ψ( , and the rate of jump is non-decreasing along the flow. Hence, according to Lemma 13, (and according to Lemma 14, the same goes for K α ). Note that the support of both probability measure K α (z) and K α (z) is [z, ∞], and that b α is non-increasing along the flow, so that (and the same goes for K α ).
Proof. Let T x be the first time of jump of X starting from x. According to Lemma 11, there exists a 1-Lipschitz functions G such that T x dist = G(T 0 ). Note that K α (ψ(x)) is the law of Now ψ is concave, and in the proof of Lemma 11 we have seen that x + G(s) ≥ s for all s ≥ 0; hence |H ′ (z)| ≤ 1 for all z ≥ 0.
Similarly, let T x be a random variable on R + with density e − t 0 (x+s)ds ∞ 0 e − u 0 (x+s)ds du , so that K α (ψ(x)) is the law of ψ x + T x . From Lemma 12 there exists a 1-Lipschitz functions G such that T x dist = G( T 0 ), and the previous argument concludes.
Proof. If T 0 is the first time of jump starting from 0 then K α (0) is the law of ψ(T 0 ). For any f ∈ A, On the other hand, if N is a standard gaussian variablen then K α (0) is the law of ψ (|N |), and for all f ∈ A As a first step, note that V ′′ (since α is non-decreasing and concave). Since j(0) = 0, it implies j(y) ≥ 0 for all y ≥ 0, in other words V ′′ 1 (z) ≥ V ′′ 0 (z). On the other hand, the last line being the very reason we decided to work with α rather than with a. As a consquence, both K α (0) and K α (0) satisfies B(1, 1) (see for instance [4,Theorem 5.4.7], applied to the diffusion with generator ∂ 2 x − V ε ∂ x ).
To sum up the consequences of the previous results, Corollary 28.
2. The invariant measure ν α of the embedded chain associated to ψ(X) satisfies B 1, The sub-commutation property has been showed in Lemma 25, and the local inequality is clear for Q α which is deterministic, and is a consequence of Lemma 26 and 27 for K α and K α .
The last step of our procedure is the study of a perturbation of ν α . Since the rate of jump of Z = ψ(X) at point z is λ α (z) = ψ −1 (z) and the operator K α is such that K α f (ψ(x)) = E (f (ψ(x + T x ))), according to Lemma 10, we have to study the function g defined by g (ψ(x)) = K α 1 λ α (ψ(x)) = E 1 x + T x Lemma 29. The function g is decreasing, and ln g is 2 so that g(z) = h ψ −1 (z) . Since h is decreasing and ψ −1 is increasing, g is decreasing.
Moreover, as | (ln g) ′ (z)| = α (ψ −1 (s))|(ln h) ′ ψ −1 (z) | and α ≤ 1, it is sufficient to prove ln h is 2 h(x) On the other hand (ln h) ′ ≤ 0; thus ln g is 1 h(0) -Lipschitz, and We have in mind to apply to ν α and g the perturbation Lemma 31 of the Appendix. In that aim we need to bound g(m α ), where m α is the median of ν α , and ν α (g −1 ). In fact, note that ν α , which is the invariant measure of the embedded chain associated to the process ψ(X), is also the image through the function ψ of the invariant measure of µ e , the invariant measure of the embedded chain associated to the initial process X. In particular if m e is the median of µ e then m α = ψ (m e ). Keeping the notation h(x) = g (ψ(x)), we have g(m α ) = h(m e ) and ν α (g −1 ) = µ e (h −1 ).

Lemma 30. We have
Proof. Recall that, keeping the notations of Section 3.2, if T x is the first time of jump of the process starting from x and E is a standard exponential variable, then T x dist = Λ −1 x (E). In the present case Λ x (t) = t 0 (x + u)du, so that T From this, 1 − e −2 ≥ 5 6(x + 2) .
Hence h(0) h(m e ) ≤ √ π 2 × 6 5 (m e + 2) Finally, if Y is a random variable with measure µ e , We can now bring the pieces together.
As we have already noticed, α ≤ √ 2 1−δ a, so that the conditions of Corollary 24 are fulfilled, and Proposition 3 is proved.  Furthermore, in that case, the optimal c (namely the smallest c such that B(2, c) holds) is such that In the present case, for all x ≥ m, ∞ x g(t)ν(t)dt