Ivan Gentil
email: gentil@ceremade.dauphine.fr

Cyril Imbert
email: imbert@math.univ-montp2.fr

The Lévy-Fokker-Planck equation: Φ-entropies and convergence to equilibrium

Keywords: Fokker-Planck equation, Lévy operator, Φ-entropy inequalities, entropy production method, logarithmic Sobolev inequalities, fractional Laplacian Mathematics Subject Classification: 46N20, 47G20, 35K15

In this paper, we study a Fokker-Planck equation of the form ut = I[u] + div(xu) where the operator I, which is usually the Laplacian, is replaced here with a general Lévy operator. We prove by the entropy production method the exponential decay in time of the solution to the only steady state of the associated stationnary equation.

Introduction and main results

The equation at stake. In this paper, we are interested in the long time behaviour of the following generalized Fokker-Planck equation:

∂ t u = I[u] + div(u∇V ) x ∈ R d , t > 0, (1)
submitted to the initial condition:

u(0, x) = u 0 (x) x ∈ R d (2)
where u 0 is nonnegative and in L 1 (R d ) and V is a given proper potential for which there exists a nonnegative steady state (see below). The operator I is a Lévy operator

I[u](x) = div (σ∇u)(x)-b•∇u(x)+ R d
(u(x+z)-u(x)-∇u(x)•zh(z))ν(dz) [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF] with parameters (b, σ, ν) where b = (b i ) ∈ R d , σ is a symmetric semi-definite d × d matrix σ = (σ i,j ) and ν denotes a nonnegative singular measure on R d that satisfies ν({0}) = 0 and min(1, |z| 2 )ν(dz) < +∞;

h is a truncature function and we fix it on this article: for any z ∈ R d , h(z) = 1/(1 + |z| 2 ).

Remark that this equation is conservative: for any t > 0, u(t, x)dx = u 0 (x)dx and we will therefore assume without loss of generality that we have u 0 (x)dx = 1. The diffusion equation ( 1) is a natural generalization of the well-known Fokker-Planck equation given by

∂ t u = ∆u + div(u∇V ) x ∈ R d , t > 0, (5)
and since I defined by ( 3) is the infinitesimal generator of a Lévy process, we refer to [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF] as the Lévy-Fokker-Planck equation.

Motivations and goals. As pointed out by Biler and Karch in [START_REF] Biler | Generalized Fokker-Planck equations and convergence to their equilibria[END_REF], Equation ( 1) is studied at least for two important reasons. First, it is deeply connected with stochastic differential equations driven by Lévy stable noise, see e.g. [START_REF] Schertzer | Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises[END_REF].

Secondly, in the case of a quadratic potential V (x) = 1 2 |x| 2 and the fractional Laplacian I = -(-∆) α 2 (even for general α-stable Lévy process -see Section 2 for a definition), it permits to describe the long time behaviour of the solution of

∂ t U + (-∆) α 2 [U ] = 0 (6)
with a nonnegative initial condition U 0 in L 1 (R d ). Indeed, the reader can check that the function u(t, x) = R d (t)U (τ (t), xR(t)) satisfies ( 1) as soon as one chooses R(t) = e t and τ (t) = (e αt -1)/α and U 0 (x) = u 0 (x). Consequently, if K(t, x) denotes the green function associated with the fractional Laplacian, u(t, x) = e dt u 0 e αt -1 α x -y K e t , y dy.

On the one hand, U vanishes as t → +∞ and on the other hand, u is expected to tend towards the steady state of [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF], that is to say towards the nonnegative stationary solution u ∞ of (1) with the quadratic potential

V (x) = 1 2 |x| 2 : I[u ∞ ] + div(u ∞ x) = 0 (7)
such that u ∞ dx = u 0 dx = 1. We see that dealing with the quadratic potential is of particular interest and we will be able to say more in this case.

In full generality, we expect that the solution u of (1) converges towards the unique nonnegative solution of:

I[u ∞ ] + div(u ∞ ∇V ) = 0 such that u ∞ = u 0 = 1. Function u ∞ is
called a steady state and conditions on V must ensure that such u ∞ exists and it is nonnegative.

Known results. Consider a smooth convex function Φ : R + → R and u ∞ positive such that u ∞ dx = 1 and define the Φ-entropy: for any nonnegative function f ,

Ent Φ u∞ (f ) := Φ (f ) u ∞ dx -Φ f u ∞ dx .
Jensen's inequality gives that Ent Φ u∞ (f ) ≥ 0. Let u 0 be a nonnegative initial condition of (1) or (5) then the maximum principle ensures that

E Φ (u 0 )(t) = Ent Φ u∞ u(t) u ∞
is well defined for t > 0.

In the case of the classical Fokker-Planck equation [START_REF] Bakry | L'hypercontractivité et son utilisation en théorie des semigroupes[END_REF], by using functional inequalities as Poincaré, logarithmic Sobolev or Φ-entopy inequalities, and under proper assumptions on the Potential V such as the Bakry-Emery criterion (Γ 2 ), one obtains exponential decays to zero of E Φ (u 0 ). Then the solution u of (5) converges towards the steady state u ∞ in the sense of Φ-entropy. Methods to prove such results are usually based on entropy/entropy-production tools. See [START_REF] Bakry | L'hypercontractivité et son utilisation en théorie des semigroupes[END_REF][START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF][START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF][START_REF] Chafaï | Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities[END_REF] for different methods and applications.

For our equation ( 1), Biler and Karch prove in [START_REF] Biler | Generalized Fokker-Planck equations and convergence to their equilibria[END_REF] that E Φ (u 0 ) decreases in time for general confinement potentials V ; they also prove that for Φ(r) = r 2 /2 and for the quadratic potential V (x) = 1 2 |x| 2 , there exists C = C(u 0 , I) and ε such that:

E |.| 2 /2 (u 0 )(t) ≤ C 0 e -εt . (8)
To prove the latter result, they assume that the symbol ψ satisfies for some real number α ∈ (0, 2]:

0 < lim inf ξ→0 ψ(ξ) |ξ| α ≤ lim sup ξ→0 ψ(ξ) |ξ| α < +∞, 0 < inf ψ(ξ) |ξ| 2 . (9)
In particular, the second assumption says more or less that there is a nontrivial Gaussian part (σ = 0).

Main results. The results of Biler and Karch we just described are the starting point of this paper. We tried to understand if one can generalize them and obtain an exponential decay for any convex function Φ under a sharper form. Our two main contributions are:

1. to exhibit the so-called Fisher information associated with the Φ-entropies E Φ , 2. and to take advantage of this result to prove an exponential decay of the Φ-entropies for a class of convex functions Φ and for a larger class of operators.

Let us describe our results more precisely. Let us consider the Bregman distance associated with Φ:

∀(a, b) ∈ R + , D Φ (a, b) := Φ(a) -Φ(b) -Φ (b)(a -b) ≥ 0. ( 10
)
Our first result states that for all initial nonnegative datum u 0 : [START_REF] Schertzer | Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises[END_REF] where v(t, x) = u(t,x) u∞(x) and ν is the Lévy measure appearing in the definition of the operator I, see [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF]. The first result of Biler and Karch is a straightforward consequence of this formula and they were very close to it (see their proof of the entropy decreasing property). In the particular case of Φ(x) = x 2 , the right hand side of ( 11) can be seen as the opposite of a Dirichlet form (see Chapter 3 of [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF]).

∀t ≥ 0, d dt E Φ (u 0 )(t) = -Φ (v)∇v • σ∇v u ∞ dx -D Φ (v(t, x), v(t, x -z)) ν(dz)u ∞ (x)dx
The main result of this paper is the following.

Theorem 1 (Exponential decay to equilibrium). Assume that V (x) = 1 2 |x| 2 and the operator I is the infinitesimal generator of a Lévy process whose Lévy measure is denoted ν. We assume that ν has a density N with respect to the Lebesgue measure and that N satisfies

R d \B ln |z| N (z) dz < +∞ ( 12
)
where B is the unit ball in R d . Then there exists a steady state u ∞ , i.e. a nonnegative solution of (7)

satisfying u ∞ dx = 1. If moreover N is even and for all z ∈ R d , +∞ 1 N (sz)s d-1 ds ≤ CN (z) (13)
for some constant C ≥ 0, then for any smooth convex function Φ such that

(a, b) → D Φ (a + b, b) (r, y) → Φ (r)y • σy are convex on {a + b ≥ 0, b ≥ 0} R + × R 2d (14)
the Φ-entropy of the solution u of (1)-( 2) goes to 0 exponentially. Precisely, for any nonnegative initial datum u 0 such that Ent Φ u∞ u0 u∞ < ∞, one gets:

∀t ≥ 0, Ent Φ u∞ u(t) u ∞ ≤ e -t C Ent Φ u∞ u 0 u ∞ ( 15
)
with C appearing in (13). In addition, u ∞ is the unique nonnegative solution of (7).

Condition (13) is fullfilled by any α-stable Lévy process and in this case, C = α -1 . It is also satisfied by multifractal operators. See the next section for definitions. Furthermore, it should be compared with the first part of (9). Indeed, freely speaking, it says that the symbol lies between two α-stable ones for small ξ.

Conditions (14) appear in [START_REF] Chafaï | Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities[END_REF] (see (H1) and (H2) in this paper). Basic but more important examples of such convex functions are given by Φ

(x) = x p -1 -p(x -1) with p ∈ (1, 2], Φ(x) = x log x -x + 1. When p = 2, D Φ (a, b) = (a -b) 2 and in the latter case, D Φ (a, b) = a ln a b + (b -a). In Theorem 4,4 of [9] it is proved that condition (14) is equivalent to either Φ is affine or Φ is C 2 , Φ > 0 on R + and 1/Φ is concave on R + .
The paper is organized as follows. In Section 2, we recall basic facts about Lévy operators and associated Lévy measures. We also recall a modified logarithmic Sobolev inequality due to Wu and Chafaï (Theorem 2). Section 3 is devoted to derivating with respect to time the Φ-entropies for general potentials V . The last section is devoted to the proof of Theorem 1, a discussion about Condition (13) and some examples.

Notation. x • y denotes the usual scalar product in R d .

Preliminaries 2.1 Lévy operators

Let us next recall basic definitions about Lévy operators and introduce notations. See [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF][START_REF] Bertoin | Lévy processes[END_REF] for further details.

Characteristic exponents and Lévy measures. The characteristic exponent of a Lévy process (X t ) t≥0 is the function ψ such that the characteristic function φ Xt (ξ) of (X t ) t≥0 is exp(tψ(ξ)). The Lévy-Khinchine formula states that ψ can be written under the following form

ψ(ξ) = -σξ • ξ + ib • ξ + a(ξ) (16)
where a is associated with the pure jump part of the Lévy process

a(ξ) = e iz•ξ -1 -i(z • ξ)h(z) ν(dz).
Recall that h(z) = 1/(1 + |z| 2 ) and the Lévy measure satisfies (4). The matrix σ characterizes the diffusion (or Gaussian) part of the operator (with eventually σ = 0), while b and ν characterises the drift part and the pure jump part, respectively. A Lévy operator I is the infinitesimal generator associated with the Lévy process and the Lévy-Khinchine formula implies that it has the form (3).

The pseudodifferential point of view. It is convenient to introduce the operator I g associated with the Gaussian part,

I g (u) = div (σ∇u) -b • ∇u
and the operator I a associated with the pure jump part

I a (u) = R d (u(x + z) -u(x) -∇u(x) • zh(z))ν(dz).
The operator I a can be seen as a pseudodifferential operator of symbol a:

I a (u) = F(a × F -1 u)
where F stands for the Fourier transform. We choose the probabilistic convention in defining, for all function w ∈ L 1 (R d ):

∀ξ ∈ R d , ŵ(ξ) = F(w)(ξ) = e ix•ξ w(x)dx. ( 17
)
Multifractal and α-stable Lévy operators. Lévy operators whose characteristic exponent is positively homogenous of index α ∈ (0, 2] are called αstable. The fractional Laplacian corresponds to a particular α-stable Lévy process whose characteristic exponent is ψ(ξ) = |ξ| α . Note that if α = 2 the Lévy operator has no jump part. They play a central role in this paper since they are the operators for which we have equality in Condition (13). See the discussion in the last section.

Lévy operators whose characteristic exponent can be written as ψ(ξ) = n i=1 ψ i (ξ) where ψ i is α i -homogenous with α i ∈ (0, 2], are often referred to as multifractal Lévy operators. These operators are typically those who satisfy Condition (13).

A modified logarithmic Sobolev inequality

The functional inequality we will need in the sequel is given in the following theorem. It was proved by Wu and Chafaï.

Theorem 2 ( [START_REF] Wu | A new modified logarithmic Sobolev inequality for Poisson point processes and several applications[END_REF][START_REF] Chafaï | Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities[END_REF]). Assume that a smooth function Φ satisfies (14) and consider an infinitely divisible law µ. Then for all smooth positive functions v,

Ent Φ µ (v) ≤ Φ (v)∇v • σ∇vµ(dx) + D Φ (v(x), v(x + z))ν µ (dz)µ(dx) (18)
where ν µ and σ denote respectively the Lévy measure and the diffusion matrix associated with µ.

Remark that the drift of the law plays no role in this functional inequality. Eq. ( 18) is proved in [START_REF] Wu | A new modified logarithmic Sobolev inequality for Poisson point processes and several applications[END_REF] for Φ(x) = x 2 or Φ(x) = x log x and in this general form in [START_REF] Chafaï | Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities[END_REF] but only for a pure jump Lévy process. The two articles are generalization of the logarithmic Sobolev inequality for Poisson measure given in [START_REF] Ané | On logarithmic Sobolev inequalities for continuous time random walks on graphs[END_REF]. For the completeness of this article, we give a sketch of the proof of this theorem in Appendix.

The important special case Φ(r) = r 2 /2. A simple computation shows that the Bregman distance in this case is D Φ (a, b) = (a -b) 2 . In this case, the Φentropy Ent Φ u∞ reduces to the variance. One can next consider the "carré du champ" operator Γ I associated with the operator I defined for smooth functions u and v by:

Γ I [u, v] = I[uv] -uI[v] -vI[u] = ∇u • σ∇v + (u(x + z) -u(x))(v(x + z) -v(x))ν(dz),
We shed some light on the fact that one can obtain in that case Inequality (18) by using Γ 2 associated with I = I µ defined as follows:

Γ I 2 (u, v) = I[Γ I [u, v]] -Γ I [u, I[v]] -Γ I [v, I[u]].
Inequality (18) can be obtained by remarking that (assume σ = 0 for simplicity):

Γ I 2 (u, u) = (u(x + z + z ) -u(x + z) -u(x + z ) + u(x)) 2 ν(dz)ν(dz ) ≥ 0,
with ν the Lévy measure associated to I. This computation was already done by Chafaï and Malrieu in a nice but unpublished note [START_REF] Chafaï | A note on functional inequalities for some lvy processes[END_REF].

Φ-Entropy and associated Fisher information

In this section, we are interested in the (time) derivative of E Φ (u 0 )(t) in the case where a steady state is given. Precisely, the following theorem is proved.

Proposition 1. Assume that the initial condition u 0 is nonnegative and satisfies Ent Φ u∞ u0 u∞ < ∞ with u ∞ , a solution of [START_REF] Biler | Generalized Fokker-Planck equations and convergence to their equilibria[END_REF]. Then for any convex smooth function Φ : R + → R and any t ≥ 0, the solution u of (1)-( 2) satisfies [START_REF] Schertzer | Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises[END_REF] where v(t, x) = u(t,x) u∞(x) and ν is the Lévy measure associated to I.

Remark 1. In the special case Φ(x) = x 2 /2, Proposition 1 can be reformulated as follows:

d dt E Φ (u 0 )(t) = -Γ I [v, v]u ∞ dx ≤ 0.
As in the case of the classical Fokker-Planck equation, the energy does not depend on the potential V .

In order to prove Proposition 1, since the Φ-entropy involves the function v(t, x) = u(t,x) u∞(x) , its derivative makes appear ∂ t v and it is natural to ask ourselves which partial differential equation v satisfies. A simple computation gives:

∂ t v = 1 u ∞ I[u ∞ v] + div(u ∞ v∇V ) = 1 u ∞ (I[u ∞ v] -I[u ∞ ]v) + ∇V • ∇v =: Lv. ( 19
)
In the case where I[u] = ∆u (i.e. σ is the identity matrix, b = 0 and a = 0), Equation (19) writes:

∂ t v = ∆v -∇V • ∇v
and is known as the Ornstein-Uhlenbeck equation. This is the reason why we will refer to Equation ( 19) as the Lévy-Ornstein-Uhlenbeck equation. We next give a simpler formulation for the Lévy-Ornstein-Uhlenbeck operator.

Lemma 1 (Lévy-Ornstein-Uhlenbeck equation). If the integrodifferential operator in the right-hand side of (19) is denoted L, we have for all smooth functions w 1 and w 2 :

w 1 Lw 2 u ∞ dx = ( Ǐ[w 1 ] -∇V • ∇w 1 ) w 2 u ∞ dx
where Ǐ is the Lévy operator whose parameters are (-b, σ, ν) with ν(dz) = ν(-dz). This can be expressed by the formula: L * = Ǐ -∇V • ∇ where duality is understood with respect to the measure u ∞ dx.

Proof. The main tool is the integration by parts for the operator I, for any smooth functions u, v one gets

vI[u]dx = u Ǐ[v]dx.
If w 1 and w 2 are two smooth functions on R d , then:

w 1 Lw 2 u ∞ dx = w 1 (I[u ∞ w 2 ] -I[u ∞ ]w 2 + u ∞ ∇V • ∇w 2 )dx = w 2 Ǐ[w 1 ] u ∞ dx -I[u ∞ ]w 1 w 2 dx -div(u ∞ w 1 ∇V )w 2 dx = u ∞ ( Ǐ[w 1 ] -∇V • ∇w 1 )w 2 dx.
The proof of Proposition 1. By using Lemma 1 with v = u/u ∞ , we get:

d dt E Φ (u 0 )(t) = Φ (v) ∂ t v u ∞ dx = Φ (v) Lv u ∞ dx = Ǐ[Φ (v)] v u ∞ dx -∇V • ∇(Φ (v)) v u ∞ dx.
If now one remarks that rΦ (r) = (rΦ (r) -Φ(r)) , we get:

d dt E Φ (u 0 )(t) = v Ǐ[Φ (v)]u ∞ dx -∇V • ∇(vΦ (v) -Φ(v))) u ∞ dx = v Ǐ[Φ (v)]u ∞ dx + div(u ∞ ∇V )(vΦ (v) -Φ(v))dx = v Ǐ[Φ (v)]u ∞ dx -I[u ∞ ](vΦ (v) -Φ(v))dx = (v Ǐ[Φ (v)] -Ǐ[vΦ (v)] + Ǐ[Φ(v)])u ∞ dx d dt E Φ (u 0 )(t) = (v Ǐg [Φ (v)] -Ǐg [vΦ (v)] + Ǐg [Φ(v)])u ∞ dx + (v Ǐa [Φ (v)] -Ǐa [vΦ (v)] + Ǐa [Φ(v)])u ∞ dx = -Φ (v)∇v • σ∇v u ∞ dx + v(x)(Φ (v(x + z)) -Φ (v(x))) -v(x + z)Φ (v(x + z)) +v(x)Φ (v(x)) + Φ(v(x + z)) -Φ(v(x))) ν(dz) u ∞ (x)dx d dt E Φ (u 0 )(t) = -Φ (v)∇v • σ∇v u ∞ dx - Φ(v(x)) -Φ(v(x + z)) -Φ (v(x + z)) ×(v(x) -v(x + z)) ν(dz) u ∞ (x)dx.
Then the definitions of the Bregman distance and of the Lévy measure ν give

d dt E Φ (u 0 )(t) = -Φ (v)∇v•σ∇v u ∞ dx- D Φ (v(x), v(x-z))ν(dz) u ∞ (x)dx.

The proof of Theorem 1, discussion and examples

To prove the first part of the Theorem, the existence of the steady state, we need to state the following lemma.

Lemma 2. Assume that the Lévy measure ν has a density N with respect to the Lebesgue measure and that it satisfies [START_REF] Wu | A new modified logarithmic Sobolev inequality for Poisson point processes and several applications[END_REF]. There then exists a steady state u ∞ , i.e. a solution of [START_REF] Biler | Generalized Fokker-Planck equations and convergence to their equilibria[END_REF]. Moreover, it is an infinitely divisible measure whose characteristic exponent A is defined by:

A(ξ) = -ξ • σξ + ib • ξ + 1 0 a(sξ) ds s . (20)
Moreover, parameters of the characteristic exponent

A are (σ, b -b A , N ∞ dx) where b A = 1 0 z (1 -τ 2 )|z| 2 (1 + τ 2 |z| 2 )(1 + |z| 2 ) dτ N (z)dz, (21)
and

N ∞ (z) = ∞ 1 N (tz)t d-1 dt. (22)
Note that the Lévy measure ν ∞ associated to the characteristic exponent A has a density N ∞ with respect to the Lebesgue measure.

Remark 2. In the general case, Condition (13) precisely says that N ∞ ≤ CN which can be written in term of measures as follows: ν ∞ ≤ Cν.

Proof. Let us start as in [START_REF] Biler | Generalized Fokker-Planck equations and convergence to their equilibria[END_REF]. At least formally, the Fourier transform û∞ of any steady state u ∞ satisfies

ψ(ξ)û ∞ + ξ • ∇û ∞ = 0
so that û∞ = exp(-A) with A such that:

∇A(ξ) • ξ = ψ(ξ).
The solution of this equation is precisely given by (20). It is not clear that A is well defined and is the characteristic exponent of an infinitely divisible measure;

this is what we prove next. This will imply in particular that F -1 (exp(-A)) is a nonnegative function (see [START_REF] Bertoin | Lévy processes[END_REF]).

Define the nonnegative N ∞ by Equation ( 22). This integral of a nonnegative function is finite since for any R > 0, if dσ denotes the uniform measure on the unit sphere S d-1 we get, We conclude that for any r ≥ R > 0 and almost every D on the unit sphere (where the set of null measure depends only on R), Hence we have min(1, |z| 2 )N ∞ (z)dz < +∞. We conclude that it is a Lévy measure. Now consider the associated characteristic exponent:

r d N ∞ (rD) = ∞ r N (τ D)τ d-1 dτ < +∞ so that N ∞ (z)
Ã(ξ) = ib • ξ -σξ • ξ + (e iz•ξ -1 -i(z • ξ) h(z))N ∞ (z)dz. Now compute: Ã(ξ) -ib • ξ + σξ • ξ = ∞ 1 (e iz•ξ -1 -i(z • ξ) h(z))N (sz)s d-1 ds dz = ∞ 1 e iz• ξ s -1 -i z • ξ s h z s N (z)dz ds s = ∞ 1 a ξ s ds s -iξ • ∞ 1 z s h z s -h(z) N (z)dz ds s = A(ξ) -iξ • b A
where b A is defined by (21). Properties (4) of the Lévy measure ν implies that b A is well defined. We conclude that A is the characteristic exponent of an infinitely divisible law u ∞ whose drift is b -b A , whose Gaussian part is σ and whose Lévy measure is N ∞ (z)dz.

Proof of Theorem 1. The proof of the first part is exactly given by Lemma 2.

We now turn to the second part of the theorem. Proposition 1 gives for t ≥ 0,

d dt Ent Φ u∞ (v(t, •)) = -Φ (v(t, •))∇v • (t, •) • σ∇v(t, •)u ∞ dx - D Φ (v(t, x), v(t, x -z))ν(dz) u ∞ dx = -Φ (v(t, •))∇v • (t, •) • σ∇v(t, •)u ∞ dx - D Φ (v(t, x), v(t, x + z))ν(dz) u ∞ dx
where we used the fact that ν is even. It is now enough to prove the following inequality

Ent Φ u∞ (v) ≤ C Φ (v(t, •))∇v(t, •) • σ∇vu ∞ dx +C D Φ (v(x), v(x + z))ν(dz)u ∞ (x)dx
for some constant C not depending on v and Gronwall's lemma permits to conclude. But this inequality is a direct consequence of (18) for the infinitely divisible law u ∞ . Remark also that if one considers another nonnegative steady state ũ∞ such that Ent Φ u∞ (ũ ∞ ) < +∞ for some Φ strictly convex and satisfying the assumptions of the theorem, then Ent Φ u∞ (ũ ∞ ) = 0 and we conclude that ũ∞ = u ∞ .

Discussion of Condition (13) and examples

Let us discuss the condition we impose in order to get exponential decay. We point out that equality in this condition holds true only for α-stable operators and we give a necessary condition on the behaviour of the Lévy measure at infinity if one knows that it decreases faster than |x| -d . Proposition 2.

• Equality N ∞ = N/λ if and only if ψ is positively homogenous of index λ ∈ (0, 2], i.e.

ψ(tξ) = t λ ψ(ξ) for any t > 0, ξ ∈ R d .
In this case, we get A = ψ/λ and b A = 0. Note that in the extreme case, where λ = 2, then we get N ∞ = N/2 = 0.

• If |x| d N (x) → 0 as |x| → +∞, then the densities N and N ∞ satisfy:

N = -div(xN ∞ ).
• In this case, Condition (13) is equivalent to:

N ∞ (tx) ≤ N ∞ (x)t -d-1/C if t ≥ 1 N ∞ (tx) ≥ N ∞ (x)t -d-1/C if 0 < t ≤ 1 (23)
Proof. The first item simply follows from the definition of A.

Let us first prove the second item. To prove the third item, use the first one to rewrite (13) as follows:

x • ∇N ∞ (x) + (d + 1/C)N ∞ (x) ≤ 0.

Integrate over [1, t] for t ≥ 1 and [t, 1] for t ≤ 1 to get the result.

Example 1. In R, the Lévy measure 1 |z| e -|z| does not satisfy Condition (13). Indeed, it is equivalent to:

A Appendix

This is a sketch of the proof given by Chafaï. The difference is that we treat here the case of a Lévy operator with eventually a diffusion part (σ = 0 in (16)).

Proof of Theorem 2. Let µ be the infinite divisible law. There then exists ψ a characteristic exponent such that the characteristic function of µ is exp ψ. Now consider K(t, •) the law associated with exp(tψ). If I defined by (3), we know that in this case the solution of

∂ t u = I[u] u(0, x) = u 0 (x)
is K(t, •) u 0 . Let v be a smooth and positive function on R. Define next for s ∈ [0, t]: ψ(s) = K(s, •) φ(K(t -s, •) v).

It is convenient to write g for K(t -s, •) v.

One gets, for s ∈ [0, t], ψ Then choose t = 1 and x = 0 in Inequality (24) and get:

Ent Φ K ψ (v) ≤ Φ (v)∇v • σ∇vK ψ (dx)+ D Φ (v(x), v(x + z))ν µ (dz) K ψ (dx).

N

  (τ D)τ d-1 dτ dσ(D) = |y|≥R N (y)dy < +∞.

1 0I|z| 2 N 1 I

 121 is well-defined almost everywhere outside B R . Choose now a sequence R n → 0 and conclude. Let us define I(r) = |D|=1 N (rD)dσ(D) and I ∞ in an analogous way. The previous equality implies that: r d I ∞ (r) = +∞ r I(τ )τ d-1 dτ . We conclude that: |z|≤1 |z| 2 N ∞ (z)dz = ∞ (r)r d+1 dr = (z)dz < +∞ |z|≥1 N ∞ (z)dz = +∞ ∞ (r)r d-1 dr = )τ d-1 dτ dr = |z|≥1 ln |z|N (z)dz.

N

  (x) = -( lim t→+∞ t d N (tx)) + 1 d N (1 × x) = -N (tx))dt = -dN ∞ (x) -x • ∇N ∞ (x) = -div(xN ∞ ).

+∞ 1 e

 1 -|x|(s-1) s ds ≤ Cand the monotone convergence theorem implies that the left hand side of this inequality goes to +∞ as |x| → +∞.

  = K(s, •) I(Φ(g)) -Φ (g)(I(g)) = K(s, •) Φ (g)∇g • σ∇g + K(s, •) D Φ (g(• + z), g(•))ν µ (dz) .Using the fact that (a, b) → D Φ (a + b, a) and (x, y) → Φ (x)y • σy are convex, one gets by Jensen's inequality,ψ (s) ≤ K(t, •) (Φ (v)∇v • σ∇v) + K(t, •) D Φ (v(• + z), v(•))ν µ (dz) .Integrate over s ∈ [0, t], we obtain for t ≥ 0 and x ∈ R d ,K(t, •) (Φ(v))(x) -Φ(K(t, •) v)(x) ≤ tK(t, •) (Φ (v)∇v • σ∇v)(x) +tK(t, •) D Φ (v(• + z), v(•))ν µ (dz) (x). (24)