Uniform semigroup spectral analysis of the discrete, fractional \&classical Fokker-Planck equations

In this paper, we investigate the spectral analysis (from the point of view of semi-groups) of discrete, fractional and classical Fokker-Planck equations. Discrete and fractional Fokker-Planck equations converge in some sense to the classical one. As a consequence, we first deal with discrete and classical Fokker-Planck equations in a same framework, proving uniform spectral estimates using a perturbation argument and an enlargement argument. Then, we do a similar analysis for fractional and classical Fokker-Planck equations using an argument of enlargement of the space in which the semigroup decays. We also handle another class of discrete Fokker-Planck equations which converge to the fractional Fokker-Planck one, we are also able to treat these equations in a same framework from the spectral analysis viewpoint, still with a semigroup approach and thanks to a perturbative argument combined with an enlargement one. Let us emphasize here that we improve the perturbative argument introduced in [7] and developed in [11], relaxing the hypothesis of the theorem, enlarging thus the class of operators which fulfills the assumptions required to apply it.

1. Introduction 1.1.Models and main result.In this paper, we are interested in the spectral analysis and the long time asymptotic convergence of semigroups associated to some discrete, fractional and classical Fokker-Planck equations.They are simple models for describing the time evolution of a density function f = f (t, x), t ≥ 0, x ∈ R d , of particles undergoing both diffusion and (harmonic) confinement mechanisms and write (1.1) ∂ t f = Λf = Df + div(xf ), f (0) = f 0 .
The diffusion term may either be a discrete diffusion for a convenient (at least nonnegative and symmetric) kernel κ.It can also be a fractional diffusion |x − y| d+α dy, with α ∈ (0, 2), χ ∈ D(R d ) radially symmetric satisfying the inequality 1 B(0,1) ≤ χ ≤ 1 B(0,2) , and a convenient normalization constant c α > 0. It can finally be the classical diffusion The main features of these equations are (expected to be) the same: they are mass preserving, namely positivity preserving, have a unique positive stationary state with unit mass and that stationary state is exponentially stable, in particular for any solution associated to an initial datum f 0 with vanishing mass.Such results can be obtained using different tools as the spectral analysis of selfadjoint operators, some (generalization of) Poincaré inequalities or logarithmic Sobolev inequalities as well as the Krein-Rutman theory for positive semigroup.The aim of this paper is to initiate a kind of unified treatment of the above generalized Fokker-Planck equations and more importantly to establish that the convergence (1.3) is exponentially fast uniformly with respect to the diffusion term for a large class of initial data which are taken in a fixed weighted Lebesgue or weighted Sobolev space X.
We investigate three regimes where these diffusion operators are close and for which such a uniform convergence can be established.In Section 2, we first consider the case when the diffusion operator is discrete where k is a nonnegative, symmetric, normalized, smooth and decaying fast enough kernel and where we use the notation k ε (x) = k(x/ε)/ε d , ε > 0. In the limit ε → 0, one then recovers the classical diffusion operator D 0 = ∆.
In Section 3, we next consider the case when the diffusion operator is fractional Df = D ε f := −(−∆) (2−ε)/2 f, ε ∈ (0, 2), so that in the limit ε → 0 we also recover the classical diffusion operator D 0 = ∆.In Section 4, we finally consider the case when the diffusion operator is a discrete version of the fractional diffusion, namely where (κ ε ) is a family of convenient bounded kernels which converges towards the kernel of the fractional diffusion operator k 0 := c α | • | −d−α for some fixed α ∈ (0, 2), in particular, in the limit ε → 0, one may recover the fractional diffusion operator D 0 = −(−∆) α/2 .
In order to write a rough version of our main result, we introduce some notation.We define the weighted Lebesgue space L 1 r , r ≥ 0, as the space of measurable functions f such that f x r ∈ L 1 , where x 2 := 1 + |x| 2 .For any f 0 ∈ L 1 r , we denote as f (t) the solution to the generalized Fokker-Planck equation (1.1) with initial datum f (0) = f 0 and then we define the semigroup S Λ on X by setting S Λ (t)f 0 := f (t).

Theorem 1.1 (rough version).
There exist q > 0 and ε 0 ∈ (0, 2) such that for any ε ∈ [0, ε 0 ], the semigroup S Λε is well-defined on X := L 1 r and there exists a unique positive and normalized stationary solution G ε to (1.1).Moreover, there exist a < 0 and C ≥ 1 such that for any f 0 ∈ X, there holds Our approach is a semigroup approach in the spirit of the semigroup decomposition framework introduced by Mouhot in [10] and developed subsequently in [7,4,12,6,5].Theorem 1.1 generalizes to the discrete diffusion Fokker-Planck equation and to the discrete fractional diffusion Fokker-Planck equation similar results obtained for the classical Fokker-Planck equation in [4,6] (Section 2) and for the fractional one in [12] (Section 4).It also makes uniform with respect to the fractional diffusion parameter the convergence results obtained for the fractional diffusion equation in [12] (Section 3).It is worth mentioning that there exists a huge literature on the long-time behaviour for the Fokker-Planck equation as well as (to a lesser extend) for the fractional Fokker-Planck equation.We refer to the references quoted in [4,6,12] for details.There also probably exist many papers on the discrete diffusion equation since it is strongly related to a standard random walk in R d , but we were not able to find any precise reference in this PDE context.1.2.Method of proof.Let us explain our approach.First, we may associate a semigroup S Λε to the evolution equation (1.1) in many Sobolev spaces, and that semigroup is mass preserving and positive.In other words, S Λε is a Markov semigroup and it is then expected that there exists a unique positive and unit mass steady state G ε to the equation (1.1).Next, we are able to establish that the semigroup S Λε splits as ε ≈ e tTε , T ε finite dimensional, S 2 ε = O(e at ), a < 0, in these many weighted Sobolev spaces.The above decomposition of the semigroup is the main technical issue of the paper.It is obtained by introducing a convenient splitting where B ε enjoys suitable dissipativity property and A ε enjoys some suitable B ε -power regularity (a property that we introduce in Section 2.4 (see also [5]) and that we name in that way by analogy with the B ε -power compactness notion introduced by Voigt [13]).It is worth emphasizing that we are able to exhibit such a splitting with uniform (dissipativity, regularity) estimates with respect to the diffusion parameter ε ∈ [0, ε 0 ] in several weighted Sobolev spaces.
As a consequence of (1.5), we may indeed apply the Krein-Rutman theory developed in [9,5] and exhibit such a unique positive and unit mass steady state G ε .Of course for the classical and fractional Fokker-Planck equations the steady state is trivially given through an explicit formula (the Krein-Rutman theory is useless in that cases).A next direct consequence of the above spectral and semigroup decomposition (1.5) is that there is a spectral gap in the spectral set Σ(Λ ε ) of the generator Λ ε , namely and next that an exponential trend to the equilibrium can be established, namely for any initial datum f 0 ∈ X with vanishing mass.Our final step consists in proving that the spectral gap (1.7) and the estimate (1.8) are uniform with respect to ε, more precisely, there exists λ * < 0 such that λ ε ≤ λ * for any ε ∈ [0, ε 0 ] and C ε can be chosen independent to ε ∈ [0, ε 0 ].
A first way to get such uniform bounds is just to have in at least one Hilbert space Eε , and then (1.8) essentially follows from the fact that the splitting (1.6) holds with operators which are uniformly bounded with respect to ε ∈ [0, ε 0 ].It is the strategy we use in the case of the fractional diffusion (Section 3) and the work has already been made in [12] except for the simple but fundamental observation that the fractional diffusion operator is uniformly bounded (and converges to the classical diffusion operator) when it is suitable (re)scaled.
A second way to get the desired uniform estimate is to use a perturbation argument.Observing that, in the discrete cases (Sections 2 and 4), for a suitable operator norm, we are able to deduce that ε → λ ε is a continuous function at ε = 0, from which we readily conclude.We use here again that the considered models converge to the classical or the fractional Fokker-Planck equation.In other words, the discrete models can be seen as (singular) perturbations of the limit equations and our analyze takes advantage of such a property in order to capture the asymptotic behaviour of the related spectral objects (spectrum, spectral projector) and to conclude to the above uniform spectral decomposition.This kind of perturbative method has been introduced in [7] and improved in [11].In Section 4, we give a new and improved version of the abstract perturbation argument where some dissipativity assumptions are relaxed with respect to [11] and only required to be satisfied on the limit operator (ε = 0).

Comments and possible extensions.
Motivations.The main motivation of the present work is rather theoretical and methodological.Spectral gap and semigroup estimates in large Lebesgue spaces have been established both for Boltzmann like equations and Fokker-Planck like equations in a series of recent papers [10,7,4,9,2,1,12,6,8].The proofs are based on a splitting of the generator method as here and previously explained, but the appropriate splitting are rather different for the two kinds of models.The operator A ε is a multiplication (0-order) operator for a Fokker-Planck equation while it is an integral (−1-order) operator for a Boltzmann equation.More importantly, the fundamental and necessary regularizing effect is given by the action of the semigroup S Bε for the Fokker-Planck equation while it is given by the action of the operator A ε for the Boltzmann equation.Let us underline here that in Section 4, we exhibit a new splitting for fractional diffusion Fokker-Planck operators (different from the one introduced in [12]) in the spirit of Boltzmann like operators (the operator A ε is an integral operator whereas it was a multiplication operator in [12] and in Section 3).Our purpose is precisely to show that all these equations can be handled in the same framework, by exhibiting a suitable and compatible splitting (1.6) which does not blow up and such that the time indexed family of operators A ε S Bε (or some iterated convolution products of that one) have a good regularizing property which is uniform in the singular limit ε → 0.
Probability interpretation.The discrete and fractional Fokker-Planck equations are the evolution equations satisfied by the law of the stochastic process which is solution to the SDE (1.9) For two trajectories X t and Y t to the above SDE associated to some initial data X 0 and Y 0 , and p ∈ [1, 2), we have We fix now Y t as a stable process for the SDE (1.9).Denoting by f ε (t) the law of X t and G ε the law of Y t , we classically deduce the Wasserstein distance estimate (1.10) In particular, for p = 1, the Kantorovich-Rubinstein Theorem says that (1.10) is equivalent to the estimate Estimates (1.10) and (1.11) have to be compared with (1.8).Proceeding in a similar way as in [9,6] it is likely that the spectral gap estimate (1.11) can be extended (by "shrinkage of the space") to a weighted Lebesgue space framework and then to get the estimate in Theorem 1.1 for any a ∈ (−1, 0).
Singular kernel and other confinement term.We also believe that a similar analysis can be handle with more singular kernels than the ones considered here, the typical example should be k(z) = (δ −1 + δ 1 )/2 in dimension d = 1, and for confinement term different from the harmonic confinement considered here, including other forces or discrete confinement term.In order to perform such an analysis one could use some trick developed in [9] in order to handle the equal mitosis (which uses one more iteration of the convolution product of the time indexed family of operators A ε S Bε ).
Linearized and nonlinear equations.We also believe that a similar analysis can be adapted to nonlinear equations.The typical example we have in mind is the Landau grazing collision limit of the Boltzmann equation.One can expect to get an exponential trend of solutions to its associated Maxwellian equilibrium which is uniform with respect to the considered model (Boltzmann equation with and without Grad's cutoff and Landau equation).

Kinetic like models.
A more challenging issue would be to extend the uniform asymptotic analysis to the Langevin SDE or the kinetic Fokker-Planck equation by using some idea developed in [1] which make possible to connect (from a spectral analysis point of view) the parabolic-parabolic Keller-Segel equation to the parabolic-elliptic Keller-Segel equation.The next step should be to apply the theory to the Navier-Stokes diffusion limit of the (in)elastic Boltzmann equation.These more technical problems will be investigated in next works.
1.4.Outline of the paper.Let us describe the plan of the paper.In each section, we treat a family of equations in a uniform framework, from a spectral analysis viewpoint with a semigroup approach.In Section 2, we deal with the discrete and classical Fokker-Planck equations.Section 3 is dedicated to the analysis of the fractional and classical Fokker-Planck equations.Finally, Section 4 is devoted to the study of the discrete and fractional Fokker-Planck equations.
1.5.Notations.For a (measurable) moment function m : R d → R + , we define the norms and the associated weighted Lebesgue and Sobolev spaces L p (m) and W k,p (m), we denote H k (m) = W k,2 (m) for k ≥ 1.We also use the shorthand L p r and W 1,p r for the Lebesgue and Sobolev spaces L p (ν) and W 1,p (ν) when the weight ν is defined as ν We denote by m a polynomial weight m(x) := x q with q > 0, the range of admissible q will be specified throughout the paper.
In what follows, we will use the same notation C for positive constants that may change from line to line.Moreover, the notation A ≈ B shall mean that there exist two positive constants Acknowledgments.The research leading to this paper was (partially) funded by the French "ANR blanche" project Stab: ANR-12-BS01-0019.The second author has been partially supported by the fellowship l'Oréal-UNESCO For Women in Science.

From discrete to classical Fokker-Planck equation
In this section, we consider a kernel k as well as the positivity condition: there exist κ 0 , ρ > 0 such that We define k ε (x) := 1/ε d k(x/ε), x ∈ R d for ε > 0, and we consider the discrete and classical Fokker-Planck equations The main result of the section reads as follows.(1) For any ε > 0, there exists a positive and unit mass normalized steady state

3).
(2) There exist explicit constants a 0 < 0 and ε 0 > 0 such that for any ε ∈ [0, ε 0 ], the semigroup S Λε (t) associated to the discrete Fokker-Planck equation (2.3) satisfies: for any f ∈ L 1 r and any a > a 0 , r , where we have denoted The method of the proof consists in introducing a suitable splitting of the operator Λ ε as Λ ε = A ε + B ε , in establishing some dissipativity and regularity properties on B ε and A ε S Bε and finally in applying the version [9,5] of the Krein-Rutman theorem as well as the perturbation theory developed in [7,11,5].

Uniform boundedness of
Lemma 2.2.For any p ∈ [1, ∞], s ≥ 0 and any weight function ν ≥ 1, the operator A ε is bounded from W s,p into W s,p (ν) with norm independent of ε.
Proof.For any f ∈ L p (ν), we have thanks to the Young inequality and because The proof for the case s > 0 is similar and it is thus skipped.
Proof.We split the operator in several pieces ε , and we estimate each term 1 ) and we consider ε ∈ (0, ε 1 ].We first deal with T 1 .We observe that using the convexity of Φ.We then compute where we have performed a change of variables to get the last equality.From a Taylor expansion, we have where for some constant C ∈ (0, ∞).The term involving the gradient of m p gives no contribution because of (2.1) and we thus obtain (2.6) We now treat the second term T 2 .Proceeding as above and thanks to (2.5) again, we have Using the mean value theorem , for some θ, θ ′ ∈ (0, 1), and the estimates As far as T 3 is concerned, we just perform an integration by parts: (2.8) The estimates (2.6), (2.7) and (2.8) together give where p ′ = p/(p − 1) and we have denoted As a conclusion, for such a choice of constants, we obtain (2.4).We refer to [4,6] for the proof in the case ε = 0.
Proof.The case s = 0 is nothing but Lemma 2.3 applied with p = 2.We now deal with the case s = 1.We consider f t a solution to From the previous lemma, we already know that (2.10) 1 2 We now want to compute the evolution of the derivative of f t : which in turn implies that 1 2 Concerning T 1 , using the proof of Lemma 2.3, we obtain Then, to deal with T 2 , we first notice that using Jensen inequality and (2.1), we have We thus obtain using that k ∈ L 1 2q : The term T 2 is then treated using Cauchy-Schwarz inequality, Young inequality and the fact that |∂ x (χ c R )| is bounded by a constant depending only on R: (2.12) for any ζ > 0 as small as we want.
The term T 3 is handled using an integration by parts and with the fact that |∂ 2 x (χ c R )| is bounded with a constant which only depends on R: (2.13) Combining estimates (2.11), (2.12) and (2.13), we easily deduce To conclude the proof in the case s = 1, we introduce the norm Combining (2.10) and (2.14), we get Using the same strategy as in the proof of Lemma 2.3, if a > d/2 − q + 1, we can choose M , R large enough and ζ, ε 0 , η small enough such that we have on R d The higher order derivatives are treated with the same method introducing a similar modified H s (m) norm.
2.4.Uniform B ε -power regularity of A ε .In this section we prove that A ε S Bε and its iterated convolution products fulfill nice regularization and growth estimates.
We introduce the notation (2.15) Lemma 2.5.There exists a constant K > 0 such that for any ε > 0, the following estimate holds: Proof.
Step 1.We prove that the assumptions made on k imply for some constant K > 0. On the one hand, we have Moreover, performing a Taylor expansion, using the normalization condition (2.1) and the fact that k ∈ L 1 3 (R d ), we have We then deduce that (2.17) holds with K = 1 in a small ball ξ ∈ B(0, δ).On the other hand, for any ξ = 0, we have where We then deduce that (2.17) holds with K = C/η in the set B(0, δ) c .
Step 2. From the normalization condition (2.1), we have As a consequence, using Plancherel formula and the identity Then, we use again Plancherel formula to obtain We conclude to (2.16) by using (2.17).
We now introduce the following notation λ := 1/(2K) > 0 and go into the analysis of regularization properties of the semigroup A ε S Bε (t).Lemma 2.6.Consider s 1 < s 2 ∈ N and q > d/2 + s 2 .We suppose that k ∈ L 1 2q+1 .Let M , R and ε 0 so that the conclusion of Lemma 2.4 holds in both spaces H s 1 (m) and H s 2 (m).Then, for any a ∈ (max{d/2−q+s 2 , −λ}, 0), there exists n ∈ N such that for any ε ∈ [0, ε 0 ], we have the following estimate for some constant C a > 0.
Proof.We first give the proof for the case (s 1 , s 2 ) = (0, 1).We consider a ∈ (max{d/2 − q + 1, −λ}, 0), α 0 and α 1 such that a > α 0 > α 1 > max{d/2 − q + 1, −λ} and f t := S Bε (t)f with f ∈ L 2 (m), i.e. that satisfies From the proof of Lemma 2.4, there exists ε 0 such that for any ε ∈ (0, ε 0 ], we have 1 2 where we have used that M ≤ 1/(2ε 2 ) for any ε ∈ (0, ε 0 ].Using Lemma 2.5, we obtain Multiplying this inequality by e −2α 0 t , it implies that and thus, integrating in time In particular, we obtain We now want to estimate Using dissipativity properties of B ε and boundedness of A ε , we get We deal with I 2 using the fact that M ∂ x (χ R ) is compactly supported, Young inequality and dissipativity properties of B ε : Finally, for I 3 , we use (2.18) to obtain All together, we have proved Consequently, using Cauchy-Schwarz inequality, we have Fom the dissipativity of B ε in H 1 (m) proved in Lemma 2.4 and the fact that A ε is bounded in H 1 (m), we also have Using the two last estimates together, we deduce that for any t ≥ 0 We have thus proved which corresponds to the case (s 1 , s 2 ) = (0, 1).Using the same strategy, we can easily obtain that for any s ≥ 2, and then conclude the proof of the lemma in the case ε > 0.
We refer to [4,6] for the proof in the case ε = 0.
Proof.We first introduce the formal dual operators of A ε and B ε : We use the same computation as the one used to deal with T 1 is the proof of Lemma 2.3 and Cauchy-Schwarz inequality: We then notice that the second term equals 0 and we use Young inequality and the fact that where I ε is defined in (2.15).We also have the following inequality: If we denote φ t := S B * ε (t)φ, we thus have 1 2 Multiplying this inequality by e −bt , we obtain and integrating in time, we get We now estimate Using Young inequality and (2.20), we conclude that As in the proof of Lemma 2.6, for any s ≥ 1, we can then establish that for some b ′ ≥ 0, and by duality Taking ℓ > d/2, so that we can use the continuous Sobolev embedding and using the fact that A ε is compactly supported combined with Lemma 2.3, we get for some b ′′ ≥ 0. To conclude the proof, we use [4, Lemma 2.17].Indeed, up to take more convolutions, we are able to recover a good rate in the last estimate.We refer to [4,6] for the proof in the case ε = 0.
Step 1.We first deal with A ε in the case s = 0. Using that χ ∈ D(R d ) and Concerning the first derivative, writing that and using that ∂ x χ R is uniformly bounded as well as χ R , we obtain the result.We omit the details of the proof for higher order derivatives.
Step 2. In order to prove the second part of the result, we just have to prove Using (2.1), we have A Taylor expansion of f gives We then observe that, because of (2.1), the integral in the y variable of the gradient term cancels and the contribution of the second term is precisely ∆f (x).We deduce that Consequently, using Jensen inequality and the fact that k ∈ L 1 2q+3 , we get This concludes the proof of the second part in the case s = 0.The proof for s > 0 follows from the fact that the operator ∂ x commutes with Λ ε − Λ 0 .
Proof.First, we have signf which ends the proof of the Kato inequality.We consider f ≤ 0 and denote f (t) := S Λε (t)f .We define the function β(s) = s + = (|s| + s)/2.Using Kato's inequality, we have from which we deduce f t ≤ 0 for any t ≥ 0.
The operator −Λ ε satisfies the following form of the strong maximum principle.
Lemma 2.10.Any nonnegative eigenfunction associated to the eigenvalue 0 is positive.In other words, we have Proof.We define x) e λt with generator D. Thanks to the Duhamel formula By assumption, there exists , and then Using that lower bound, we obtain Repeating once more the argument, we get the same lower estimate with i = 3, u 3 = 7/4, κ 3 > 0 and v 3 = 3/2.By an induction argument, we finally get f > 0 on R d .
We are now able to prove Theorem 2.1.We suppose that the assumptions of Theorem 2.1 hold in what follows and thus consider r > d/2 and also r 0 > max(r + d/2, 5 + d/2).
Proof of part (1) in Theorem 2.1.Using Lemmas 2.2-2.4-2.3,2.9, 2.10 and the fact that Λ * ε 1 = 0, we can apply Krein-Rutman theorem which implies that for any ε > 0, there exists a unique It also implies that for any ε > 0, there exists a ε < 0 such that in X = L 1 r or X = H s r 0 for any s ∈ N, there holds Proof of part (2) in Theorem 2.1.
We now have to establish that estimate (2.21) can be obtained uniformly in ε ∈ [0, ε 0 ].In order to do so, we use a perturbation argument in the same line as in [7,11] to prove that our operator Λ ε has a spectral gap in H 3 r 0 which does not depend on ε.First, we introduce the following spaces: r 0 , notice that r 0 > d/2 + 5 implies that the conclusion of Lemma 2.4 is satisfied in the three spaces X i , i = −1, 0, 1.
One can notice that we also have the following embedding We now summarize the necessary results to apply a perturbative argument (obtained thanks to Lemmas 2.8, 2.2, 2.3, 2.4 and 2.6 and from [4,6]).
(ii) For any a > a 0 and ℓ ≥ 0, there exists C ℓ,a > 0 such that (iii) For any a > a 0 , there exist n ≥ 1 and C n,a > 0 such that (iv) There exists a function η(ε) − −− → ε→0 0 such that where 0 is a one dimensional eigenvalue.
Using a perturbative argument as in [11], from the facts (i)-(v), we can deduce the following proposition: Proposition 2.11.There exist a 0 < 0 and ε 0 > 0 such that for any ε ∈ [0, ε 0 ], the following properties hold in X 0 = H 3 r 0 : (1) Σ(Λ ε ) ∩ D a 0 = {0}; (2) for any f ∈ X 0 and any a > a 0 , To end the proof of Theorem 2.1, we have to enlarge the space in which the conclusions of the previous Proposition hold.To do that, we use an extension argument (see [4] or [7, Theorem 1.1]) and Lemmas 2.2, 2.3-2.4 and 2.6-2.7.Our "small" space is H 3 r 0 and our "large" space is L 1 r (notice that r 0 > r + d/2 implies the embedding H 3 r 0 ⊂ L 1 r ).

From fractional to classical Fokker-Planck equation
In this part, we denote α := 2 − ε ∈ (0, 2] and we deal with the equations We here recall that the fractional Laplacian ∆ α/2 f is defined for a Schwartz function f through the integral formula (1.2).Moreover, the constant c α in (1.2) is chosen such that which implies that c α ≈ (2 − α).By duality, we can extend the definition of the fractional Laplacian to the following class of functions: In particular, one can define (−∆) α/2 m when q < α.
Theorem 3.13.There exists a constant a 0 < 0 such that for any α ∈ (0, 2), ), there holds Σ(L α ) ∩ D a 0 = {0}; (2) the following estimate holds: for any a > a 0 , 3.2.Splitting of L α and uniform estimates.The proof is based on the splitting of the operator L α as L α = A + B α where A is the multiplier operator Af := M χ R f , for some M, R > 0 to be chosen later, and an extension argument taking advantage of the already known exponential decay in L 2 (G As a straightforward consequence of the definition of A, we get the following estimates. Lemma 3.14.Consider s ∈ N and p ≥ 1.The operator is uniformly bounded in α from W s,p (ν) to W s,p with ν = m or ν = G −1/2 α .We next establish that B α enjoys uniform dissipativity properties.Lemma 3.15.For any a > −q, there exist M > 0 and R > 0 such that for any α ∈ [α 0 , 2], B α − a is dissipative in L 1 (m).
Proof.We just have to adapt the proof of Lemma 5.1 from [12] taking into account the constant c α .Indeed, we have We can then show that thanks to the rescaling constant c α , I α (m)/m goes to 0 at infinity uniformly in α ∈ [α 0 , 2).As a consequence, if a > −q, since (x • ∇m)/m goes to −q at infinity, one may choose M and R such that for any α ∈ [α 0 , 2), which gives the result.
Lemma 3.16.For any a > a 0 where a 0 is defined in Theorem 3.13, ).
We finally establish that AS Bα enjoys some uniform regularization properties.

Spectral analysis.
Before going into the proof of Theorem 3.12, let us notice that we can make explicit the projection Π α onto the null space N (L α ) through the following formula: Π α f = f G α .Moreover, since the mass is preserved by the equation ∂ t f = L α f , we can deduce that Π α (S Lα (t)f ) = Π α f for any t ≥ 0.

From discrete to fractional Fokker-Planck equation
Let us fix α ∈ (0, 2).We consider the equations (4.24) where We here recall that for α ∈ (0, 2), the fractional Laplacian on Schwartz functions is defined through the formula (1.2).Since α is fixed in this part, we can get rid of the constant c α and consider that it equals 1.The main theorem of this section reads: Theorem 4.18.Assume 0 < r < α/2.
(1) For any ε > 0, there exists a positive and unit mass normalized steady state G ε ∈ L 1 r (R d ) to the discrete fractional Fokker-Planck equation (4.24).(2) There exist an explicit constant a 0 < 0 and a constant ε 0 > 0 such that for any ε ∈ [0, ε 0 ], the semigroup S Λε (t) associated to the discrete and fractional Fokker-Planck equations (4.24) satisfies: for any f ∈ L 1 r and any a > a 0 , r .The method of the proof is similar to the one of Section 2. We introduce a suitable splitting Λ ε = A ε + B ε , establish some dissipativity and regularity properties on B ε and A ε S Bε and apply the Krein-Rutman theory revisited in [9,5].However, let us emphasize that we introduce a new splitting for the fractional operator (a different one from Section 3 and from [12]) and we also develop a new perturbative argument in the same line as [7,11,5] but with some less restrictive assumptions on the operators A ε and B ε , requiring that they are fulfilled only on the limit operator (i.e. for ε = 0).
We split these operators into several parts: for any ε ≥ 0, (4.26) where the constants η ∈ [ε, 1], R > 0 and 0 < L ≤ 1/ε will be chosen later.One can notice that given the facts that η ≥ ε and L ≤ 1/ε, we have for any ε > 0, A ε = A 0 =: A. Finally, we denote for any ε ≥ 0, Lemma 4.19.Consider p ∈ (1, ∞) and q ∈ (0, α/p).The following convergence holds: Step 1.We first consider the case s = 0 and we introduce the notation To deal with T 1 , we perform a Taylor expansion of f of order 2 and we use that χ(z) = 1 if |z| ≤ 1, in order to get From Hölder inequality applied with the measure µ ε (dz) := 1 |z|≤1 k 0,ε (z) |z| 2 dz, we have where p ′ = p/(p − 1) is the Hölder conjugate exponent of p.Using now Jensen inequality, we get by Lebesgue dominated convergence theorem.To treat T 2 , we first notice that the term involving ∇f (x) gives no contribution, because k 0,ε χ ≡ 0 for ε ∈ (0, 1/2), so that performing similar computations as for T 1 , we have by the Lebesgue dominated convergence theorem again.As a consequence, we obtain Step 2. We now consider the case s = −2, and we recall that by definition where p ′ = p/(p − 1) and because (Λ ε − Λ 0 ) * = Λ ε − Λ 0 (where Λ * stands for the formal dual operator of Λ).For sake of simplicity, we introduce the notation (4.27) We then estimate the integral in the right hand side of the previous equality: Moreover, (4.28) We deduce that To deal with J 1 , we use the step 1 of the proof which gives us The term J 2 is split into two parts: We first notice that for |z| ≤ 1, which implies that Since 0 < q < 2, |D 2 m| ≤ C and 1/m p ′ ≤ C in R d , we thus deduce using Hölder inequality and a change of variable, and we obtain that J 22 is bounded from above by which implies, using Hölder inequality and a change of variable, Finally, we handle J 3 performing a Taylor expansion of φ: Hölder inequality and a change of variable, As a consequence, we obtain that which concludes the proof.
Proof.First, one can notice that (4.29) the proof is hence immediate in the case s = t = 0 using Young inequality: We now deal with the case (s, t) = (0, 2).First, we have for ℓ = 1, 2 and for any (i, j, k) such that i which concludes the proof in the case (s, t) = (0, 2).Finally, arguing by duality, we have which proves the estimate in the case (s, t) = (−2, 0).
Notice that there exists C L > 0 depending on L such that |∇m p (y + θz)| ≤ C L y pq−1 for any y ∈ R d , |z| ≤ 2L.We hence obtain which leads to As a consequence, we obtain We estimate the term involving B 4 ε using that ξ R (x, y) ≥ χ R (x), and we get Finally, using integration by parts, we have Gathering all the previous estimates and denoting First, since ϕ m : x → d(1 − 1/p) − x • ∇m p (x)/p m p (x) is a continuous function, we can bound it by above by a constant C R depending on R on {|x| ≤ R} for any R > 0. We denote ℓ := d(1 − 1/p) − q which is the limit of ϕ m as |x| → ∞.One can also notice that A ε η,L := 2η≤|z|≤L k ε (z) dz → ∞ as ε → 0 and η → 0. We first choose ε 1 > 0, η ≥ ε 1 , L ≤ 1/ε 1 and R > 0, so that we have Up to make decrease the value of η, we can then choose ε 0 < ε 1 such that for any ε ∈ [0, ε 0 ], As a conclusion, for this choice of constants, for any x ∈ R d and ε ∈ [0, ε 0 ], we have ψ ε η,L,R (x) ≤ a, which yields the result.Lemma 4.22.Consider q ∈ (0, α/2).There exists b ∈ R such that for any s ∈ N, B 0 − b is hypodissipative in H s (m).
Proof.Step 1.We first treat the case s = 0. We write B 0 = Λ 0 − A 0 and we compute Concerning T 1 , we have Since one can prove that I 0 (m 2 )/m 2 goes to 0 at infinity (cf Lemma 5.1 from [12]) and is thus bounded in R d , we can deduce that there exists C ∈ R + such that We observe that We split the last term into two pieces, that we estimate in the following way: We recall that the homogeneous Sobolev space Ḣs for s ∈ R is the set of tempered distributions u such that u belongs to L 1 loc and and that for s ∈ (0, 1), there exists a constant c 0 > 0 such that |x − y| d+2s dx dy from which we deduce the following identity: As a consequence, up to change the value of C, we have proved Next, we compute Concerning T 3 , we use Lemma 4.20 and Cauchy-Schwarz inequality: . As a consequence, gathering the three previous inequalities, we have Step 2. We now consider b > b 0 and we prove that for any s ∈ N, B 0 − b is hypodissipative in H s (m).For s ∈ N * , we introduce the norm which is equivalent to the classical H s (m) norm.We use again the fact that B 0 = Λ 0 −A 0 and we only deal with the case s = 1, the higher order derivatives being treated in the same way.First, we have Then, we can notice that where the last inequality is obtained thanks to inequality (4.29) as in the proof of Lemma 4.20.We deduce that Then, doing the same computations as in the case s = 0, we obtain with , and also . Finally, using Cauchy-Schwarz inequality, we have As a consequence, we have We now introduce f t the solution to the evolution equation We now use the following interpolation inequality which concludes the proof in the case s = 1.
We now introduce the operator B 0,m defined by Corollary 4.23.Consider q such that 2q < α.There exists b ∈ R such that for any s ∈ N, B 0,m − b is hypodissipative in H s .
Proof.The proof comes from Lemma 4.22 and is immediate noticing that the norms defined on H s (m) by Lemma 4.24.Consider q such that 2q < α.There exists b ∈ R such that for any s Proof.We introduce the dual operator of B 0,m defined by: where ω := m −1 .We now want to prove that B * 0,m is hypodissipative in H s .
Step 1.We consider first the case s = 0 and we compute We have Next, using (4.29), we have A 0 (m φ) L 2 ≤ C A 0 (|φ|) L 2 and thus from Lemma 4.20.Let us now estimate T 1 .
Case α < 1.We write Let us point out here that from (4.31), we have Next, using a Taylor expansion, there exists θ ∈ (0, 1) such that (4.35) Concerning T 13 , we have from (4.30) |m(y) − m(x)| ≤ C x − y q min x q/2 , y q/2 , from which we deduce All together, we have thus proved Case α ∈ [1, 2).We write where we recall that T ν is defined in (4.27).We have again Arguing similarly as for T 12 in (4.35), but using a Taylor expansion at order 2 instead of order 1, we obtain Next, we split T 13 into two parts: where we have used (4.30), we thus obtain: Concerning T 14 , we use Young inequality which implies that for any ζ > 0, Consequently, taking ζ > 0 small enough, we have We hence conclude that Step 2. We now consider b > b 0 and we prove that for any s ∈ N, B * 0,m − b is hypodissipative in H s .As in (4.32), for s ∈ N * , we introduce the norm which is equivalent to the classical H s norm.We only deal with the case s = 1, the higher order derivatives are treated in the same way.First, using the identity (4.28) (with k 0 instead of k 0,ε ), we notice that where Before going into the computation of ∂ x (B * 0,m φ), we also notice that where A 0,m satisfies thanks to (4.29).Consequently, we have and We have from the step 1 of the proof Moreover, we easily obtain that The term J 2 is first separated into two parts: where we recall that T ∂xm is defined in (4.27).The term J 21 is treated as T 12 is the step 1 of the proof.Concerning J 22 , as for T 13 , we split it into two parts: because q < α/2 < 1.We hence deduce that Concerning J 3 , we perform a Taylor expansion of φ and use the fact that |∇(∂ x m)| ω ∈ L ∞ (R d ): (4.36) where we have used Jensen inequality and Young inequality.We use a change of variable for the first term of the RHS of (4.36), which implies that J 3 ≤ C φ 2 Ḣ1 .We deal with J 4 splitting it into two parts (|x − y| ≤ 1 and |x − y| ≥ 1) and using the same method as for T 12 and T 13 in the step 1 of the proof, we obtain To deal with J 5 , we proceed exactly as for J 3 and we obtain J 5 ≤ C φ 2 Ḣ1 .Summarizing the previous inequalities and using (4.33), we obtain that for any ζ > 0, We now fix 0 < r < α/2 as in the assumptions of Theorem 4.18.We also introduce r 0 ∈ (r, α/2) and m 0 (x) := x r 0 .From Lemma 4.21 applied with p = 1, there exists a < 0 such that B ε − a is dissipative in L 1 (m 0 ) for any ε ∈ [0, ε 1 ] (or equivalently, B ε,m 0 − a is dissipative in L 1 where B ε,m 0 is defined as B 0,m in (4.34)).From Lemma 4.21 applied with p = 2, Corollary 4.23 and As an immediate consequence, there holds Proof.We know that the operators AR Bε (z) : X 0 → X 1 (from Lemmas 4.20 and 4.25) and R Λ 0 (z) : X 1 → X 1 (previous works from [4,6]) are bounded for any z ∈ Ω and that the operators Λ ε − Λ 0 : X 1 → X 0 are small as ε → 0 uniformly in z ∈ Ω (Lemma 4.19).Because 0 is a simple eigenvalue, we have for some C > 0. We introduce the constant C a θ > 0 (coming from Lemmas 4.20 and 4.25) such that AS Bε (t) B(X 0 ,X 1 ) ≤ C a θ e a θ t .
Since Λ ε −z is invertible for ℜe z large enough and J ε (z) is uniformly locally bounded in Ω ε , we deduce that Λ ε − z is invertible in Ω ε , and its inverse is its right-inverse J ε (z).