1 Introduction

Fractional Fokker-Planck equation

Abstract.

This paper deals with the long time behavior of solutions to a “fractional Fokker-Planck” equation of the form where the operator stands for a fractional Laplacian. We prove an exponential in time convergence towards equilibrium in new spaces. Indeed, such a result was already obtained in a space with a weight prescribed by the equilibrium in [6]. We improve this result obtaining the convergence in a space with a polynomial weight. To do that, we take advantage of the recent paper [7] in which an abstract theory of enlargement of the functional space of the semigroup decay is developed.

Mathematics Subject Classication (2010): 47G20 Integro-differential operators; 35B40 Asymptotic behavior of solutions; 35Q84 Fokker-Planck equations.

Keywords: Fractional Laplacian; Fokker-Planck equation; spectral gap; exponential rate of convergence; long-time asymptotic.

1. Introduction

1.1. Model and main result

For , we consider the following generalization of the Fokker-Planck equation:

(1.1)

with an initial data . In the sequel, we will use the shorthand notations

The operator is a fractional Laplacian, we first define it on the space of Schwartz functions and we then extend the definition to others functions. We refer to Section 2 for the exact definition and for properties.

We also define here weighted spaces in the following way: for some given Borel weight function on , let us define , , as the Lebesgue space associated to the norm

We now state our main result concerning the equation (1.1).

Theorem 1.1.

Let us consider . For any (where will be defined in Corollary 4.5) and for any initial data , the solution  of the equation (1.1) satisfies the following decay:

where and for some constant .

1.2. Known results

The main references to mention here are the papers [4] and [6]. In these two papers, “Lévy-Fokker-Planck equations” (the fractional Laplacian is replaced by a Lévy operator) are studied using the entropy production method. There is a proof of existence and uniqueness of a nonnegative steady state of mass of the associated stationary equation. Then, in a weighted space with a weight prescribed by the equilibrium, a convergence (with an exponential rate) of the solution of the full equation towards equilibrium is obtained. Let us give more details about these results. We first introduce the main tools used.

Consider a smooth convex function and positive such that and define the -entropy: for any nonnegative function ,

Jensen’s inequality gives that . Let be an initial condition of a Lévy-Fokker-Planck equation or of the classical Fokker-Planck equation:

(1.2)

Then, let us introduce the quantity which is well-defined for any .

In the case of the classical Fokker-Planck equation (1.2), by using functional inequalities as Poincaré, logarithmic Sobolev or -entropy inequalities, one obtains exponential decays to zero of . Then, the solution of (1.2) converges towards the steady state of mass in the sense of -entropy. Methods to prove such results are usually based on entropy/entropy-production tools. See [3, 1, 2, 5] for different methods and applications.

In [4], Biler and Karch study Lévy-Fokker-Planck equations where the Lévy operators are Fourier multipliers associated to symbols satisfying for some real number

They prove that there exist and such that

which means that the solution converges towards equilibrium at an exponential rate in where we denote the only steady state of mass . They deduce a similar result in and finally, under some more restrictive regularity and decay assumptions on , they prove that the exponential convergence holds in .

In [6], taking advantage of the paper [4], Gentil and Imbert prove an exponential decay of the -entropies for a class of convex functions and for a larger class of operators which includes the fractional Laplacian.

In the present paper, we only consider the equation (1.1) but we are able to enlarge the space where we have a decay towards equilibrium with minimal assumptions on . If we compare our result to the one obtained in [4] for others operators defined above, we have to underline the fact that the result of convergence of the solution towards equilibrium in from [4] requires additional assumptions on ( must have finite moments of a large order), it is not the case in our main result where is only supposed to belong to with .

1.3. Method of proof and outline of the paper

The main outcome of the present paper is a result of decay towards equilibrium with an exponential rate of convergence in (with ) for solutions of our equation (1.1). To do that, we adopt the same strategy as the one adopted in [7] by Gualdani, Mischler and Mouhot for the classical Fokker-Planck equation. Let us explain in more details this strategy. It is based on the theory of enlargement of the functional space of the semigroup decay developed in [7]. It enables to get a spectral gap in a larger space when we already have one in a smaller space. It applies to operators which can be splitted into two parts, with bounded and dissipative. Moreover, if we denote the semigroup associated to the operator , the semigroup is required to have some regularization properties. The fact that we can use this theory for our operator is based on two facts:

  • we know from [6] that our operator has a spectral gap in where is the only steady state of mass of (1.1),

  • we are able to get a splitting satisfying the previous properties using computations based on properties of the fractional Laplacian.

In section 2, we recall some technical tools about the fractional Laplacian that are useful in order to get a splitting of the operator. In section 4, we state results from [6] which are necessary to apply the abstract theorem of enlargement of spectral gap, which is reminded in Section 3. Finally, in Section 5, we apply this theorem to obtain our main result on the convergence towards equilibrium of the solution of (1.1) in with .

Acknowledgements We would like to thank Stéphane Mischler and Robert Strain for enlightened discussions and their help.

2. Preliminaries on the fractional Laplacian

In this section, we recall some elementary properties of the fractional Laplacian that we will need through this paper. The usual reference for this kind of operators is Landkof’s book [9].

2.1. Definition on

Let us consider . The fractional Laplacian is an operator defined on by:

(2.1)

This definition has to be understood in the sense of principal value:

Due to the singularity of the kernel, the right hand-side of (2.1) is not well defined in general. However, when , the integral is not really singular near . Indeed, since , both and are bounded. We hence deduce the following inequality:

When , we can also write the fractional Laplacian with a non principal value integral. For any , we have

(2.2)

and this integral is well defined.

We can extend the integral definition of the fractional Laplacian to the following set of functions:

In particular, we can define when .

2.2. Fractional Laplacian and Fourier transform

Let us remind a well-known fact about the Fourier transform of the fractional Laplacian of a Schwartz function.

Lemma 2.1.

There exists such that for any , we have:

If is a Schwartz function, there is a singularity at in the Fourier transform of . It implies a lack of decay at infinity for itself, is not a Schwartz function. We can prove that decays at infinity as .

We now mention a very useful property of the fractional Laplacian which can be seen as a sort of integration by parts.

Lemma 2.2.

Let us consider and two Schwartz functions. Then, we have

If , we can also prove that

2.3. Fractional Laplacian and fractional Sobolev spaces

Most of the time, fractional Sobolev spaces are defined in the following way: for is the set of functions such that is also in . We remind here an equivalent definition which is going to be useful in what follows.

Lemma 2.3.

Let us consider . We have:

We also have the following fact:

for some .


3. Theorem of enlargement of the functional space of the semigroup decay

3.1. Notations

For a given real number , we define the half complex plane

For some given Banach spaces and we denote by the space of bounded linear operators from to and we denote by or the associated norm operator. We write when . We denote by the space of closed unbounded linear operators from to with dense domain, and in the case .

For a Banach space and we denote by , , its semigroup, by its domain, by its null space and by its range. We also denote by its spectrum, so that for any belonging to the resolvent set the operator is invertible and the resolvent operator

is well-defined, belongs to and has range equal to . We recall that is said to be an eigenvalue if . Moreover, an eigenvalue is said to be isolated if

In the case when is an isolated eigenvalue, we may define the associated spectral projector by

with . Note that this definition is independent of the value of as the application , is holomorphic. For any isolated, it is well-known (see [8] paragraph III-6.19) that , so that is indeed a projector.

When moreover the so-called “algebraic eigenspace” is finite dimensional we say that is a discrete eigenvalue, written as . In that case, is a meromorphic function on a neighborhood of , with non-removable finite-order pole .

Finally for any such that

where are distinct discrete eigenvalues, we define without any risk of ambiguity

We shall also need the following definition on the convolution of semigroups. Consider some Banach spaces , and . For two given functions

the convolution is defined by

When and , is defined recursively by and for any .

3.2. The abstract theorem

Let us now present an enlargement of the functional space of a quantitative spectral mapping theorem (in the sense of semigroup decay estimate). The aim is to enlarge the space where the decay estimate on the semigroup holds. The version stated here comes from [7, Theorem 2.13] and [7, Lemma 2.17].

Theorem 3.1.

Let , be two Banach spaces such that with dense and continuous embedding, and consider , with and . We assume:

  1. generates a semigroup and

    (with if and if ) and is dissipative on .

  2. There exist such that (with corresponding restrictions and on ) and some constants , , and so that

    1. and are dissipative respectively on and ,

    2. and ,

    3. satisfies

Then the following estimate on the semigroup holds:

Remark 3.2.

The assumption (2)-(iii) implies that for any , there exist some constructive constants , such that


4. Semigroup decay in where is the steady state

4.1. Preliminaries on steady states

We recall results obtained in [6] about existence of steady states. They prove such a theorem for a more general equation than ours:

where . The operator is a Lévy operator defined as:

where is a symmetric semi-definite matrix, and denotes a nonnegative singular measure on that satisfies and .

The fractional Laplacian corresponds to a particular Lévy operator. Indeed, with , and , we obtain the fractional Laplacian. In this particular case, the proof of existence of steady states of (1.1) is easier, we hence give a sketch of a proof of it (it is adapted from the proof of [6, Theorem 1]).

We suppose that is an equilibrium of the equation (1.1). At least formally, we have:

(4.1)

We do the following computation in order to take the Fourier transform of (4.1)

We deduce that an equilibrium satisfies

which implies that for a constant .

In the remaining part of the paper, we denote the only steady state of (1.1) of mass : .

Remark 4.1.

In the case of the classical Fokker-Planck equation (1.2), the steady state is a Maxwellian, it is hence a Schwartz function. In our case, the steady state is not anymore a Schwartz function because its Fourier transform has a singularity at . If we denote a smooth function which is nonnegative, supported on and such that for , we can write the following decomposition of :

We see that the second part of the right-hand side is a Scwhartz function and the first one induces a singularity at . We can hence prove that

4.2. Decay properties in

We again use results obtained in [6]. We just use them in our particular case, the fractional Laplacian.

For a convex function, we introduce on as:

which is nonnegative on .

We will not prove the next two lemmas which are going to enable us to prove the decay towards equilibrium in . The first one is [6, Proposition 1] and the second one is [6,  Theorem 2] and comes from [5].

Lemma 4.2.

Consider a nonnegative initial data for the equation (1.1) which satisfies . Then, for any smooth convex function and for any , the solution satisfies

where .

Lemma 4.3.

Let us suppose that is a smooth convex function such that

Then, for any smooth function , we have:

for some .

We can now state the main theorem ([6, Theorem 1]) of this section, its proof is a direct consequence of the two previous lemmas and the Gronwall lemma.

Theorem 4.4.

Consider a nonnegative initial data such that . We then have:

In what follows, we denote . We now give a corollary of this theorem which gives the decay property in the space i.e .

Corollary 4.5.

Consider a nonnegative initial data such that is finite. Then, there exist and such that:

Proof.

Theorem 4.4 applied with gives the result in and the conclusion of the Corollary is a direct consequence of it because of Remark 4.1. ∎


5. Semigroup decay in

5.1. Splitting of the operator

We would like to get a splitting of our operator into two operators which satisfies hypothesis of Theorem 3.1 with and with . In what follows, we denote , .

Lemma 5.1.

Consider where is defined in Corollary 4.5. There exist two operators and which satisfy the following conditions:

  • ,

  • and ,

  • is dissipative on and .

Proof.

We are going to estimate the integral with a Schwartz function. The inequality obtained will also hold for any because of the density of in . We split the integral into two parts:

As far as is concerned, we introduce the function on which is convex and its derivative is . We also introduce the notation . Let us do the following computation:

where the last inequality comes from the convexity of . We hence deduce that

because of Lemma 2.2.

Let us now deal with . Performing integrations by parts, we obtain:

We now introduce . Let us study the behavior of at infinity. First, tends to as tends to infinity. Then, we prove that tends to as tends to infinity. We use both representations (2.1) and (2.2) to split into two parts:

Concerning , using a Taylor expansion, we obtain:

from which we deduce that

(5.1)

Concerning , let us introduce the function on . Using the fact that is -Hölder continuous on because , we obtain for any , :

for some . We deduce the following inequalities:

Finally, we obtain the following estimate on :

(5.2)

where we notice that the integrals are convergent because .

Gathering (5.1) and (5.2), we deduce that tends to at infinity. Finally, we obtain:

We introduce the smooth function () which is nonnegative, supported on and such that for . For any , we may find and large enough so that

(5.3)

Indeed, if we choose large enough such that for any , and , we have (5.3).

We then introduce and . We finally obtain:

which implies that is dissipative on .

Let us now check that is dissipative on :

where the last inequality comes from Corollary 4.5. We thus deduce that is dissipative on using that . To conclude that we also have the dissipativity of on , we use the fact that there exists such that on .

We can now conclude. This splitting fulfills conditions (i), (ii) and (iii) of Lemma 5.1. Indeed, it is immediate to check assumption (ii) because is a truncation operator. ∎

5.2. Regularization properties of

We are now going to show that there exists such that has a regularizing effect. In order to get such a result, we are going to use the negative term in the computations done to get the dissipativity of . Let us state a result which is going to be useful to get an estimate on this negative term.

Lemma 5.2 (Fractional Nash inequality).

Consider . There exists a constant such that for any , we have: