Nonlinear Fokker–Planck equations with fractional Laplacian and McKean–Vlasov SDEs with L´evy–Noise

This work is concerned with the existence of mild solutions to nonlinear Fokker–Planck equations with fractional Laplace operator ( − ∆) s for s ∈ (cid:0) 12 , 1 (cid:1) . The uniqueness of Schwartz distributional solutions is also proved under suitable assumptions on diﬀusion and drift terms. As applications, weak existence and uniqueness of solutions to McKean– Vlasov equations with L´evy–Noise, as well as the Markov property for their laws are proved. MSC: 60H15


Introduction
We consider here the nonlinear Fokker-Planck equation (NFPE) where β : R → R, D : R d → R d , d ≥ 2, and b : R → R are given functions to be made precise later on, while (−∆) s , 0 < s < 1, is the fractional Laplace operator defined as follows.Let S ′ := S ′ (R d ) be the dual of the Schwartz test function space S := S(R d ).Define where F stands for the Fourier transform in R d , that is, (F extends from S ′ to itself.)NFPE (1.1) is used for modelling the dynamics of anomalous diffusion of particles in disordered media.The solution u may be viewed as the transition density corresponding to a distribution dependent stochastic differential equation with Lévy forcing term.
Here, we shall study the existence of a mild solution to equation (1.1) (see Definition 1.1 below) and also the uniqueness of distributional solutions.As regards the existence, we shall follow the semigroup methods used in [6]- [9] in the special case s = 1.Namely, we shall represent (1.1) as an abstract differential equation in L 1 (R d ) of the form where A is a suitable realization in L 1 (R d ) of the operator A 0 (u) = (−∆) s β(u) + div(Db(u)u), u ∈ D(A 0 ), where div is taken in the sense of Schwartz distributions on R d .
Like in the present work, the results obtained in [16] are based on the Crandall & Liggett generation theorem of nonlinear contraction semigroups in L 1 (R d ).However, the approach used in [16] cannot be adapted to cope with equation (1.1).In fact, the existence and uniqueness of a mild solution to (1.1) reduces to prove the m-accretivity in L 1 (R d ) of the operator A 0 , that is, (I + λA 0 ) −1 must be nonexpansive in L 1 (R d ) for all λ > 0. If D ≡ 0 and β(u) = |u| m−1 u, m > (d − 2s) + /d, this follows as shown in [16] (see, e.g., Theorem 7.1) by regularity However, such a property might not be true in our case.For instance, if , and D sufficiently regular ([9, Theorem 2.2]).To circumvent this situation, following [6] (see Section 2) we have constructed here an maccretive restriction A of A 0 and derive so via the Crandall & Liggett theorem a semigroup of contractions S(t) such that u(t) = S(t)u 0 is a mild solution to (1.1).In general, that is if A = A 0 , this is not the unique mild solution to (1.1).However, as shown in Theorem 3.1 below, under Hypotheses (j) (resp.(j) ′ ), (jj), (jjj) (see Section 3), for initial conditions in L 1 ∩ L ∞ it is the unique bounded, distributional solution to (1.1).For initial conditions in L 1 , the uniqueness of mild solutions to (1.1) as happens for s = 1 ( [9]) or for D ≡ 0, s ∈ (0, 1), as shown in [16], in the case of the present paper remains open.One may suspect, however, that one has in this case as for s = 1 (see [12], [14]) the existence of an entropy, resp.kinetic, solution to (1.1) for u 0 ∈ L 1 ∩ L ∞ .But this remains to be done.Let us mention that there is a huge literature on the well-posedness of equation (1.1) for the case s = 1, in particular when D ≡ 0. We refer the reader e.g. to [3]- [9], [11], [12], [14], [15], [23] and the references therein.
In Section 4, we apply our results to the following where L is a d-dimensional isotropic 2s-stable process with Lévy measure dz/|z| d+2s (see (4.7) below).We prove that provided u(0, •) is a probability density in L ∞ , by our Theorem 2.4 and the superposition principle for nonlocal Kolmogorov operators (see [25,Theorem 1.5], which is an extension of the local case in [18] and [29]) it follows that (1.11) has a weak solution (see Theorem 4.1 below).Furthermore, we prove that our Theorem 3.1 implies that we have weak uniqueness for (1.11) among all solutions satisfying (see Theorem 4.2).As a consequence, their laws form a nonlinear Markov process in the sense of McKean [22], thus realizing his vision formulated in that paper (see Remark 4.3).We stress that for the latter two results β is allowed to be degenerate, if D ≡ 0. We refer to Section 4 for details.
McKean-Vlasov SDEs for which (L t ) in (1.11) is replaced by a Wiener process (W t ) have been studied very intensively following the two fundamental papers [22], [31].We refer to [19], [28] and the monograph [13] as well as the references therein.We stress that (1.11) is of Nemytskii type, i.e. distribution density dependent, also called singular McKean-Vlasov SDEs, so there is no weak continuity in the measure dependence of the coefficients, as usually assumed in the literature.This (also in case of Wiener noise) is a technically more difficult situation.Therefore, the literature on weak existence and uniqueness for (1.11) with Lévy noise is much smaller.In fact, since the diffusion coefficient is allowed to depend (nonlinearly) on the distribution density, except for [25], where weak existence (but not uniqueness) is proved for (1.11), if D ≡ 0 and β(r) := |r| m−1 r, m > (d − 2σ) + /d, we are not aware of any other paper adressing weak well-posedness in our case.If in (1.11) the Lévy process (L t ) is replaced by a Wiener process (W t ), we refer to [3], [4], [5], [6] for weak existence and to [7]- [9] for weak uniqueness, as well as the references therein.Denote by H σ (R d ) = H σ , 0 < σ < ∞, the standard Sobolev spaces on R d in L 2 and by H −σ its dual space.By C b (R) denote the space of continuous and bounded functions on R and by C 1 (R) the space of differentiable functions on R. For any T > 0 and a Banach space X , C([0, T ]; X ) is the space of X -valued continuous functions on [0, T ] and by L p (0, T ; X ) the space of Xvalued L p -Bochner integrable functions on (0, T ).We denote also by C ∞ 0 (O), O ⊂ R d , the space of infinitely differentiable functions with compact support in O and by D ′ (O) its dual, that is, the space of Schwartz distributions on O.

Existence of a mild solution
To begin with, let us construct the operator A : D(A) ⊂ L 1 → L 1 mentioned in (1.4).To this purpose, we shall first prove the following lemmas.
Proof of Lemma 2.1.We shall first prove the existence of a solution y = y λ ∈ D(A 0 ) to the equation for f ∈ L 1 .To this end, for ε ∈ (0, 1] we consider the approximating equation where, for r ∈ R, β ε (r) := β(r) + εr and Clearly, we have Now, let us assume that f ∈ L 2 and consider the approximating equation where F ε,λ : L 2 → S ′ is defined by where (εI − ∆) s : S → S is defined as usual by Fourier transform and then it extends by duality to an operator (εI − ∆) s : S ′ → S ′ (which is consistent with (1.2)).
In particular, it follows that where Then, since β ε (y ε ) ∈ H 1 , by (2.9) we get (2.36) Setting by Hypotheses (iii), (iv) we have and hence, since y ε ∈ H 1 , the left hand side of (2.36) is equal to which, because (y ε , β ε (y ε )) 2 ≥ 0 and H 1 ⊂ H s , by (2.10) and Hypothesis (iv) implies that λ|(εI−∆) Since where Since obviously for all u ∈ H s (⊂ Ḣs ), ε ∈ (0, 1], )|r|, we conclude from (2.35) and (2.38) that (along a subsequence) as ε → 0 where the second statement follows, because By [1, Theorem 1.69], it follows that as ε → 0 so (selecting another subsequence, if necesary) Since β −1 (the inverse function of β) is continuous, it follows that as ε → 0 Recalling that y ε solves (2.9), we can let ε → 0 in (2.9) to find that (2.41) Since β ∈ Lip(R), the operator (A 0 , D(A 0 )) defined in (1.5) is obviously closed as an operator on L 1 .Again defining for y as in (2.41) it follows by (2.22) and Fatou's lemma that for (2.42) Hence J λ extends continuously to all of L 1 , still satisfying (2.42) for all f 1 , f 2 ∈ L 1 .Then it follows by the closedness of (A 0 , D(A 0 )) on L 1 that J λ (f ) ∈ D(A 0 ) and that it solves (2.41) for all f ∈ L 1 .Hence, Lemma 2.1 is proved except for (2.3) and (2.4).However, (2.3) is obvious, since by (2.1) it is equivalent to which in turn is equivalent to Now let us prove (2.4).We may assume that f ∈ L 1 ∩L ∞ and set y : where g s 1 is as in the Appendix.Then, clearly, where the last statement follows from (2.39).Furthermore, and, as n → ∞, (2.45) Again it is easy to see that J λ (L 1 ) is independent of λ ∈ (0, λ 0 ) and that Therefore, we have Lemma 2.3.Under Hypotheses (i)-(iv), the operator A defined by (2.45) is m-accretive in L 1 and (I +λA Here, D(A) is the closure of D(A) in L 1 .
Then, by the Crandall & Liggett theorem (see, e.g., [2], p. 131), we have that the Cauchy problem (1.4), that is, Moreover, by (2.5) and the exponential formula Let us show now that u = S(t)u 0 is a Schwartz distributional solution, that is, (1.10) holds.
By (1.6)-(1.9),we have Taking into account that, by (1.6) and (i) ) as h → 0 for each t > 0, we get that (1.10) holds.This together with Remark 2.2 implies the following existence result for equation (1.1).
Theorem 2.4.Assume that Hypotheses (i)-(iv) hold.Then, there is a C 0 -semigroup of contractions S(t) : and then all above assertions remain true, if we drop the assumption β ∈ Lip(R) from Hypothesis (i), and additionally we have that Remark 2.5.It should be emphasized that, in general, the mild solution u given by Theorem 2.4 is not unique because the operator A constructed in Lemma 2.3 is dependent of the approximating operator A ε y ≡ (εI + (−∆) s )β ε (y) + div(D ε b ε (y)y) and so u = S(t)u 0 may be viewed as a viscosity-mild solution to (1.1).However, as seen in the next section, this mild solution -which is also a distributional solution to (1.1) -is, under appropriate assumptions on β, D and b, unique in the class of solutions u ∈ L ∞ ((0, T )×R d ), T > 0.
We note that by (3.16) and Parseval's formula we have and This yields because 2s ≥ 1.
Next, by (3.13), (3.15) and (3.18), we have where This yields Taking into account that, by (3.4), we have where C > 0 is independent of ε and lim In particular, by (3.17), it follows that ) be two distributional solutions to the equation where u 0 is a measure of finite variation on R d and β(0) 0 := 0. If (3.1) holds, then y 1 ≡ y 2 .
Proof.We note first that Then, we have z t + (−∆) s w + div(Db(u)z) = 0, and so, arguing as in the proof of Theorem 3.1, we get that y 1 ≡ y 2 .The details are omitted.As is easily checked, Hypotheses (i)-(iv) imply that condition (1.18) in [25] holds.Furthermore, it follows by Theorem 2.4 that µ(dx) := u(t, x)(dx), t ≥ 0, solves the Fokker-Planck equation (1.10) with u 0 := u 0 (x)dx.Hence, by [ is the standard space of Lebesgue pintegrable functions on R d .We denote by L p loc the corresponding local space and by | • | p the norm of L p .The inner product in L 2 is denoted by (•, •) 2 .