Uniqueness of degenerate Fokker – Planck equations with weakly differentiable drift whose gradient is given by a singular integral ∗

In this paper we prove the uniqueness of solutions to degenerate Fokker–Planck equations with bounded coefficients, under the additional assumptions that the diffusion coefficient has W 1,2 loc regularity, while the gradient of the drift coefficient is merely given by a singular integral.


Introduction
This short note is motivated by the work of Röckner and Zhang [21], where they proved the uniqueness of solutions to degenerate Fokker-Planck equations with bounded coefficients, satisfying a pointwise inequality.Before going to the details, we first introduce some notations.Let σ : [0, T ] × R d → R d ⊗ R m and b : [0, T ] × R d → R d be measurable functions.Define the second order differential operator where ∂ i ϕ(x) = ∂ϕ ∂xi (x) and ∂ ij ϕ(x) = ∂ 2 ϕ ∂xi∂xj (x), 1 ≤ i, j ≤ d.We consider the Fokker-Planck equation where L * t is the adjoint operator of L t .Here is the rigorous meaning of this equation: where the initial condition means that µ t weakly * converges to µ 0 as t tends to 0. If µ t is absolutely continuous with respect to the Lebesgue measure with the density function u t for all t ∈ [0, T ], then the density function u t solves the PDE below in the weak sense: ∂ t u t = L * t u t , u| t=0 = u 0 .
We can now recall the main result of Röckner and Zhang [21].They assume that the coefficients σ and b are bounded and for any R > 0 and a.e.x, y ∈ B(R) := {z ∈ R d : |z| ≤ R}, where f R ∈ L q ([0, T ] × B(R)) for some q ≥ 1.Under these conditions, they proved the uniqueness of solutions to (1.3), in an integrability class depending on q, with probability density ρ as the initial value u 0 .Their method is based on the natural connection between Fokker-Planck equations and stochastic differential equations (SDE), see Subsection 2.1 for more details.We mention that (1.4) is satisfied when b t ∈ W 1,q loc and σ t ∈ W 1,2∨q loc with q > 1 for a.e.t ∈ [0, T ], but is in general not so when q = 1.Our purpose in this work is to generalize Röckner and Zhang's result to cover the case that b t ∈ W 1,1 loc .Indeed, by employing Bouchut and Crippa's estimate (see Theorem 2.15 of the current paper), we can treat more general situation where the drift coefficient b has a gradient given by a singular integral.
Here are our assumptions on the coefficients σ and b.
Assumption 1.1.Assume that (H1) the functions σ and b are essentially bounded; H3) for a.e.t ∈ (0, T ) and for every i, j = 1, . . ., d, we have where S i jk are singular integral operators of fundamental type in R d (see Definition 2.13 for the precise meaning) and the functions g i jk ∈ L 1 ((0, T ) × R d ) for all i, j = 1, . . ., d and k = 1, . . ., m 0 .In vectorial form, the above identity can be written as S jk (g jk (t)) holds in D (R d ), for a.e.t ∈ (0, T ), (1.6) in which S jk is a vector consisting of d singular integral operators and for each j = 1, . . ., d and k = 1, . . ., m 0 , we have Some comments on the assumptions are in order.We assume σ and b are bounded because we shall make use of a representation formula by Figalli (see [16,Theorem 2.6] or Theorem 2.5 below), where such boundedness condition are imposed on the coefficients.The assumption (H2) on σ is natural, and it has already been used in [18,21,20].
The motivation for considering the condition (H3) on the drift b comes from the recent developments in the DiPerna-Lions theory, especially the papers [9,10] by Bouchut and Crippa, where the authors established the existence and uniqueness of flows associated to such vector field b.This theory has its origin in the celebrated work of DiPerna and Lions [13], who proved that if b is a W 1,1 loc vector field with bounded divergence, then there exists a unique flow of measurable maps generated by b which leaves the Lebesgue measure quasi-invariant.Ambrosio [1] extended the main result in [13] to the case where the vector field has only BV spatial regularity, see [2,3] for more details.In the recent preprint [5], Ambrosio and Trevisan developed the DiPerna-Lions theory in a rather general setting, that is, on metric measure spaces.This theory is indirect in the sense that the authors first established the well-posedness of the corresponding first order linear PDE (transport equation or continuity equation), from which they deduced the results on ODE.See [4,14] for the developments in the infinite dimensional Wiener space.Crippa and De Lellis [12] obtained some a-priori estimates on the flow in the Lagrangian formulation, which enables them to give a direct construction of the flow (see [23,15] for the extension to the stochastic setting).While this approach works very well when the vector field b has W 1,p loc regularity with p > 1, it is not so for the case p = 1.This motivates Bouchut and Crippa to further develop the direct method to cover the case b ∈ W 1,1 loc .Indeed, they are able to deal with more general vector fields b whose gradient is given by a singular integral, cf.[10].Remark that this family of functions include the Sobolev space W 1,1 , but does not contain the BV class, nor is contained in it.
We can now state the main result of this paper.
Theorem 1.2 (Uniqueness of Fokker-Planck equations).Under the assumptions (H1)-(H3), for any given probability density function ρ on R d , there is at most one weak solution u t to the Fokker-Planck equation We recall some known results concerning the uniqueness of Fokker-Planck equations.Let P(R d ) be the set of probability measures on R d .In the non-degenerate case, it was shown in [6] that if in addition the diffusion coefficient σ is Lipschitz continuous and the drift vector field b is locally integrable and coercive, then the uniqueness holds for (1.2) in P(R d ) when the initial measure has finite entropy.On the other hand, Le Bris and Lions [18] established the well-posedness of degenerate Fokker-Planck type equations with coefficients fulfilling quite general Sobolev regularity, by extending the DiPerna-Lions theory to this setting.In [20], we slightly generalize the main result in [18] to the case where the drift b has only BV spatial regularity, in the spirit of [1].
The study of Fokker-Planck equations in the infinite dimensional setting can be found in [7,19].Bogachev Under such conditions, one can prove some a-priori estimates on the solution u to (1.3), see e.This paper is organized as follows.In Section 2, we first recall some well known results on the connection between Fokker-Planck equations and SDEs, then we introduce the pointwise estimate of weakly differentiable functions with gradient given by a singular integral.Finally we prove in Section 3 our main result by following the arguments in [21,10].

Preliminary results
In this section we recall some known results which are necessary for proving our main result.

Connection between Fokker-Planck equations and SDEs
This subsection mainly follows the beginning parts of [21, Sections 1 and 2].We first introduce some notations.Denote by W m T = C([0, T ]; R m ) the space of continuous functions from [0, T ] to R m .Let F m t be the canonical filtration generated by coordinate process W t (w) = w t , w ∈ W m T .We write ν for the standard Wiener measure on Let µ t be the distribution of X t .Then it is well known that, by Itô's formula, µ t is a distributional solution to the Fokker-Planck equation (1.2).
Recall that P(R d ) is the set of probability measures on (R d , B(R d )).Here are two well known notions of solutions to (2.1) in the theory of SDEs, which are stated in detail to fix notations.
Definition 2.2 (Weak solution).Let µ 0 ∈ P(R d ).The SDE (2.1) is said to have a weak solution with initial law µ 0 if there exist a filtered probability space (Ω, G, (G t ) 0≤t≤T , P ), on which are defined a (G t )-adapted continuous process X t taking values in R d and an m-dimensional standard (G t )-Brownian motion W t , such that X 0 is distributed as µ 0 and a.s., We denote this solution by Ω, G, (G t ) 0≤t≤T , P ; X, W .
The next result can be found in the proof of [17, Chap.IV, Theorem 1.1].

Proposition 2.3. Given two weak solutions
The assertion below is a special case of [17, Chap.IV, Proposition 2.1].Proposition 2.4 (Existence of martingale solution implies that of weak solution).Let µ 0 ∈ P(R d ) and P µ0 be a martingale solution of SDE (2.1).Then there exists a weak solution (Ω, G, (G t ) 0≤t≤T , P ; X, W ) to SDE (2.1) such that P • X −1 = P µ0 .
Finally we remind the following result which is an easy consequence of Figalli's representation theorem (see [16,Theorem 2.6]) for solutions to the Fokker-Planck equation (1.2).Theorem 2.5.Assume that σ and b are two bounded measurable functions.Given µ 0 ∈ P(R d ), let µ t ∈ P(R d ) be a measure-valued solution to equation (1.2) with initial value µ 0 .Then there exists a martingale solution P µ0 to SDE (2.1) with initial law µ 0 such that for all ϕ ∈ C ∞ c (R d ), one has

Elements from harmonic analysis and Bouchut and Crippa's estimate
In this subsection we first recall some basic facts in harmonic analysis, and then we introduce the pointwise estimate of Bouchut and Crippa on weakly differentiable functions whose gradients are given by singular integrals.The main reference is [10, Sections 2-4].

Weak Lebesgue spaces
Denote by L d the Lebesgue measure on R d , and B(R) the ball in R d centered at the origin with radius R.
and denote by M p (O) the totality of measurable functions u defined on O such that It is worth mentioning that M p (O) is not a Banach space, since ||| • ||| M p (O) is not subadditive and hence not a norm.From the simple inequality below The following result (see [ and for p = ∞, (2.4)

Maximal functions
We first introduce the notion of local maximal functions.Let R > 0 and u : where B(x, r) is the ball centered at x of radius r > 0. ) (2.7) Moreover, if u belongs to the Sobolev space W 1,1 loc (R d ), then there exist a constant C d > 0 and a negligible set N ⊂ R d such that for all x, y ∈ N c with |x − y| ≤ R, one has We shall also need the so-called grand maximal function which is an important tool in the theory of Hardy spaces.Denote by L ∞ c (R d ) the space of bounded functions with compact support.

Definition 2.9 (Grand maximal function). Given a family of functions {ρ
we define the grand maximal function of u relative to {ρ α } α as ) (i) Compared to the definition (2.5) of the local maximal function, we move the absolute value outside the integral sign.This allows some kind of cancellation effect when the grand maximal function is composed with the singular integral operator, see [10, Section 3] for more details.

Singular integral operators
We now recall some facts on singular kernels and singular integral operators, see [22, Chap.II] for details.Let S(R d ) be the Schwartz space and S (R d ) the space of tempered distributions.
Definition 2.11 (Singular kernel).We call K a singular kernel on R d if Theorem 2.12 (Calderón-Zygmund).Let K be a singular kernel.For u ∈ L 2 (R d ), define Su = K * u in the sense of multiplication in the Fourier variable.Then for every p ∈ (1, ∞), the following strong estimate holds: (2.10) when p = 1, the weak estimate below holds: As a direct consequence of the above theorem, for any 1 < p < ∞, we can extend the domain of S to the whole L p (R d ) with values in L p (R d ), and the inequality (2.10) holds for all u ∈ L p (R d ); furthermore, S can be extended to the whole of L 1 (R d ) with values in M 1 (R d ), and the estimate (2.11) holds for all u ∈ L 1 (R d ).The operator S constructed in this way is called the singular integral operator associated to the singular kernel K.
Following the terminology of [10], we introduce a special class of singular kernels.Definition 2.13 (Singular kernel of fundamental type).We say that K is a singular kernel of fundamental type if it possesses the following properties:

Bouchut and Crippa's estimate
Now we are ready to introduce the important pointwise estimate of Bouchut and Crippa on weakly differentiable functions whose gradient is given by a singular integral.First of all, we present the following result (cf.[10,Theorem 3.3]) on the cancellation effect between the singular integral and the maximal function introduced in Definition 2.9.
Theorem 2.14.Let K be a singular kernel of fundamental type as in Definition 2.13 and set Su = K * u for u ∈ L 2 (R N ).Let {ρ α } α be a family of kernels satisfying supp(ρ α ) ⊂ B(1) and ρ α L 1 (R d ) ≤ Q 1 for every α. (2.12) Assume that for every ε > 0 and every α, it holds every ε > 0 and every α. (2.13) Then we have where C 0 and C 1 are constants in Definition 2.13; ( where S jk are singular integral operators of fundamental type as in Definition 2.13 and g jk ∈ L 1 (R d ) for all j = 1, . . ., d and k = 1, . . ., m 0 .Then there exists a nonnegative function (2.17) Moreover, the function U is explicitly given by where the maximal function relative to a family of kernels is defined in Definition 2.9, and the kernel h satisfies h(y) dy = 1 and supp(h) ⊂ B(1/2). (2.20) At the beginning of the proof of [10, Proposition 4.2], it has been checked that Theorem 2.14 now applies to the singular kernels S jk and the family of mollifiers Λ ξ,j , since they verify the conditions (2.12) and (2.13).We would like to mention that, in Section 3, we actually use the smooth version of the above theorem, that is, In this case, (2.17) holds for all x, y ∈ R d (cf. Step 1 of the proof of [10, Proposition 4.2]).

Proof of the main result
This section is devoted to the proof of Theorem 1.2, which is quite long and will be divided into several steps.
Proof of Theorem 1.2.We follow the idea of the proof of [21 Then by Theorem 2.5, there exist two martingale solutions P (i) µ0 , i = 1, 2 to the SDE (2.1) with the same initial probability distribution µ 0 , such that for all ϕ ∈ C ∞ c (R d ), Applying Proposition 2.4, we obtain two weak solutions Ω (i) , G (i) , (G Finally by Proposition 2.3, we can find a common filtered probability space (Ω, G, (G t ) 0≤t≤T , P ), on which are defined a standard m-dimensional (G t )-Brownian motion W and two continuous (G t )adapted processes Y (i) (i = 1, 2), such that P Y s dW s for all t ≤ T.
Taking expectation on both sides with respect to P yields (3.3) In the sequel we shall estimate the two terms separately.
Step 1.We first deal with the simpler term Then by (H1), for a.e.s ∈ [0, T ], σ ε s ∈ C ∞ b (R d ) for every ε ∈ (0, 1).By the triangular inequality, we have To estimate the first term, we shall use (2.8).Note that σ ε s is now smooth, hence the inequality (2.8) holds without the exceptional set N .Thus Recall that Y (i) s has the same law with X (i) s , which is distributed as u Consequently, where in the third inequality we have used (2.7).Note that the bound is independent of ε ∈ (0, 1).In the same way,  (3.6) The estimate of the term I 1,2 is analogous to that of I 2,2 : Finally we deal with the term I 1,1 .Fix any η > 0. From (H3), we have S jk g jk (s) * χ ε .
Moreover, for the finite family {g jk ; 1 we can find C η > 0 and a set A η ⊂ (0, T )×R d with finite measure such that for every j = 1, . . ., d and k = 1, . . ., m 0 , we have the decomposition below: jk (s, x) + g jk (s, x) jk ⊂ A η and g (2) Now by Theorem 2.15 (see in particular the remark after it), where jk  It remains to estimate the quantity I 1,1,1 defined in (3.10).We have (3.12) For simplicity of notations, we denote by ψ s (x) the integrand on the right hand side.
Using the simple inequality  Combining this estimate with (3.13) and applying (2.4), we get in which we have used the fact that the function s → s 1 + log + (C/s) is nondecreasing on [0, ∞).Substituting this inequality into (3.12)finally leads to Combining the above estimate with (3.10) and (3.11), we obtain g. [18, Section 5.2, p.1289] for more details.(ii) The proof of Theorem 1.2 follows the line of arguments in [21, Theorem 1.1].A close look at the proof reveals that this method allows us to prove the pathwise uniqueness of solutions to SDE (2.1), once we have some a-priori estimates on the distributions of solutions, cf.[11, Theorem 1.1].

Definition 2 . 6 .
Let O ⊂ R d be an open set and u a measurable function (possibly vector valued) defined on O.For any

Remark 2 . 10 .
the space of smooth functions with compact support, the same definition applies for distributions u ∈ D (R d ), more precisely, M {ρ α } u(x) Here are two comments on the above definition.

≥
R with the convention that inf ∅ = T .Since the coefficients σ and b are bounded, it is clear that lim R→∞ τ R (ω) = T almost surely.(3.2)

condition nor Lipschitz continuity on σ, and the drift b has only very weak differentiability which is not included in the BV class. Remark 1.3. Before finishing
this section, we give the following two remarks:(i) This paper is only concerned with the uniqueness of solutions to the Fokker-Planck equation (1.3).To show the existence of solutions to (1.3), one usually needs some assumptions on the divergence of the coefficients, for instance The following properties of the local maximal function are well known, see for instance [12, Lemmas A.2 and A.3].In the sequel, C with subscripts d, p and so on means it is a positive constant depending on these parameters.Fix any R, ρ > 0. If u ∈ L 1 loc (R d ), then it holds .15) Finally we can introduce Bouchut and Crippa's pointwise estimate (see [10, Proposition 4.2]).Let u ∈ L 1 loc (R d ) and assume that for every j = 1, . . ., d, it holds