Riccardo Bonalli
email: riccardo.bonalli@cnrs.fr.

Alessandro Rudi
email: alessandro.rudi@inria.fr.

NON-PARAMETRIC LEARNING OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH FAST RATES OF CONVERGENCE *

Keywords: Non-parametric system identification with guarantees, non-linear stochastic differential equations, discrete-time observation, Reproducing Kernel Hilbert Space

We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of non-linear stochastic differential equations, which relies upon discretetime observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker-Planck equation to such observations, yielding theoretical estimates of learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may in principle be profitably leveraged to enable efficient numerical implementation.

Here, X(t) ∈ R n denotes the (stochastic) state of dimension n ∈ N of the system at time t ∈ [0, T ], with T > 0 a fixed time horizon, (W t ) t∈[0,T ] is a n-dimensional Wiener process, whereas b : [0, T ] × R n → R n and a : [0, T ] × R n → Sym ++ (n) are regular enough drift and diffusion coefficients, with Sym ++ (n) ⊆ R n×n the subset of symmetric positive-definite matrices. Equations such as (1.1) may be profitably leveraged to accurately model complex phenomena in a wide range of applications, such as aerospace, finance, and robotics to name a few [START_REF] Ridderhof | Nonlinear Uncertainty Control with Iterative Covariance Steering[END_REF][START_REF] Yong | Stochastic Controls: Hamiltonian Systems and HJB equations[END_REF][START_REF] Bonalli | Sequential convex programming for non-linear stochastic optimal control[END_REF].

In practice, both coefficients a and b might be completely unknown, e.g., a might model external perturbations due to many different physical phenomena which affect the motion of the system, as it occurs in aerospace and robotics. Therefore, appropriate stochastic system identification procedures for both a and b must be devised.

1.1. Related Work. The problem of stochastic system identification has been investigated for several decades. The earliest methods mainly address either discretetime models [START_REF] Lennart | System Identification -Theory For the User[END_REF][START_REF] Nelles | Nonlinear System Identification[END_REF] or continuous-time models that are however linear in the state variable [START_REF] Garnier | Identification of Continuous-Time Models from Sampled Data[END_REF], which thus do not fit the identification setting introduced by (1.1).

More recently, identification of non-linear stochastic differential equations such as (1.1) have seen an important surge of interest. More specifically, besides some likelihood-based [START_REF] Liptser | Statistics of Random Processes: General Theory[END_REF][START_REF] Dacunha-Castelle | Estimation of the Coefficients of a Diffusion from Discrete Observations[END_REF] or Kalman filtering-based estimation methods [START_REF] Surace | Online Maximum-Likelihood Estimation of the Parameters of Partially Observed Diffusion Processes[END_REF][START_REF] Bhudisaksang | Online Drift Estimation for Jump-Diffusion Processes[END_REF][START_REF] Sharrock | Joint Online Parameter Estimation and Optimal Sensor Placement for the Partially Observed Stochastic Advection-Diffusion Equation[END_REF], which assume continuous-time observation of the stochastic state, the vast majority of the existing methods leverage more realistic discrete-time observations of X. Efficient paradigms range from non-parametric estimation [START_REF] Genon-Catalot | Asymptotic Equivalence of Nonparametric Diffusion and Euler Scheme Experiments[END_REF][START_REF] Comte | Drift Estimation on Non Compact Support for Diffusion Models[END_REF][START_REF] Lavenant | Towards a Mathematical Theory of Trajectory Inference[END_REF] and non-parametric Bayesian estimation [START_REF] Van Der Meulen | Consistent Nonparametric Bayesian Inference for Discretely Observed Scalar Diffusions[END_REF][START_REF] Nickl | Nonparametric Bayesian Posterior Contraction Rates for Discretely Observed Scalar Diffusions[END_REF], to maximum likelihood and quasi-likelihood methods [START_REF] Florens-Zmirou | Approximate Discrete-Time Schemes for Statistics of Diffusion Processes[END_REF][START_REF] Yoshida | Polynomial Type Large Deviation Inequalities and Quasi-Likelihood Analysis for Stochastic Differential Equations[END_REF][START_REF] Masuda | Convergence of Gaussian Quasi-Likelihood Random Fields for Ergodic Lévy driven SDE Observed at High Frequency[END_REF][START_REF] Uchida | Quasi Likelihood Analysis of Volatility and Nondegeneracy of Statistical Random Field[END_REF], generalized methods of moments [START_REF] Gallant | Estimation of Stochastic Volatility Models with Diagnostics[END_REF][START_REF] Gallant | The Relative Efficiency of Method of Moments Estimators[END_REF][START_REF] Nielsen | Parameter Estimation in Stochastic Differential Equations: an Overview[END_REF], and online gradient descent-based methods [START_REF] Kushner | Stochastic Approximation and Recursive Algorithms and Applications[END_REF][START_REF] Nakakita | Parametric Estimation of Stochastic Differential Equations via Online Gradient Descent[END_REF], among others. On the one hand, although the first group of these works provides learning rates which become increasingly tighter when the number of observations of the state grows, estimates of the error between the unknown drift and diffusion coefficients and the learned ones are obtained by assuming diffusion coefficients are either known or have specific parametric structures. On the other hand, although the second group of the aforementioned works rely on generally milder assumptions to be implemented, they require working with given families of finite-dimensional parametric drift and diffusion coefficients, which might hinder the identification process in the case the family of parametric coefficients is not rich enough. More importantly, it is worth noting that, to the best of our knowledge, no work has so far investigated how to leverage the regularity, i.e., high order continuous differentiability of drift and diffusion coefficients to improve learning rates. Finally, from a numerical standpoint, learning-based methods, such as scalable gradient methods [START_REF] Li | Scalable Gradients for Stochastic Differential Equations[END_REF] and infinitely deep Bayesian neural networks [START_REF] Xu | Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equations[END_REF], have shown remarkable performance on complex stochastic differential equations. Unfortunately, it seems extremely challenging to endow these latter paradigms with theoretical guarantees of convergence, motivating investigation of numerically efficient identification methods for (1.1) which enjoy guarantees of accuracy under mild assumptions.

1.2. Outline and Contributions. We propose a non-parametric, Reproducing Kernel Hilbert Space (RKHS)-based learning paradigm for the identification of drift and diffusion coefficients of the non-linear stochastic differential equation (1.1), which relies upon discrete-time observation of the state. In particular, motivated by classical likelihood-based methods [START_REF] Nielsen | Parameter Estimation in Stochastic Differential Equations: an Overview[END_REF] we propose a two-step, discrete-time observation-based scheme which entails fitting the Fokker-Planck equation related to (1.1).

Under assumptions of smoothness for the unknown drift and diffusion coefficients, we provide theoretical estimates of learning rates which become increasingly tighter when the number of observations of the state grows. In particular, given the nature of our data set, which essentially depends on observations of the state process, we consider the error between solutions to (1.1) generated with the learned drift and diffusion coefficients and the unknown trajectories of (1.1) is a "good metric" to test the accuracy of our identification method. Importantly, unlike other works, under this accuracy metric we can additionally prove our learning rates become tighter when the regularity (in a Sobolev sense) of the unknown drift and diffusion coefficients is higher. Finally, from a numerical standpoint, our method being kernel-based, offline pre-processing may be successfully leveraged to enable efficient implementations.

Our method is composed of two steps which are informally summarized below.

A. Learning the laws of the stochastic differential equation through independent discrete-time observation of the state.

We assume there exist regular enough drift b : [0, T ] × R n → R n and diffusion a : [0, T ] × R n → Sym ++ (n), and a stochastic process X : [0, T ] × Ω → R n in some filtered probability space (Ω, G, F , P), such that X solves (1.1) with drift b and diffusion a. In addition, we may assume we can sample X at M ∈ N times 0 = t 1 < • • • < t M = T , and more specifically that at every time t ℓ , ℓ = 1, . . . , M , we have access to N ∈ N samples X ℓ,1 X(t ℓ , ω 1 ), . . . , X ℓ,N X(t ℓ , ω N ) of the solution X which have been independently drawn from the law P X(t ℓ ,•) .

In such setting, as first step we propose to approximate the unknown densities p : [0, T ] × R n → R of the laws of X through the (random) RKHS-based model:

p(t, x) M ℓ=1 c ℓ (t) g ℓ (x), where g ℓ (x) 1 N N j=1 ρ R (x -X ℓ,j ),
for appropriate coefficients c ℓ : [0, T ] → R and radial mappings ρ R , R > 0. Let µ X denote the probability measure which is generated by the process X. For every tuple of precision parameters 0 < ε, δ < 1, if the unknown drift b and diffusion a are regular enough, by leveraging RKHS approximation theory we can show appropriate values for M , N , and R (which depend on ε uniquely) may be selected so that the following learning rate holds with probability µ X at least 1 -δ:

(1.2)

T 0 ∂ p ∂t (t, •) - ∂p ∂t (t, •) 2 L 2 + p(t, •) -p(t, •) 2 H 2 dt = O log 1 δε ε 2 ,
where • L 2 and • H 2 denote the norms of the Hilbert spaces L 2 (R n , R) and H 2 (R n , R), respectively. Moreover, the higher the degree of smoothness of b and a is, the lower the values of M , N , and R needed to achieve this precision become.

The main benefit which comes with the model p consists of computing accurate finite-dimensional approximations of the laws of solutions to (1.1) without a priori resorting to conservative families of parametric densities. In particular, the model p being kernel-based, one may considerably reduce the computational effort by resorting to prior offline computations, which essentially boil down to simply inverting a M ×M matrix. As a byproduct, (1.2) provides a quantitative estimate of the approximation error which is key to derive theoretical guarantees for the accuracy of our identification method in the next step.

B. Learning finite-dimensional models for the drift and diffusion coefficients by fitting approximated solutions to the Fokker-Planck equation.

If the unknown drift b and diffusion a are regular enough, the unknown densities p satisfy the following Fokker-Planck equation: where (L a,b t ) * is the dual operator of the Kolmogorov generator L a,b t ϕ(y)

(1.
1 2 n i,j=1 a ij (t, y) ∂ 2 ϕ ∂y i ∂y j (y) + n i=1 b i (t, y) ∂ϕ ∂y i (y), ϕ ∈ C 2 (R n , R).
Given the results at the previous step, it is then natural to learn models of the drift b

Q : [0, T ] × R n → R n and the diffusion a Q : [0, T ] × R n → Sym ++ (n)
which "best" match the Fokker-Planck equation when evaluated at p, that is as solutions to the following finite-dimensional convex optimization problem:

(1.4) min

( aQ, bQ)∈HQ T 0 ∂ p ∂t (t, •) -(L aQ, bQ t ) * p(t, •) 2 L 2 dt + λ ( a Q , b Q ) 2 H ,
where H Q is an appropriate finite-dimensional subspace of a RKHS H with norm • H , in which the unknown drift and diffusion coefficients lie, whereas λ > 0 is a regularization weight to be appropriately selected. In particular, problem (1.4) being kernel-based, it can be efficiently solved by combining prior offline computations of numerical integrals with efficient tools from finite-dimensional convex optimization. Thanks to estimate (1.2), one shows that, for appropriate choices of the subspace H Q and the regularization weight λ, the solution ( a Q , b Q ) ∈ H Q to problem (1.4) satisfies the following, with probability µ X at least 1 -δ:

T 0 ∂ p ∂t (t, •) -(L aQ, bQ t ) * p(t, •) 2 L 2 dt + λ ( a Q , b Q ) 2 H = = O log 1 δε ε 2 . (1.5)
By combining estimate (1.5) with energy-type estimates for parabolic partial differential equations, we may infer theoretical estimates of learning rates for the identification of drift and diffusion coefficients of non-linear stochastic differential equations, which we informally summarize as follows.

As we mentioned previously, since our data consists of observation of the state process, the error between the unknown densities and the densities stemming from the learned coefficients is a "good metric" with which the convergence of an identification algorithm for stochastic differential equations which leverage observations of the state process may be tested. To better formalize this metric, let X aQ, bQ and p aQ, bQ respectively denote the solutions to (1.1) and ( 1

.3) with coefficients ( a Q , b Q ) ∈ H H .
We thus define the following metric to test the accuracy:

E( a Q , b Q ) p aQ, bQ -p 2 L 2 = T 0 p aQ, bQ (t, •) -p(t, •) 2 L 2 dt.
By adopting this metric, our main result on the accuracy of our learning method may be summarized in the following meta-theorem:

Meta-Theorem 1.1. Assume the unknown drift b and diffusion a coefficients are regular enough. The following estimate holds true with probability µ X at least 1 -δ:

E( a Q , b Q ) = O log 1 δε ε 2 .
In particular, for every regular enough function f : [0, T ] × R n → R:

(1.6) E T 0 f (t, X(t)) dt = E T 0 f (t, X aQ, bQ (t)) dt + O log 1 δε ε .
Although the estimates provided in this meta-theorem are informal and need some clarification (see Section 5 for more rigorous statements), they show the aforementioned two-step identification method enjoys practical theoretical guarantees of accuracy, i.e., estimate (1.6): an observation/regulation metric f computed at the unknown state solution to (1.1) with unknown drift b and diffusion a may be rather observed through the process solution to (1.1) with model drift b Q and diffusion a Q up to an error O log 1 δε ε . Such result has important implications in observation and regulation of stochastic differential equations [START_REF] Lavenant | Towards a Mathematical Theory of Trajectory Inference[END_REF], and it may represent a good starting result to develop paradigms for the identification of controlled stochastic differential equations, which are crucial for the control of complex autonomous systems.

Paper Organization.

The paper is organized as follows. After gathering basic notation and preliminary results in Section 2, in Section 3 we summarize both classical and less classical results about stochastic differential equations and corresponding relationships with the Fokker-Planck equation. To ease the reading of this section, we moved a more detailed description of the aforementioned results to Section 7 and their technical proofs to Appendix A. Similarly, in Section 4 we expose the results in RKHS theory which we leverage in this work. Our main contributions are contained in Section 5, which in particular details the methodologies we exposed at the previous steps 1) and 2) and corresponding learning rates. Finally, in Section 6 we provide concluding remarks and some perspectives.

Notation and Preliminary

Results. We fix the dimension n ∈ N of the state space and a time horizon T > 0 for the identification process. We denote by Sym + (n) and Sym ++ (n) the subsets of R n×n of symmetric semi-positive-definite and symmetric positive-definite matrices, respectively.

We assume stochastic differential equations are defined on a given filtered probability space (Ω, G F T , F (F t ) t∈[0,T ] , P), which is complete, and the noise is generated by a F -adapted Wiener process W : [0, T ] × Ω → R n (e.g., we may consider the canonical process in the space Ω = C([0, T ], R n ), equipped with the Wiener measure). Moreover, we introduce the complete metric space (S, d), where

S C([0, T ], R n ), d(w 1 , w 2 ) sup t∈[0,T ] w 1 (t) -w 2 (t) ,
and equip it with the Borel sigma-algebra B(S) induced by the metric d.

For any ℓ ∈ N, r, s > 0, and any A ⊆ R ℓ , we denote by H r,s ([0, T ] × R n , A) the Sobolev (Hilbert) space of functions whose image is in A, and whose weak time derivatives are defined up to order r and whose weak space derivatives are defined up to order s; in particular, we denote

H r ([0, T ] × R n , A) H r,r ([0, T ] × R n , A
). Finally, we introduce the following Hilbert space and corresponding positive convex cone, in which we will assume the unknown drift and diffusion coefficients lie (this latter requirement will be made more explicit shortly):

H m H d(m) ([0, T ] × R n , R n×n ) × H d(m) ([0, T ] × R n , R n ), H + m (a, b) : [0, T ] × R n → Sym + (n) × R n : (a, b) ∈ H m ,
for every m ∈ N, where d(m) 2(m + 1) + n 2 ∈ N is the unique integer greater than 2m + 1 such that d(m) -n 2 = 2m + 1. The choice of taking the same exponent d(m) for both the drift and the diffusion coefficients has been made without loss of generality for the sake of conciseness. The following regularity result will be crucial: Theorem 2.1. There exists a constant C > 0, which depends on m uniquely, such that every (a, b) ∈ H + m satisfies the following properties: 1. Differentiability:

a ∈ C 2m+1 ([0, T ] × R n , R n×n ), b ∈ C 2m+1 ([0, T ] × R n , R n ).

Boundness of functions and their derivatives:

2m+1 i=0 D i (t,y) a L ∞ + D i (t,y) b L ∞ ≤ C (a, b) Hm ,
where D i (t,x) denotes the differential of order i with respect to (t, y). In particular, the following refined bound holds:

2 i=0 D i (t,y) a L ∞ + D i (t,y) b L ∞ ≤ C (a, b) H d(m)-2m+1 .
The proof of this result makes use of classical embedding arguments and it is reported in Appendix A for the sake of completeness.

3. Stochastic Differential and Fokker-Planck Equations. In this section, we summarize both classical and less classical results about stochastic differential equations, and corresponding relationships with the Fokker-Planck equation. In particular, we propose a minimally detailed discussion for the sake of conciseness, reporting a more structured exposition in Section 7 for the sake of completeness.

3.1. The Fokker-Planck equation. From now on, we fix m ∈ N (to be specified later) and a constant α > 0. Since we will need to work with diffusion coefficients which are never trivial, for every (a, b) ∈ H + m we will rather replace a with the mapping

a + αI : [0, T ] × R n → Sym ++ (n) with a + αI = √ a + αI √ a + αI.
At this step, we fix a non-negative density p 0 ∈ L 2 (R n , R) which will serve as appropriate initial condition, and we denote by µ 0 ∈ P(R n ) the associate probability measure.

Motivated by [START_REF] Stroock | Multidimensional Diffusion Processes[END_REF][START_REF] Figalli | Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients[END_REF], we recall the following notions of stochastic differential equation and its solutions (though they differ from the ones in [START_REF] Stroock | Multidimensional Diffusion Processes[END_REF][START_REF] Figalli | Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients[END_REF]):

Theorem-Definition 3.1. There exists a measurable mapping X : R n × Ω → S such that each process X x (t, ω) X(x, ω)(t) is F -progressively measurable for every x ∈ R n , and such that the following Stochastic Differential Equation (SDE)

SDE x    dX x (t) = b(t, X x (t)) dt + (a + αI)(t, X x (t)) dW t , P X x (0) = x = 1,
holds in (Ω, G, F , P) for µ 0 -almost every x ∈ R n . We then say that X solves or is solution to SDE (with coefficients (a, b) ∈ H + m ). The solution X to SDE is unique in the following sense: if a measurable mapping Y : R n × Ω → S solves SDE, then it holds that X(x, •) = Y (x, •) a.s., for µ 0 -almost every x ∈ R n .

For the well-posedness of Theorem-Definition 3.1, see Section 7. Solutions to SDE share a close relationship with the solutions to the Fokker-Planck equation, which we introduce next. For every (a, b) ∈ H + m , we denote the Kolmogorov generator by L a,b t ϕ(y)

1 2 n i,j=1 (a + αI) ij (t, y) ∂ 2 ϕ ∂y i ∂y j (y) + n i=1 b i (t, y) ∂ϕ ∂y i (y), ϕ ∈ C 2 (R n , R).
Let X : R n × Ω → S be solution of SDE with coefficients (a, b) ∈ H + m . Thanks to specific regularity properties of X (we refer to Section 7 for these latter), we may define the curve µ : [0, T ] → P(R n ) of probability measures

(3.1) µ t (A) R n Ω 1 {Xx(t)∈A} dP µ 0 (dx), A ∈ B(R n ),
and note that µ is narrowly continuous, i.e., for every ϕ ∈ C b (R n , R), the mapping

t ∈ [0, T ] → R n ϕ(y)µ t (dy) = R n Ω ϕ(X x (t)) dP µ 0 (dx) ∈ R
is continuous. By combining this latter property with the results in [START_REF] Figalli | Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients[END_REF], we introduce the following notions of Fokker-Planck equation, its solutions, and additional relationship between these solutions and the solutions to SDE (see also Section 7):

Theorem-Definition 3.2. The curve µ : [0, T ] → P(R n ) defined through (3.1) is the unique narrowly continuous curve satisfying the Fokker-Planck Equation

FPE    d dt R n ϕ(y)µ t (dy) = R n L a,b t ϕ(y)µ t (dy), ϕ ∈ C ∞ c (R n , R), µ t=0 = µ 0 .
We then say that µ solves or is solution to FPE (with coefficients (a, b) ∈ H + m ). If X denotes the solution to SDE, the following representation formula holds:

(3.2) R n ϕ(y)µ t (dy) = R n Ω ϕ(X x (t)) dP µ 0 (dx), for t ∈ [0, T ], ϕ ∈ C c (R n , R).

Absolutely continuous solutions to the Fokker-Planck equation.

Thanks to the regularity of the coefficients (a, b) ∈ H + m which is offered through Theorem 2.1, the solution µ to FPE is absolutely continuous, i.e., it takes the form

µ t (A) = A p(t, y) dy, A ∈ B(R n ),
for an appropriate p : [0, T ] × R n → R. To elucidate this property, we first introduce broader definitions of FPE and corresponding solutions, which encompass Theorem-Definition 3.2 as a sub-case (see Theorem 3.4 below) and will be useful in our analysis:

Definition 3.3. Let f ∈ L 2 ([0, T ] × R n , R) and p ∈ L 2 (R n , R). A (regular enough) function p : [0, T ] × R n → R is said to solve or be solution to the non- homogeneous Fokker-Planck Equation (with coefficients (a, b) ∈ H + m ), if FPE f              d dt R n ϕ(y)p(t, y) dy = = R n L a,b t ϕ(y)p(t, y) + f (t, y)ϕ(y) dy, ϕ ∈ C ∞ c (R n , R), p(0, •) = p(•).
The next theorem gathers important properties of the solution to FPE f , additionally showing the solution to FPE is absolutely continuous. We refer the reader to Section 7 for exhaustive presentation and proof of these results.

Theorem 3.4. For every (a, b) ∈ H + m , f ∈ L 2 ([0, T ] × R n , R), and p ∈ L 2 (R n , R), there exists a unique mapping p ∈ C(0, T ; L 2 (R n , R))∩H 0,1 ([0, T ]×R n , R) which solves FPE f with coefficients (a, b) ∈ H + m .
In addition, the following estimate holds:

p(t, •) 2 L 2 + t 0 p(t, •) 2 H 1 dt ≤ ≤ C (a, b) Hm p 2 L 2 + t 0 f (s, •) 2 L 2 ds , t ∈ [0, T ], (3.3)
where C (a, b) Hm > 0 is a constant which continuously depends on (a, b) H + m . Assume f = 0 and p = p 0 . Then, it holds that

p(t, •) ≥ 0, R n p(t, y) dy = 1, t ∈ [0, T ],
and therefore, if for every t ∈ [0, T ] we define

µ t (A) A p(t, y) dy, A ∈ B(R n ),
then the curve µ : [0, T ] → P(R n ) is narrowly continuous and solves FPE. Finally, if in addition p 0 ∈ H 2m+1 (R n , R), then the function p satisfies

p ∈ H m+1,2(m+1) ([0, T ] × R n , R)
and the following Strong Fokker-Planck Equations (with coefficients (a, b) ∈ H + m ):

SFPE      ∂p ∂t (t, y) = (L a,b t ) * p(t, y), a.e. (t, y) ∈ [0, T ] × R n , p(0, •) = p 0 (•),
where (L a,b t ) * denotes the dual operator of the Kolmogorov generator L a,b t . Combining Theorem 3.4 with the representation formula (3.2) allows us to introduce criteria to establish satisfactory guarantees for our learning approach in the context of observation and regulation of stochastic differential equations (a more exhaustive presentation is provided in Section 7). More specifically, let f ∈ L 2 ([0, T ]×R n , R) be an observation/regulation integral metric and (a, b) ∈ H + m . By denoting X a,b and p a,b respectively the solution to SDE and to SFPE with coefficients (a, b) ∈ H + m , we are interested in studying the accuracy with which the following observation/regulation metric is approximated by our learning approach:

(3.4) E µ0×P T 0 f (t, X a,b x (t)) dt = T 0 R n f (t, y)p a,b (t, y) dydt.
To give precise estimates of the approximation error for (3.4), the following corollary of Theorem 3.4 will be crucial (see Section 7 for a proof):

Corollary 3.5. For every (a 1 , b 1 ), (a 2 , b 2 ) ∈ H + m , it holds that

E µ0×P T 0 f (t, X a1,b1 x (t)) dt -E µ0×P T 0 f (t,X a2,b2 x (t)) dt ≤ f L 2 p a1,b1 -p a2,b2 L 2 . (3.5)
4. Useful Results from RKHS Theory. In this section, we list classical results about Sobolev Spaces of functions with scattered zeros and Reproducing Kernel Hilbert Spaces (RKHS), which will be extensively leveraged in the following sections and for which we mainly refer to [START_REF] Aronszajn | Theory of reproducing kernels[END_REF][START_REF] Wendland | Scattered Data Approximation[END_REF][START_REF] Steinwart | Support vector machines[END_REF]. Let r, s ∈ N such that r > s/2.

Every function u ∈ H r (R s , R) satisfies (4.1) C u L ∞ ≤ u H r = J r (•)(F u)(•) L 2 ,
where J r (z) (1 + z 2 ) r/2 /(2π) s/2 , z ∈ R s denotes the Bessel potential and F u the Fourier transform of u, whereas the constant C > 0 depends on s uniquely. Let ℓ ∈ N, D ⊆ R s be an open and bounded domain, and u ∈ H r (D, R) be a function such that u| X = 0, where X ℓ {x 1 , . . . , x ℓ } ⊆ R s is a finite set of ℓ given points. For the fill distance, which is defined by

h X ℓ ,D sup x∈D min i=1,...,ℓ
x -x i , the following inequality holds true, e.g., [START_REF] Wendland | Scattered Data Approximation[END_REF]Theorem 11.32

], (4.2) u H ν ≤ Ch r-ν X ℓ ,D u H r , 0 ≤ ν ≤ r,
where the constant C > 0 depends on ν, r, and D uniquely. Inequality 4.2 may be interpreted as follows: if u is zero on a well distributed set of points over D, i.e., h X ℓ ,D is small, and is very regular, i.e., r is large, then any norm • H ν , 0 ≤ ν ≤ r, e.g., the L 2 norm for q = 0, is small over the whole domain D. Given a set D, a RKHS H D is a separable Hilbert space of functions u : D → R such that the following reproducing property holds true: Definition 4.1 (Reproducing property [START_REF] Aronszajn | Theory of reproducing kernels[END_REF]). For every point x ∈ D there exists a reproducing function

k x ∈ H D such that f (x) = f, k x HD , f ∈ H D .
We denote K D : D × D → R, the reproducing kernel associated to H D , i.e.,

K D (x 1 , x 2 ) k x1 , k x2 HD , x 1 , x 2 ∈ D.
The following crucial result holds true, see, e.g., [START_REF] Steinwart | Support vector machines[END_REF]:

Theorem 4.2. Given a reproducing kernel for a RKHS H D , by the reproducing property, the reproducing function k x ∈ H D for any x ∈ D corresponds to

k x = K D (x, •) ∈ H D .
We recall that the Sobolev space H r (D, R) is a RKHS for r > s/2, and D ⊆ R s be either R s or an open domain with Lipschitz boundary and s ∈ N. In this case, the associated reproducing kernel has a known closed form, see, e.g., [START_REF] Wendland | Scattered Data Approximation[END_REF].

Learning Stochastic Differential Equations.

In this section, we finally introduce and study the problem of learning drift and diffusion coefficients of a stochastic differential equation. For this, from now on we assume the following hypothesis, which naturally stems from our setting, to hold true (see also Definition 3.1):

(A) Let m ∈ N with m ≥ 1, α > 0, and p 0 ∈ H 2m+1 (R n , R). There exist (a * , b * ) ∈ H + m and a (unique) solution X to SDE with coefficients (a * , b * ) ∈ H + m .
Thanks to the results we gathered in Section 3, one readily checks that Assumption (A) yields the following characterization of the mapping X:

Corollary 5.1. With the notation L t L a * ,b * t , there exists a unique function p ∈ H m+1,2(m+1) ([0, T ] × R n , R) which is non-negative and such that A p(t, x) dx = µ t (A) R n Ω 1 {Xx(t)∈A} dP µ 0 (dx), for all A ∈ B(R n ),
and which satisfies:

∂p ∂t (t, y) = L * t p(t, y), a.e. (t, y) ∈ [0, T ] × R n .
To achieve our goal, we proceed along three successive steps:

1. First, we approximate the unknown curve of densities p from given samples {X xj (t ℓ , ω j )} ℓ=1,...,M,j=1,...,N of the unknown solution X to SDE, through a finite-dimensional RKHS-based model p. Thanks to this, all the quantities appearing in the learning problems we define in the next steps can be actually (tractably) computed. As is customary in many applications, we assume our data are collected by independently sampling X at pre-defined times. 2. Second, we approximate the unknown coefficients (a * , b * ) ∈ H + m through additional coefficients ( a, b) ∈ H + m which "best" match SFPE when evaluated at p, through an appropriately well-posed infinite-dimensional learning problem. We then combine appropriate estimates with (3.3) to show the solution to SFPE with coefficients ( a, b) well-approximates the unknown p.

3. Third, we make the learning problem at the second step "tractable" by approximating the coefficients ( a, b)

through a finite-dimensional RKHS-based model ( a Q , b Q ).
We then combine the estimates we obtained at the second step with (3.3) to show the solution to SFPE with coefficients ( a Q , b Q ) wellapproximates the unknown p. Finally, this latter property is combined with (3.5) to show theoretical error bounds in the context of observation and regulation of stochastic differential equations.

Approximating solutions to SDE via RKHS-based models.

We start by defining our data set. We assume sampling happens at M ∈ N fixed times 0 = t 1 < • • • < t M = T , which are equally spaced for simplicity, i.e., t ℓ T (ℓ -1)/(M -1). Then, we assume at each time t ℓ , ℓ = 1, . . . , M , we have access to

N ∈ N samples X ℓ,1 X x1 (t ℓ , ω 1 ), . . . , X ℓ,N X xN (t ℓ , ω N )
of the solution X to SDE, which have been independently drawn from the same probability µ t ℓ , with density p(t ℓ , •) (see Corollary 5.1). Now, for every t ∈ [0, T ] the family of probability measures given by

µ t,k µ t ⊗ • • • ⊗ µ t k-times : B(R n × • • • × R n k-times ) → [0, 1], k ∈ N
may be extended to a unique probability measure µ N t : B N → [0, 1] thanks to Kolmogorov (extension) theorem, and by definition, for every ℓ = 1, . . . , M and N ∈ N the samples {X ℓ,j } j=1,...,N may be seen as independent random variables in the probability space ((R n ) N , B N , µ N t ℓ ), with equal density p(t ℓ , •). Finally, by one further application of Kolmogorov theorem, we extend the family of probability measures

{µ N s1 ⊗ • • • ⊗ µ N s k } 0≤s1≤•••≤s k ≤T to a unique probability measure µ X : (B N ) [0,T ] → [0, 1], so that, with an abuse of notation, µ X | t1,...,tM = µ N t1 ⊗ • • • ⊗ µ N
tM for every M ∈ N, and for every M, N ∈ N the samples {X ℓ,j } ℓ=1,...,M,j=1,...,N may be seen as (not necessarily i.i.d.) random variables in the probability space

((R n ) N ) [0,T ] , (B N ) [0,T ] , µ X .
Below, the assessment "with probability at least" will be meant with respect to µ X .

At this step, fix M, N ∈ N and define the following model (random) density

p(t, x) M ℓ=1 c ℓ (t) g ℓ (x), where g ℓ (x) 1 N N j=1 ρ R (x -X ℓ,j ) with ρ R (x) R n/2 x -n/2 B n/2 (2πR x ), R > 0 and B n/2 is the Bessel function of order n/2, while c ℓ (t) e ⊤ ℓ G -1 v(t)
with {e 1 , . . . , e M } the canonical basis of R M and we define v(t) (K m+1 (t, t 1 ), . . . , K m+1 (t, t M )). The notation G is used for the Gram matrix with elements G j,k K m+1 (t j , t k ), where K m+1 denotes the Sobolev kernel of smoothness degree m + 1 (see, e.g., [START_REF] Wendland | Scattered Data Approximation[END_REF]Page 133] for an explicit formula). Finally, for every u

∈ H 1,2 ([0, T ] × R n , R) we denote L(u) T 0 ∂p ∂t (t, •) - ∂u ∂t (t, •) 2 L 2 + p(t, •) -u(t, •) 2 H 2 dt.
Our result on the approximation of p via p is as follows:

Theorem 5.2. There exists a constant C > 0, which depends on n, m, and T uniquely, such that the following learning rate for the random model p holds with probability at least 1 -δ, for every M, N ∈ N such that M ≥ 2T :

L( p) ≤ C p 2 H m+1,2 + p 2 H 1,2(m+1) × × M -2m + R -4m + R n log 4M δ 2 N -1 .
In particular, up to overloading C, for every ε > 0 small enough we have that

L( p) ≤ C p 2 H m+1,2 + p 2 H 1,2(m+1) log 1 δε ε 2 , if (as closest integers) M = ε -1/(4m) /4, N = ε -(2+n/(2m))
, and R = ε -1/(4m) .

Proof. For the sake of clarity, we divide the proof in several steps.

1)

Preliminaries. With obvious notation, fix r, s ≥ 1 and define the operator

P M : H r ([0, T ], R) → H r ([0, T ], R), via (P M u)(t) = M ℓ=1 c ℓ (t)u(t ℓ ). For u ∈ H r,s ([0, T ]× R n , R), we denote by u x the function u x (•) = u(•, x), x ∈ R n , and by ũ the function ũ(t, x) = (P M u x )(t). Now, note that the function v x := u x -P M u x satisfies v x (t ℓ ) = 0 for any ℓ = 1, . . . , M .
From the results we recalled in Section 4, we can bound the norm of functions with scattered zeros as follows: for every 0 ≤ σ ≤ r there exists a constant C r,σ > 0 such that (5.1)

u x -P M u x H σ ≤ C r,σ M σ-r u x H r ,
for almost every x ∈ R n . Below, we will often implicitly overload the constant C r,σ . Moreover, we recall that, for α, β ≥ 0,

D α t D β x ũ(t, x) = D α t D β x (P M u x )(t) = D α t P M (D β x u)(t, x),
so that by leveraging (5.1) with σ = α, for 0 ≤ α ≤ r and 0 ≤ ν ≤ s, yields

T 0 D α t u(t, •) -D α t ũ(t, •) 2 H ν dt = (5.2) = |β|≤ν T 0 R n (D α t D β x u(t, x) -D α t D β x ũ(t, x)) 2 dxdt = |β|≤ν R n T 0 (D α t D β x u(t, x) -D α t P M D β x u(t, x)) 2 dxdt ≤ |β|≤ν R n D β x u(•, x) -P M D β x u(•, x) 2 H α dx ≤ |β|≤ν R n C 2 r,α M 2(α-r) D β x u(•, x) 2 H r dx = C 2 r,α M 2(α-r) u 2 H r,ν .
At this step, by decomposing p -p = (p -p) + ( p -p) and applying the triangular inequality to L( p), we obtain that

L( p) 1/2 ≤ L( p) 1/2 + A 1/2 , A T 0 ∂ p ∂t (t, •) - ∂ p ∂t (t, •) 2 L 2 + p(t, •) -p(t, •) 2 H 2 dt. Thanks to (5.2) and p ∈ H m+1,2(m+1) ([0, T ] × R n , R), we may bound L( p) by L( p) 1/2 ≤ C m+1,1 M -m p H m+1,0 + C m+1,0 M -(m+1) p H m+1,2 .
The rest of the proof is devoted to appropriately bounding A.

2) Further decomposing the term A. For this, from the definition of both p and p, for every 0 ≤ α ≤ m + 1 and 0 ≤ ν ≤ 2(m + 1) we have that

T 0 D α t p(t, •) -D α t p(t, •) 2 H ν dt 1/2 = =   T 0 M ℓ=1 D α t c ℓ (t)(p(t ℓ , •) -g ℓ (•)) 2 H ν dt   1/2 ≤ M ℓ=1 c ℓ H α p(t ℓ , •) -g ℓ (•) H ν .
In, particular, by applying this bound to the two terms in A, we infer that

A 1/2 ≤ M ℓ=1 c ℓ H 1 p(t ℓ , •) -g ℓ (•) L 2 + c ℓ L 2 p(t ℓ , •) -g ℓ (•) H 2 .
3) Bounding each term p(t ℓ , •)g ℓ (•) H ν . Fix ℓ = 1, . . . , M and 0 ≤ ν ≤ 2(m + 1). As recalled in Section 4, the norm of any u ∈ H ν (R n , R), with ν ∈ N writes

u H ν = J ν (•)(F u)(•) L 2 ,
where F u denotes the Fourier transform of u, and

J ν (z) (1 + z 2 ) ν/2 /(2π) n/2 , z ∈ R n . In particular, it holds that (5.3) F g ℓ (z) = N j=1 1 B R n R (0) (z)e 2πiz ⊤ X ℓ,j = 1 B R n R (0) (z)F g ℓ (z), z ∈ R n ,
so that, thanks to the fact that (1 -

1 B R n R (0) )1 B R n R (0) = 0 and 1 2 B R n R (0) = 1 B R n R (0) , by denoting the function p ℓ (•) p(t ℓ , •) we may compute p(t ℓ , •) -g ℓ (•) H ν = J ν (F p ℓ -F g ℓ ) L 2 = J ν (1 -1 B R n R (0) )F p ℓ + J ν 1 B R n R (0) (F p ℓ -F g ℓ ) L 2 ≤ J ν (1 -1 B R n R (0) )J -1 2(m+1) L ∞ J 2(m+1) F p ℓ L 2 + J ν 1 B R n R (0) L ∞ 1 B R n R (0) (F p ℓ -F g ℓ ) L 2 ≤ R ν-2(m+1) p ℓ H 2(m+1) + 2 ν R ν 1 B R n R (0) (F p ℓ -F g ℓ ) L 2
, where we used the fact that J 2(m+1) , J ν , and 1 B R n R (0) are radial functions, and thus

J ν (1 -1 B R n R (0) )J -1 2(m+1) L ∞ = sup r>R (1 + R 2 ) -(2(m+1)+ν)/2 ≤ R ν-2(m+1) , J ν 1 B R n R (0) L ∞ = sup 0<r≤R (1 + R 2 ) ν/2 ≤ 2 ν R ν , as soon as R ≥ 1. 4) Bounding each term 1 B R n R (0) (F p ℓ -F g ℓ ) L 2 .
For this, for every integers ℓ = 1, . . . , M and j = 1, . . . , N we define a random mapping ζ ℓ,j : (R n ) N → L 2 (R n , C) through the following expression, which holds for every

x N ∈ (R n ) N , ζ ℓ,j (x N )(z) 1 B R n R (0) (z) e 2πiz ⊤ X ℓ,j (x N ) , z ∈ R n .
Note that, for every fixed ℓ = 1, . . . , M , since they depend deterministically from X ℓ,1 , . . . , X ℓ,M , the random mappings ζ ℓ,1 , . . . , ζ ℓ,N are independent random mappings in the probability space ((R n ) N , B N , µ N t ℓ ), and they are additionally equally distributed with respect to p ℓ (•) p(t ℓ , •). In particular, for every j = 1, . . . , N we may compute

E µ N t ℓ [ζ ℓ,j ](z) = R n 1 B R n R (0) (z) e 2πiz ⊤ x p(t ℓ , x) dx = 1 B R n R (0) (z)(F p ℓ )(z), z ∈ R n .
We may also compute ess sup

x N ∈(R n ) N ζ ℓ,j (x N ) -E µ N t ℓ [ζ ℓ,j ] N 2 L 2 ≤ 4 N 2 R n 1 B R n R (0) (z) dz = 4 N 2 V n R n ,
where V n is the volume of the n-dimensional unit ball. At this step, thanks to Pinelis inequality, for every η > 0 and ℓ = 1, . . . , M , we may compute

µ X   1 B R n R (0) (F p ℓ -F g ℓ ) L 2 > 3(V n R) n/2 log 4 η √ N   ≤ ≤ µ X     1 B R n R (0) (F p ℓ -F g ℓ ) L 2 > 2 √ 2(V n R) n/2 log 2 η √ N     = µ N t ℓ     N j=1 ζ ℓ,j -E µ N t ℓ [ζ ℓ,j ] N L 2 > 2 √ 2(V n R) n/2 log 2 η √ N     ≤ η.
Therefore, for every δ > 0, by taking the union bound with η = δ/M , with probability at least 1 -δ it finally holds that max ℓ=1,...,M

1 B R n R (0) (F p ℓ -F g ℓ ) L 2 ≤ 3(V n R) n/2 log 4M δ √ N . 5) Bounding each term c ℓ H ν . By construction, each c ℓ ∈ H m+1 ([0, T ], R) is the function with minimum norm that satisfies c ℓ1 (t ℓ2 ) = δ ℓ1,ℓ2 , for ℓ 1 , ℓ 2 = 1, . . . , M .
In particular, since the function z ℓ (t) sinc(M (t -t ℓ )/T ) is analytic, and thus z ℓ ∈ H m+1 ([0, T ], R), and satisfies z ℓ1 (t ℓ2 ) = δ ℓ1,ℓ2 , for ℓ 1 , ℓ 2 = 1, . . . , M , it must hold that c ℓ H m+1 ≤ z ℓ H m+1 , for every ℓ = 1, . . . , M . Therefore, since max ℓ1=1,...,M

|c ℓ2 (t ℓ1 )| = 1, ℓ 2 = 1, . . . , M,
by applying the bound for Sobolev functions with scattered zeros recalled in 4 with h = T /M , we obtain that, for every 0 ≤ ν ≤ m + 1 there exists a constant C m+1,ν > 0 such that

c ℓ H ν ≤ C m+1,ν (T /M ) -ν + (T /M ) m+1-ν z ℓ H m+1 , ℓ = 1, . . . , M.
To conclude, we note that the Fourier transform of the extension of each z ℓ to

H m+1 (R, R) is the function (T /M )1 {M/T } .
Combining this latter result with the Fourier characterization of the norm • H m+1 yields, for every ℓ = 1, . . . , M ,

z ℓ 2 H m+1 ≤ T M M 2T -M 2T (1 + τ 2 ) m+1 dτ ≤ M T 2(m+1)
, where the last step follows from the assumption M ≥ 2T 1 . Therefore, for every ℓ = 1, . . . , M we finally obtain that (we implicitly overload the constant C m+1,ν )

c ℓ H ν ≤ C m+1,ν T M -ν , ν = 0, . . . , m + 1,
1 Indeed, under this assumption, 1 ≤ 3/4M 2 /T 2 , and thus max

0≤τ ≤M/(2T ) (1 + τ 2 ) ≤ M 2 /T 2 .
6) Gathering the previous bounds together and end of the proof. Recalling from the results in Section 4 that there exists a constant C > 0 such that

u L ∞ ≤ C u H 1 , for every u ∈ H m+1 ([0, T ], R),
we may compute, for every ℓ = 1, . . . , M ,

p(t ℓ , •) H 2(m+1) (R n ,R) ≤ p L ∞ ([0,T ],R)⊗H 2(m+1) (R n ,R) ≤ C p H 1,2(m+1) .
Combining this latter inequality with all the previous bounds finally yields

L( p) 1/2 ≤ L( p) 1/2 + A 1/2 ≤ C m+1,1 M -m p H m+1,0 + C m+1,0 M -(m+1) p H m+1,2 + A 1/2 ≤ C p H m+1,2 + p H 1,2(m+1) × × M -m + R -2m + R n/2 log 4M δ N -1/2 ,
for some appropriate constant C > 0, which holds with probability at least 1 -δ, and the sought conclusion may be easily inferred.

The main benefit offered by replacing the unknown density p with the model density p consists of the fact that integrals of this latter mapping and of its derivatives can be actually (easily) computed, enabling to correctly instantiate our first learning problem for (a * , b * ) ∈ H + m , which is our next step. 5.2. The infinite-dimensional learning problem and its fidelity. In this section, our goal consists of instantiating and analyzing the learning problem to identify the coefficients (a * , b * ) ∈ H + m . This problem computes coefficients which "best" matches SFPE when evaluated at the model density p. Below, we seek those coefficients in the whole set H + m , deferring to a later section the problem of learning (a * , b * ) ∈ H + m through finite-dimensional models. In particular, the main result contained herein consists of appropriate learning error estimates which pave the way to obtaining learning error estimates for the problem of learning (a * , b * ) ∈ H + m through finite-dimensional models. However, for pedagogical purposes, we present complete learning rates for the infinite-dimensional learning problem as well.

From now on, we fix 0 < ε, δ < 1, and select M, N ∈ N and R > 0 as claimed in Theorem 5.2, so that, with probability at least 1 -δ, it holds that:

(5.4) L( p) ≤ C(a * , b * ) log 1 δε ε 2 ,
where the constant C(a * , b * ) > 0 depends on a * and b * uniquely. To compute accurate learning rates, we leverage classical RKHS approximation theory, which is essentially well-posed for mappings which are defined on bounded domains. For this, we will make use of the following additional assumption:

(B) There exists R * > 0 such that the coefficients (a * , b * ) ∈ H + m satisfy: supp a * (t, •), b * (t, •) ⊆ B R n R * (0), for every t ∈ [0, T ].
Remark 5.3. Assumption (B) plays a key role in computing learning rates which, on the one hand, leverage classical RKHS approximation theory, and which, on the other hand, depend explicitly on the parameters defining the learning problem. In particular, the latter property can not be generally obtained by just (smoothly) restricting to balls quantities which are defined on unbounded domains. That said, our result can be shown to hold under less restricting assumptions, such as assuming the coefficients (a * , b * ) ∈ H + m decay to zero at infinity under specific rates [START_REF] Wendland | Scattered Data Approximation[END_REF], although we opted for Assumption (B) to avoid excessively tedious computations, in turn fostering a smooth exposition. In addition, it is worth mentioning Assumption (B) is often naturally verified in many applications ranging from biology to robotics, where the state space is a bounded domain.

Thanks to Assumption (B), we may restrict ourselves to the closed subspace:

H + m,R * (a, b) ∈ H + m : supp a(t, •), b(t, •) ⊆ B R n R * (0), for every t ∈ [0, T ] ⊆ H + m ,
We are now ready to define our infinite-dimensional problem for learning the coefficients (a * , b * ) ∈ H + m,R * , which writes as follows: Definition 5.4. For every positive real λ > 0, the (random) infinite-dimensional Learning Problem to learn stochastic differential equations is defined as:

LP min (a,b)∈H + m,R * L λ (a, b) T 0 ∂ p ∂t (t, •) -(L a,b t ) * p(t, •) 2 L 2 dt + λ (a, b) 2 Hm .
The well-posedness of problem LP is proven in the following proposition:

Proposition 5.5. For every λ > 0, problem LP is well-posed and has a unique solution, which is denoted by ( a, b) ∈ H + m,R * . Proof. Since the mapping

(5.5) (a, b) ∈ H m → T 0 ∂ p ∂t (t, •) -(L a,b t ) * p(t, •) 2 L 2 dt + λ (a, b) 2
Hm is strictly convex when restricted to the closed convex subset H + m,R * ⊆ H m , we just need to prove the existence of a solution to LP. For this, if (a k , b k ) k∈N ∈ H + m,R * is any minimizing sequence for LP, there must exist some constant C > 0 such that (a k , b k ) H + m ≤ C for every k ∈ N, and therefore, since H + m,R * is in particular a closed and convex subset of the Hilbert space H m , up to extracting a subsequence there exists ( a, b) ∈ H + m,R * such that (a k , b k ) k∈N converges to ( a, b) for the weak topology of H m . Now, thanks to Theorem 2.1, it is clear the mapping (5.5) is in addition continuous for the strong topology of H m . Thus, we infer that the mapping (5.5) is in particular weakly lower semi-continuous, and therefore we finally obtain that

T 0 ∂ p ∂t (t, •) -(L a, b t ) * p(t, •) 2 L 2 dt + λ ( a, b) 2 Hm ≤ ≤ lim inf k→∞ T 0 ∂ p ∂t (t, •) -(L a k ,b k t ) * p(t, •) 2 L 2 dt + λ (a k , b k ) 2 Hm = min (a,b)∈H + m,R * T 0 ∂ p ∂t (t, •) -(L a,b t ) * p(t, •) 2 L 2 dt + λ (a, b) 2 Hm ,
and the sought conclusion follows.

We now investigate the fidelity of the learning problem LP. More specifically, through the estimates of Section 3, under appropriate norms we compute bounds for the approximation error between solutions to SFPE, associated with the drift and diffusion coefficients solutions to LP, and the unknown mapping p. In turn, these bounds will endow our learning procedure with well-posedness and high fidelity.

We start with the following technical lemma:

Lemma 5.6. For every λ > 0, with probability at least 1 -δ, it holds that:

L λ ( a, b) ≤ C(a * , b * ) λ + log 1 δε ε 2 ,
where the constant C(a * , b * ) > 0 depends on a * and b * uniquely.

Proof. Thanks to Theorem 2.1 and Corollary 5.1, a routine use of Hölder and Young inequalities allows us to compute

L λ ( a, b)-λ (a * , b * ) 2 Hm ≤ ≤ T 0 ∂ p ∂t (t, •) -L * t p(t, •) 2 L 2 dt ≤ 2 T 0 ∂ p ∂t (t, •) - ∂p ∂t (t, •) 2 L 2 dt + 2 T 0 L * t ( p -p)(t, •) 2 L 2 dt ≤ 2 T 0 ∂ p ∂t (t, •) - ∂p ∂t (t, •) 2 L 2 dt + 2C(a * , b * ) T 0 p(t, •) -p(t, •) 2 H 2 dt ≤ C(a * , b * )L( p) ≤ C(a * , b * ) log 1 δε ε 2 ,
where the (overloaded) constant C(a * , b * ) > 0, which depends on a * and b * uniquely, comes from the constant in (5.4), and the conclusion follows.

We are now ready to compute error bounds between solutions to SFPE, associated with the drift and diffusion coefficients solutions to LP, and the unknown mappings p. It is important to note that, given the nature of our data set, which essentially depends on observations of the law of the state process, the error between the unknown densities and the densities stemming from the learned coefficients is a "good metric" with which the convergence of an identification algorithm for stochastic differential equations which leverage observations of the state process may be tested.

We better formalize this metric as follows. For every (a, b)

∈ H + m , let p a,b ∈ H m+1,2(m+1) ([0, T ] × R n , R) denote the unique solution to SFPE with coefficients ( a, b) ∈ H + m .
Note that the existence and uniqueness of the regular mapping p a,b ∈ H m+1,2(m+1) ([0, T ] × R n , R) as non-negative solution to SFPE, of unitary mass and with coefficients (a, b) ∈ H + m is immediate consequence of Theorem 3.4. We define the following metric to test the accuracy of our method:

E(a, b) p a,b -p 2 L 2 = T 0 p a,b (t, •) -p(t, •) 2 L 2 dt, (a, b) ∈ H + m .
Among other benefits, we will see this metric is also particularly well-suited to estimate the error which is done when computing the observation/regulation metric (3.4).

Our main result on the accuracy of our learning method writes as follows: Proof. We define

ρ p a, b -p ∈ C(0, T ; L 2 (R n , R)) ∩ H 0,1 ([0, T ] × R n , R), and
f (t, •) - ∂ p ∂t (t, •) -(L a, b t ) * p(t, •) ∈ L 2 ([0, T ] × R n , R).
It is readily seen that

             d dt R n ϕ(y)ρ(t, y) dy = = R n L a, b t ϕ(y)ρ(t, y) + f (t, y)ϕ(y) dy, ϕ ∈ C ∞ c (R n , R), ρ(0, •) = 0,
T 0 p a, b (t, •) -p(t, •) 2 L 2 dt ≤ C ( a, b) Hm C(a * , b * ) log 1 δε ε 2 .
At this step, from Lemma 5.6 in particular we obtain that

(5.7) ( a, b) 2 Hm ≤ C(a * , b * ) λ λ + log 1 δε ε 2 = 2C(a * , b * ),
as soon as λ = log 1 δε ε 2 . Therefore, up to overloading the constant C(a * , b * ), the conclusion follows from combining (5.6) with (5.7).

5.3.

The finite-dimensional learning problem and its fidelity. The learning problem we introduced in Section 5.2 (see Definition 5.4) remains difficult to numerically solve. Here, we discuss appropriate finite-dimensional approximations of LP and error bounds ensuring the fidelity of this latter approximation, ultimately making our learning approach for stochastic differential equations accurate and tractable.

We start by recalling and adapting the approximation tools we introduced in Section 4 to our framework. Let D = [0, T ] × B R n R * (0), and consider the RKHS

H D = H d(m) (D, R) with associated kernel K D . For every given set of Q ∈ N points X D (t 1 , x 1 ), . . . , (t Q , x Q ) ,
we consider the following coordinate-wise finite dimensional models to approximate the candidate drift and diffusion coefficient solutions ( a, b):

ãi,j (t, x) = Q ℓ=1 A i,j,ℓ K D ((t, x), (t ℓ , x ℓ )), (t, x) ∈ D, i, j ∈ {1, . . . , n}, bi (t, x) = Q ℓ=1 B i,ℓ K D ((t, x), (t ℓ , x ℓ )), (t, x) ∈ D, i ∈ {1, . . . , n}, (5.8)
with A i,j,ℓ , B i,ℓ ∈ R. Therefore, the finite dimensional convex subset of H + m,R * in which the tuple ( a, b) is approximated is defined as

H +,Q m,R * span A i,j,ℓ ,B i,ℓ ∈R (ã, b) : ã(t, x) ∈ Sym ++ (n), (t, x) ∈ D ⊆ H + m,R * .
Before moving to the core of this section, by leveraging the facts we recalled in Section 4, we provide a crucial approximation results for coefficients (a, b) ∈ H + m,R * . Specifically, by combining the bounds for Sobolev functions with scattered zeros we listed in Section 4 with Theorem 2.1, we obtain the following: There exist constants C 1 , C 2 , C 3 > 0 such that, for every (a, b) ∈ H + m,R * , we may find a tuple we denote by (P Q (a),

P Q (b)) ∈ H +,Q m,R * , such that (5.9) (P Q (a), P Q (b)) = ã + C 1 (a, b) Hm h 2m-1 Q I, b ,
where (ã, b) is defined as in (5.8), and that

(a, b) -(P Q (a), P Q (b)) W 2,∞ ≤ C 2 (a, b) Hm h 2m-1 Q , (P Q (a), P Q (b)) Hm ≤ (a, b) Hm + C 3 (a, b) Hm h 2m-1 Q .
Proof. First, by combining the bounds for Sobolev functions with scattered zeros in Section 4 with Theorem 2.1, we may readily claim the existence of a constant C 1 > 0 such that, for every (a, b) ∈ H m there exists (ã, b) defined as in (5.8) such that

(5.10) (a, b) -(ã, b) W 2,∞ ≤ C 1 (a, b) Hm h 2m-1 Q .
Although ã is symmetric by definition, a priori it may not be positive semidefinite. We prove next that this is actually the case due to the positive semidefinitness of a. Indeed, thanks to (5.10) we may compute, for every (t, x) ∈ D,

sup (t,x)∈D a(t, x) -ã(t, x) op ≤ sup (t,x)∈D a(t, x) -ã(t, x) Frobenius = a -ã L ∞ ≤ (a, b) -(ã, b) W 2,∞ ≤ C 1 (a, b) Hm h 2m-1 Q ,
where • op and • Frobenius respectively denote the operator and Frobenius norms, from which the positive semidefinitness of a yields ã + C 1 (a, b) Hm h 2m-1 Q I a 0. At this step, by defining P Q (a) and P Q (b) as in (5.9) we may compute

(a, b) -(P Q (a), P Q (b)) W 2,∞ ≤ ≤ (a, b) -(ã, b) W 2,∞ + C 1 (a, b) Hm h 2m-1 Q (I, 0) W 2,∞ ≤ C 2 (a, b) Hm h 2m-1 Q , where C 2 (1 + √ n)C 1 comes from (I, 0) W 2,∞ = (I, 0) L ∞ = √
n and (5.10). Finally, note that by construction (ã, b) Hm ≤ (a, b) Hm (see also Section 4), therefore we may additionally compute

(P Q (a), P Q (b)) Hm ≤ (a, b) Hm + C 1 (a, b) Hm h 2m-1 Q (I, 0) Hm ≤ (a, b) Hm + C 3 (a, b) Hm h 2m-1 Q , concluding the proof.
In the rest of the manuscript, we adopt the notation we introduced in Theorem 5.8. We are now ready to define our infinite-dimensional problem for learning the coefficients (a * , b * ) ∈ H + m,R * , which writes as follows: Definition 5.9. For every positive real λ > 0, the (random) finite-dimensional Learning Problem to learn stochastic differential equations is defined as:

LP Q min (a,b)∈H +,Q m,R * L λ (a, b) T 0 ∂ p ∂t (t, •) -(L a,b t ) * p(t, •) 2 L 2 dt + λ (a, b) 2 Hm .
Since H +,Q m,R * is a finite-dimensional convex set, the well-posedness of LP Q may be proven similarly to the well-posedness of the original problem LP, i.e., by replicating the proof of Proposition 5.5. We thus report this result below without proof: Proposition 5.10. For every λ > 0, problem LP Q is well-posed and has a unique solution, which is denoted by

( a Q , b Q ) ∈ H +,Q m,R * .
The next result is a natural extension of Lemma 5.6 to the setting of problem LP Q , and it represents the main result of this section. Lemma 5.11. For λ = log 1 δε ε 2 and Q ∈ N such that h Q ≤ 1, with probability at least 1 -δ, it holds that:

L λ ( a Q , b Q ) ≤ C(a * , b * ) log 1 δε ε 2 + h 2m-1 Q ,
where the constant C(a * , b * ) > 0 depends on a * and b * uniquely.

Proof. First, the definition of optimal solution for LP Q yields

L λ ( a Q , b Q ) ≤ T 0 ∂ p ∂t (t, •) -(L PQ( a),PQ( b) t ) * p(t, •) 2 L 2 dt + λ P Q ( a), P Q ( b) 2 Hm ≤ T 0 ∂ p ∂t (t, •) -(L PQ( a),PQ( b) t ) * p(t, •) 2 L 2 dt + 2λ(1 + τ ) ( a, b) 2 Hm ,
where the constant τ C 2 3 h 4m-2 Q comes from Theorem 5.8. Then, thanks to Lemma 5.6 and a routine use of Hölder and Young inequalities, we may continue to compute

T 0 ∂ p ∂t (t, •) -(L PQ( a),PQ( b) t ) * p(t, •) 2 L 2 dt + 2λ(1 + τ ) ( a, b) 2 Hm ≤ 2 T 0 ∂ p ∂t (t, •) -(L a, b t ) * p(t, •) 2 L 2 dt + λ ( a, b) 2 Hm + 2 T 0 (L PQ( a),PQ( b) t ) * -(L a, b t ) * p(t, •) 2 L 2 dt + 2λτ ( a, b) 2 Hm ≤ C(a * , b * ) λ + log 1 δε ε 2 + ( a, b) -P Q ( a), P Q ( b) W 2,∞ T 0 p(t, •) 2 H 2 dt + 2λτ ( a, b) 2 Hm ,
where the constant C(a * , b * ) > 0, which we will overload below, depends on a * and b * uniquely. Now, from applying Theorem 5.8 to ( a, b) Hm and the choice of λ = log 1 δε ε 2 in Lemma 5.6, we obtain that (compare with (5.7))

(5.11) ( a, b) -P Q ( a), P Q ( b) W 2,∞ ≤ C(a * , b * )h 2m-1 Q .
On the other hand, Theorem 5.2 and the choice 0 < ε, δ < 1 readily yield (5.12)

T 0 p(t, •) 2 H 2 dt ≤ C(a * , b * ).
Therefore, by combining (5.11) with (5.12) we finally infer that

( a, b) -P Q ( a), P Q ( b) W 2,∞ T 0 p(t, •) 2 H 2 dt ≤ C(a * , b * )h 2m-1 Q ,
and the conclusion easily follows from the choice λ = log 1 δε ε 2 and the fact that, thanks to h Q ≤ 1 and (5.7), it holds that 2λτ ( a, b)

Hm ≤ C(a * , b * )h 4m-2 Q ≤ C(a * , b * )h 2m-1 Q .
. Thanks to Lemma 5.11, Theorem 5.7 may be straightforwardly extended to the context of the finite-dimensional learning problem LP Q (just by replicating its proof). We thus report this result without proof in the proposition below. Theorem 5.12. Let the coefficients

( a Q , b Q ) ∈ H +,Q m,R * be the unique solution to LP Q with λ = log 1 δε ε 2 and Q ∈ N such that h Q = log 1 δε ε 2 2m-1 ≤ 1.
With probability at least 1 -δ, it holds that:

E( a Q , b Q ) ≤ C(a * , b * ) log 1 δε ε 2 ,
where the constant C(a * , b * ) > 0 depends on a * and b * uniquely.

We conclude this section with a result which summarizes our essential contributions, showing how our learning approach may be leveraged for efficient observation/regulation of stochastic differential equations. First, we recall the setting introduced at the end of Section 3. Specifically, for every coefficients (a, b) ∈ H + m , we denote by X a,b and p a,b respectively the (unique) solutions to SDE and SFPE with coefficients (a, b) ∈ H + m . By combining Theorem 5.12 with Corollary 3.5 (in particular, compare with (3.4); more details about this remark are provided in Section 7), we readily obtain the following summarizing result:

Theorem 5.13. Let m ∈ N, α > 0, R * > 0, p 0 ∈ H 2m+1 (R n , R),
• M = ε -1/(4m) /4, N = ε -(2+n/(2m)) (as closest integers), and R = ε -1/(4m) , • λ = log 1 δε ε 2 , and Q ∈ N such that h Q = min 1, log 1 δε ε 2 2m-1
, with probability at least 1 -δ, the following estimate holds:

p -p aQ, bQ L 2 ≤ C(a * , b * ) log 1 δε ε,
where

( a Q , b Q ) ∈ H +,Q m,R *
is the unique solution to the finite-dimensional learning problem LP Q , where the learning parameters have been selected as above. In particular, for every f ∈ L 2 ([0, T ] × R n , R) the following observation/regulation estimate holds:

E µ0×P T 0 f (t, X x (t)) dt -E µ0×P T 0 f (t, X aQ, bQ x (t)) dt ≤ C(a * , b * ) f L 2 log 1 δε ε.

Conclusion and Perspectives.

In this paper, we propose a Reproducing Kernel Hilbert Space-based learning paradigm for the identification of drift and diffusion coefficients of non-linear stochastic differential equations, which relies upon discrete-time observation of the state. Under assumptions of smoothness for the unknown drift and diffusion coefficients, we provide theoretical estimates of learning rates which become increasingly tighter when both the number of observations of the state and the regularity of the unknown drift and diffusion coefficients grow. Some possible improvements and perspectives are in order. Since our learning rates essentially apply to the laws of the state process, it would be interesting to understand whether and under what conditions our method may be extended to derive stronger L p norm-based learning rates. Finally, it would be interesting to investigate extensions of our work for the identification of controlled stochastic differential equations: although these models are crucial for the control of complex systems, e.g., in aerospace and robotics, methods which offer relevant guarantees of accuracy and efficiency of the identification process still require extensive investigation.

7. Details on Stochastic Differential and Fokker-Planck Equations. In this section, we provide a more structured exposition of the concepts we previously introduced in Section 3. For this, we chronologically retrace in more details every definition and result of Section 3 step by step.

7.1. The Fokker-Planck equation. We recall we fixed m ∈ N and a constant α > 0, and since we worked with diffusion coefficients which are never trivial, for every (a, b) ∈ H + m , the coefficient a has been replaced with the mapping

a + αI : [0, T ] × R n → Sym ++ (n) with a + αI = √ a + αI √ a + αI.
Also, we fixed a non-negative density p 0 ∈ L 2 (R n , R) which served as appropriate initial condition, and we denoted by µ 0 ∈ P(R n ) the associate probability measure. We recall the following notions of stochastic differential equation and its solutions:

Definition 7.1. A measurable mapping X : R n × Ω → S solves the Stochastic Differential Equation with coefficients (a, b) ∈ H + m if each process X x (t, ω) X(x, ω)(t) is F -progressively measurable for every x ∈ R n , and

SDE x    dX x (t) = b(t, X x (t)) dt + (a + αI)(t, X x (t)) dW t , P X x (0) = x = 1, holds in (Ω, G, F , P) for µ 0 -almost every x ∈ R n . A solution X to SDE is unique if, for every measurable mapping Y : R n × Ω → S which solves SDE with coefficients (a, b) ∈ H + m , it holds that X(x, •) = Y (x,
•) a.s., for µ 0 -almost every x ∈ R n . The well-posedness of Definition 7.1 is contained in the following theorem, together with other useful properties on solutions to SDE: Theorem 7.2. For every (a, b) ∈ H + m , there exists a unique measurable mapping X : R n × Ω → S which solves SDE with coefficients (a, b) ∈ H + m . In addition, the following properties hold true for the mapping X:

1. The following mapping is measurable:

(x, ω, t) ∈ R n × Ω × [0, T ] → X x (t, ω) ∈ R n .
2. For every ϕ ∈ C b ([0, T ] × R n , R), the following mapping is continuous:

t ∈ [0, T ] → R n Ω ϕ(t, X x (t)) dP µ 0 (dx) ∈ R.
Proof. Thanks to Theorem 2.1, for every x ∈ R n there exists a unique (up to stochastic indistinguishability) F -adapted process X x : [0, T ] × Ω → R n with continuous sample paths which solves SDE x , and which satisfies the following inequality

(7.1) E d(X x1 , X x2 ) 2 ≤ C (a, b) H + m x 1 -x 2 2 ,
where the constant C (a, b) H + m > 0 depends on (a, b) ∈ H + m uniquely (see, e.g., [START_REF] Gall | Brownian Motion, Martingales, and Stochastic Calculus[END_REF]) 2 . Therefore, we define the mapping X : R n × Ω → S by X(x, ω)(t) X x (t, ω). From the continuity of the sample paths of each X x , one may show that

X(x, •) -1 B S ε (w 0 ) = t∈[0,T ]∩Q X x (t) -1 B R n ε (w 0 (t)) , x ∈ R n ,
2 Here, we use the fact that the mapping y → a(t, y) + αI is Lipschitz, for every t ∈ [0, T ]. This is a straightforward consequence of Theorem 2.1 and the fact that the mapping

A ∈ Sym ++ (n) → √ A is Lipschitz on {A ∈ Sym ++ (n) : A ≤ a L ∞ + α, y ⊤ Ay ≥ α y 2 , y ∈ R n }. Indeed, the latter set is compact and the mapping A ∈ Sym ++ (n) → √ A is continuously differentiable.
from which, by leveraging a routine monotone class argument, we easily infer the measurability of the process (X(x, •)) x∈R n . To prove the measurability of X : R n × Ω → S, we rather build a measurable mapping X : R n × Ω → S such that the process ( X(x, •)) x∈R n is a modification of (X(x, •)) x∈R n . In particular, such property would imply for every x ∈ R n the existence of a subset N x ∈ F T with P(N x ) = 1, and such that

(7.2) Xx (t, ω) = X x (t, ω), t ∈ [0, T ], ω ∈ N x .
From (7.2), together with the completeness of F , we would infer each process Xx is F -adapted and has continuous sample paths (the latter property being trivially true by definition). Moreover, thanks to Theorem 2.1, a routine application of Burkholder-Davis-Gundy inequality, and the fact that each process X x satisfies SDE x , one may straightforwardly compute, for every

x ∈ R n , E sup t∈[0,T ] Xx (t) -x - t 0 b(s, Xx (s)) dr - t 0 (a + αI)(s, Xx (s)) dW s 2 ≤ ≤ CE sup t∈[0,T ] Xx (t) -X x (t) 2 + CE   T 0 b(t, Xx (t)) -b(t, X x (t)) dt 2   + CE T 0 (a + αI)(t, Xx (t)) -(a + αI)(t, X x (t)) 2 dt ≤ CE d( Xx , X x ) 2 ,
for some (overloaded) constant C > 0, and therefore by combining (7.1) with (7.2) yields that each process Xx satisfies SDE x , x ∈ R n . In turn, we showed the existence of a measurable mapping X : R n ×Ω → S which solves SDE with coefficients (a, b) ∈ H + m . By leveraging the same argument (and a routine application of Gronwäll's inequality), one also shows the uniqueness of this mapping as in Definition 7.1.

At this step, we build the aforementioned mapping X : R n × Ω → S via Kolmogorov's lemma. More precisely, combining (7.1) with Kolmogorov's lemma yields the existence of a modification ( X(x, •)) x∈R n of (X(x, •)) x∈R n such that, for every ω ∈ Ω, the mapping x ∈ R n → Xx (•, ω) ∈ S is continuous. Therefore, the mapping X : R n × Ω → S is Caratheodory, and thus measurable (see, e.g., [START_REF] Aubin | Set-Valued Analysis[END_REF]Lemma 8.2.6]).

At this step, note that the mapping

π : [0, T ] × S → R n : (t, w) → w(t) satisfies π(t 1 , w 1 ) -π(t 2 , w 2 ) ≤ w 1 (t 1 ) -w 1 (t 2 ) + d(w 1 , w 2 ), for every t 1 , t 2 ∈ [0, T ], w 1 , w 2 ∈ S.
Hence, π is continuous and property 1. follows from

X x (t, ω) = π t, X(x, ω) , for (x, ω, t) ∈ R n × Ω × [0, T ]. Moreover, if ϕ ∈ C b ([0, T ] × R n , R) and (t k ) k∈N ⊆ [0, T ] satisfies t k → t,
(y), ϕ ∈ C 2 (R n , R).
Fix (a, b) ∈ H + m , and assume we are given a measurable mapping X : R n × Ω → S which solves SDE with coefficients (a, b) ∈ H + m . For ϕ ∈ C ∞ c (R n , R), a straightforward application of Itô's formula to SDE x yields

(7.3) Ω ϕ(X x (t)) dP = ϕ(x) + Ω t 0 L a,b s ϕ(X x (s)) ds dP, t ∈ [0, T ],
which holds for µ 0 -almost every x ∈ R n . Thanks to Theorem 7.2, we may define the curve µ : [0, T ] → P(R n ) of probability measures

µ t (A) R n Ω 1 {Xx(t)∈A} dP µ 0 (dx), A ∈ B(R n ),
and note that µ is narrowly continuous, i.e., for every ϕ ∈ C b (R n , R), the mapping

t ∈ [0, T ] → R n ϕ(y)µ t (dy) = R n Ω ϕ(X x (t)) dP µ 0 (dx) ∈ R
is continuous. By combining this latter property with (7.3) and Theorem 7.2, one readily checks that the curve of probabilities µ satisfies:

Definition 7.3. A narrowly continuous curve µ : [0, T ] → P(R n ) is said to solve the Fokker-Planck Equation with coefficients (a, b) ∈ H + m if FPE    d dt R n ϕ(y)µ t (dy) = R n L a,b t ϕ(y)µ t (dy), ϕ ∈ C ∞ c (R n , R), µ t=0 = µ 0 .
Importantly, thanks to the regularity of the coefficients (a, b) ∈ H + m , FPE can have one narrowly continuous solution µ at most, as we state in the following: Proposition 7.4 (Propositions 4.1 and 4.2 in [START_REF] Figalli | Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients[END_REF]). Given any tuple of coefficients (a, b) ∈ H + m , at most one narrowly continuous curve µ : [0, T ] → P(R n ) can solve FPE with coefficients (a, b) ∈ H + m . Our previous computations show that solutions to FPE may be obtained from solutions to SDE, and we now establish this process may be inverted. Specifically, motivated by the results in [START_REF] Stroock | Multidimensional Diffusion Processes[END_REF][START_REF] Figalli | Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients[END_REF], we prove that any narrowly continuous curve µ : [0, T ] → P(R n ) solution to FPE with coefficients (a, b) ∈ H + m is associated with a unique (in the sense of Definition 7.1) measurable mapping X : R n × Ω → S which solves SDE with coefficients (a, b) ∈ H + m . We gather such result in the following theorem, which is a natural extension of [START_REF] Figalli | Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients[END_REF]Theorem 2.6] to our setting: Theorem 7.5. Let a narrowly continuous curve µ : [0, T ] → P(R n ) be solution to FPE with coefficients (a, b) ∈ H + m (which is unique thanks to Proposition 7.4). There exists a unique measurable mapping X : R n × Ω → S which solves SDE with coefficients (a, b) ∈ H + m , and which satisfies the representation formula:

R n ϕ(y)µ t (dy) = R n Ω ϕ(X x (t)) dP µ 0 (dx), for t ∈ [0, T ], ϕ ∈ C c (R n , R).
Proof. Thanks to Theorem 7.2, we already know there exists a unique measurable mapping X : R n × Ω → S which solves SDE with coefficients (a, b) ∈ H + m . To conclude we just need to show the representation formula holds true. For this, we define the curve μ : [0, T ] → P(R n ) of probability measures μt (A)

R n Ω 1 {Xx(t)∈A} dP µ 0 (dx), A ∈ B(R n ),
which is well-defined and narrowly continuous thanks to Theorem 7.2. In addition, by combining this latter property with (7.3) and Theorem 7.2, we see that the curve μ : [0, T ] → P(R n ) solves FPE, and thus Proposition 7.4 yields

R n ϕ(y)µ t (dy) = R n ϕ(y)μ t (dy) = R n Ω ϕ(X x (t)) dP µ 0 (dx),
for every t ∈ [0, T ] and ϕ ∈ C c (R n , R), and the conclusion follows.

Absolutely continuous solutions to the Fokker-Planck equation.

From what we showed, solutions to SDE may be found by solving FPE. In this section, we show the existence of narrowly continuous curves µ : [0, T ] → P(R n ) of type

µ t (A) = A p(t, y) dy, A ∈ B(R n ),
for appropriate densities p : [0, T ]× R n → R, which are solutions to FPE. Note that, if such solutions to FPE exist, then they are unique thanks to Proposition 7.4. Although such existence result is classic (see, e.g., [START_REF] Figalli | Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients[END_REF][START_REF] Breiten | Control Strategies for the Fokker-Planck Equation[END_REF]), we retrace its proof in Appendix A to characterize the constants appearing in some appropriate estimates and regularity properties which have been paramount to derive the results in Section 5.

For this, let us first recall the broader definition of solution to FPE, which in particular encompasses Definition 7.3 as a sub-case (see also Theorem 7.7 below):

Definition 7.6. Let f ∈ L 2 ([0, T ] × R n , R) and p ∈ L 2 (R n , R). A (regular enough) function p : [0, T ] × R n → R is said to solve the non-homogeneous Fokker- Planck Equation with coefficients (a, b) ∈ H + m , if FPE f              d dt R n ϕ(y)p(t, y) dy = = R n L a,b t ϕ(y)p(t, y) + f (t, y)ϕ(y) dy, ϕ ∈ C ∞ c (R n , R), p(0, •) = p(•).
We gather results on the existence, uniqueness, and energy-type estimates for solutions to FPE f , and therefore for solutions to FPE, in the following:

Theorem 7.7. For every f ∈ L 2 ([0, T ] × R n , R) and every p ∈ L 2 (R n , R), there exists a unique mapping p ∈ C(0, T ; L 2 (R n , R)) ∩ L 2 (0, T ; H 1 (R n , R)) which solves FPE f with coefficients (a, b) ∈ H + m ,

and with

∂p ∂t ∈ L 2 (0, T ; H -1 (R n , R)).
In addition, the following first parabolic estimate holds:

p(t, •) 2 L 2 + t 0 p(t, •) 2 H 1 dt ≤ ≤ C (a, b) Hm p 2 L 2 + t 0 f (s, •) 2 L 2 ds , t ∈ [0, T ], (7.4)
where C (a, b) Hm > 0 is a constant which continuously depends on (a, b) Hm . Finally, if f = 0 and p is a non-negative density in L 2 (R n , R), then

p(t, •) ≥ 0, R n p(t, y) dy = 1, t ∈ [0, T ],
and therefore, if for every t ∈ [0, T ] we define

µ t (A) A p(t, y) dy, A ∈ B(R n ),
then the curve µ : [0, T ] → P(R n ) is narrowly continuous and solves FPE.

Remark 7.8. The regularity of the mapping f may be weakened (see, e.g., [START_REF] Lions | Optimal Control of Systems Governed by Partial Differential Equations[END_REF][START_REF] Breiten | Control Strategies for the Fokker-Planck Equation[END_REF]), although the requirement f ∈ L 2 ([0, T ] × R n , R) already fits our purpose.

The proof of Theorem 7.7 is reported in Appendix A and is based on the classical Lions scheme (see, e.g., [START_REF] Lions | Optimal Control of Systems Governed by Partial Differential Equations[END_REF][START_REF] Chipot | Elements of Nonlinear Analysis[END_REF][START_REF] Evans | Partial Differential Equations[END_REF]). Among straightforward benefits, we recall Theorem 7.7 enables introducing rigorous criteria to establish satisfactory guarantees for our learning approach in many circumstances. Let us better introduce this concept below. As a matter of example, when dealing with stochastic differential equations in applications such as observation and regulation, one must often manipulate metrics

(7.5) E µ0×P T 0 f (t, X x (t)) dt = T 0 R 2n f (t, y) P Xx(t) (dy) µ 0 (dx) dt,
where the measurable mapping X : R n × Ω → S solves SDE for some coefficients (a, b) ∈ H + m , and the mapping f : [0, T ] × R n → R is regular enough. In the case the coefficients (a, b) ∈ H + m of SDE are to be learned, one does not have perfect knowledge of (7.5), and therefore the error between (7.5) and its counterpart in which (X x ) x∈R n is replaced with the solution to SDE stemming from rather learned coefficients must be estimated. This gap is filled with the following: Corollary 7.9. For any coefficients (a, b) ∈ H + m , let X a,b : R n × Ω → S denote the unique measurable mapping which solves SDE with coefficients (a, b) ∈ H + m (see Theorem 7.5). There exists an operator

O : H + m → L ∞ (0, T ; L 2 (R n , R) * ) ∼ = L ∞ (0, T ; L 2 (R n , R)) (a, b) → O a,b • (•) which, for every t ∈ [0, T ] and every ϕ ∈ C c (R n , R), satisfies O a,b t (ϕ) = R 2n ϕ(y) P X a,b x (t) (dy) µ 0 (dx).
In addition, if p a,b ∈ C(0, T ; L 2 (R n , R)) denotes the unique solution to FDE 0 with coefficients (a, b) ∈ H + m and p = p 0 (which uniquely exists thanks to Theorem 7.7), for every

f ∈ L 2 ([0, T ] × R n , R) the mapping t ∈ [0, T ] → O a,b t f (t, •) ∈ R is measurable and in L 1 ([0, T ], R), and it satisfies, for every (a 1 , b 1 ), (a 2 , b 2 ) ∈ H + m , (7.6)
T 0 O a1,b1 t f (t, •) -O a2,b2 t f (t, •) dt ≤ f L 2 p a1,b1 -p a2,b2 L 2 .
Remark 7.10. Thanks to the previous Corollary, we may provide (7.5) with a rigorous meaning by defining, for every

f ∈ L 2 ([0, T ] × R n , R), E µ0×P T 0 f (t, X a,b x (t)) dt T 0 O a,b t f (t, •) dt.
As we showed in Section 5, the estimate (7.6) enables the control of any estimation error occurring during the computation of the metric (7.5). Note that more refined estimates than (7.6) may be easily obtained by requiring more regularity on the function f (e.g., see the proof of Corollary 7.9 below).

Proof. Fix any (a, b) ∈ H + m . For t ∈ [0, T ] and ϕ ∈ L 2 (R n , R), we define O a,b t (ϕ) = lim k→∞ R 2n ϕ k (y) P X a,b x (t) (dy) µ 0 (dx),
where

(ϕ k ) k∈N ⊆ C c (R n , R
) is any sequence which converges to ϕ for the strong topology of L 2 (R n , R). This definition is well-posed thanks to Theorem 7.5: indeed, the limit above uniquely exists, given that if (ϕ

1 k ) k∈N , (ϕ 2 k ) k∈N ⊆ C c (R n , R)
are two sequences which converge to ϕ for the strong topology of L 2 (R n , R), we may compute

R 2n ϕ 1 k (y) P X a,b x (t) (dy) µ 0 (dx)- R 2n ϕ 2 k (y) P X a,b x (t) (dy) µ 0 (dx) ≤ ≤ p a,b (t, •) L 2 ϕ 1 k -ϕ 2 k L 2 → 0, k → ∞.
Similarly, one may easily prove that

O a,b t (•) ∈ L 2 (R n , R) * ∼ = L 2 (R n , R) for every t ∈ [0, T ].
In particular, note that thanks to (7.4), for t ∈ [0, T ] and ϕ ∈ L 2 (R n , R),

(7.7) |O a,b t (ϕ)| ≤ C(a, b) p 0 L 2 ϕ L 2 , for some appropriate constant C(a, b) > 0. Finally, let g ∈ L 2 ([0, T ] × R n , R) and let (g k ) k∈N ⊆ C c ([0, T ] × R n , R) converge to g for the strong topology of L 2 ([0, T ] × R n , R).
In particular, up to some subsequence, we infer that each sequence g k (t,

•) k∈N ⊆ C c ([0, T ] × R n , R) converges to g(t, •) ∈ L 2 (R n , R) for the strong topology of L 2 (R n , R),
(•) ∈ L ∞ (0, T ; L 2 (R n , R) * ).
On the other hand, (7.8) yields the measurability of the mapping t ∈ [0, T ] → O a,b t f (t, •) ∈ R, which together with (7.7) makes the mapping t ∈ [0, T ] → O a,b t f (t, •) ∈ R measurable and in L 1 ([0, T ], R). The conclusion of the proof follows from the fact that (7.6) is a straightforward consequence of (7.8) and Fatou's lemma.

Additional regularity of solutions to the Fokker-Planck equation.

In Section 5, we dealt with solutions p : [0, T ] × R n → R of FPE 0 which enjoy higher regularity properties, and specifically for which p ∈ H m+1,2(m+1) ([0, T ] × R n , R), and for which FPE 0 holds pointwise. Below, we are going to show such additional attributes stem from the properties of coefficients (a, b) ∈ H + m listed in Theorem 2.1. First, for every (a, b) ∈ H + m and almost every t ∈ [0, T ] we introduce the notation:

(L a,b t ) * u(t, y)

1 2 n i,j=1 ∂ 2 a α ij u ∂y j ∂y i (t, y) - n i=1 ∂ b i u ∂y i (t, y), u ∈ L 2 (0, T ; H 2 (R n , R)),
which denotes the dual operator of L a,b t . This operator is well-defined with image in L 2 ([0, T ] × R n , R). Indeed, thanks to Theorem 2.1, one readily shows that

b i u ∈ L 2 (0, T ; H 1 (R n , R)), with ∂ b i u ∂y r = b i ∂u ∂y r + u ∂b i ∂y r ∈ L 2 ([0, T ] × R n , R), a ij u ∈ L 2 (0, T ; H 2 (R n , R)), with ∂ a ij u ∂y r = a ij ∂u ∂y r + u ∂a ij ∂y r ∈ L 2 ([0, T ] × R n ,
∈ L 2 ([0, T ] × R n , R),
for u ∈ L 2 (0, T ; H 2 (R n , R)), i, j, r, s = 1, . . . , n, thus the well-posedness of (L a,b t ) * . Thanks to Theorem 2.1, the following higher regularity result holds: Theorem 7.11. Under the setting and notation of Theorem 7.7, if furthermore p ∈ H 2m+1 (R n , R), then the unique solution p :

[0, T ] × R n → R to FPE 0 with coefficients (a, b) ∈ H + m is additionally such that p ∈ H m+1 (0, T ; H 2(m+1) (R n , R)).
In particular, the mapping p additionally satisfies the following Strong Fokker-Planck Equations with coefficients (a, b) ∈ H + m :

SFPE      ∂p ∂t (t, y) = (L a,b t ) * p(t, y), a.e. (t, y) ∈ [0, T ] × R n , p(0, •) = p(•).
Remark 7.12. This result is classically proved under either time-independent coefficients or bounded domains (e.g., [START_REF] Evans | Partial Differential Equations[END_REF][START_REF] Lions | Non-Homogeneous Boundary Value Problems and Applications[END_REF]). Although in classical reference which consider time-dependent coefficients such as [START_REF] Lions | Non-Homogeneous Boundary Value Problems and Applications[END_REF] it is explicitly mentioned therein that their results may be extended to unbounded domains, we found such extension non-straightforward. We therefore decided to revisit and extend the regularity results for parabolic equations which are contained in [START_REF] Evans | Partial Differential Equations[END_REF]Section 6.3] to time-dependent coefficients and unbounded domains, and to report the proof in Appendix A.

• Eu| [0,T ]×R n = u, for every u ∈ H 1 ([0, T ] × R n , R), • Eu L 2 ≤ C u L 2 and Eu H 1 ≤ C u H 1 , for every u ∈ H 1 ([0, T ] × R n , R),
for instance, see [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]. For this, we may follow the proof of [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]Theorem 8.6]. More specifically, let η ∈ C ∞ (R, [0, 1]) be such that

η(t) =      1, t < T 4 , 0, t > 3 4 T,
and for every u : [0, T ] × R n → R define the mappings u -(t, y) u(t, y), 0 < t ≤ T, 0, t ≤ 0 and u + (t, y) u(t, y), 0 ≤ t < T, 0, t ≥ T.

If u ∈ H 1 ([0, T ] × R n , R), one sees that ηu + ∈ H 1 ([0, ∞) × R n ,
-η)u -∈ H 1 ((-∞, T ] × R n , R) with, for every i = 1, . . . , n, ∂(1 -η)u - ∂t = (1 -η) ∂u ∂t - + u -∂η ∂t and ∂(1 -η)u - ∂y i = (1 -η) ∂u ∂y i - .
At this step, since u = ηu+(1-η)u, we first extend ηu + by reflection at t = 0 through a mapping v 1 ∈ H 1 (R n+1 , R) which thus satisfies

v 1 L 2 ≤ C u L 2 and v 1 H 1 ≤ C u H 1 ,
and then we extend (1 -η)u -by reflection at t = T through another mapping

v 2 ∈ H 1 (R n+1 , R) which thus satisfies v 2 L 2 ≤ C u L 2 and v 2 H 1 ≤ C u H 1 ,
where the constant C > 0 only depends on η. The conclusion easily follows if we define the linear and bounded operator E as Eu v 1 + v 2 .

A.2. Proofs of Section 7. Below, we will make use of the notation a α a+αI.

Proof of Theorem 7.7. The proof is standard and based on the classical Lions scheme (see, e.g., [START_REF] Lions | Optimal Control of Systems Governed by Partial Differential Equations[END_REF][START_REF] Chipot | Elements of Nonlinear Analysis[END_REF][START_REF] Evans | Partial Differential Equations[END_REF]), therefore we mainly focus on deriving the constant C (a, b) Hm = C α, (a, b) Hm in (7.4) and the properties of p when f = 0 and p is a non-negative density in L 2 (R n , R). For this, consider the Gelfand triple

V H 1 (R n , R) = H 1 0 (R n , R) ֒→ H L 2 (R n , R) ֒→ V * = H -1 (R n , R)
and define the t-measurable bilinear form Thanks to Theorem 2.1, one may prove that the form A α,a,b is continuous and semicoercive (see also the computations for the estimate (7.4) below). Therefore, thanks to a straightforward modification of the proof of [START_REF] Chipot | Elements of Nonlinear Analysis[END_REF]Theorem 11.7], there exists a unique solution p ∈ C(0, T ; H) ∩ L 2 (0, T ; V ) to the variational problem (A.1)

A α,a,b : [0, T ] × V × V → R (t; u, v) → n i,j=1 R n 1 2 a α ij (t,
     d dt (p(t, •), ϕ) H + A α,a,b (t; p(t, •), ϕ) = (f (t, •), ϕ) H , ϕ ∈ V, p(0, •) = p(•),
which in addition satisfies ∂p ∂t ∈ L 2 (0, T ; V * ). At this step, thanks to Theorem 2.1

a α ij (t, •)p(t, •) ∈ H 1 (R n , R),
≤ f (t, •) 2 H 2 + p(t, •) 2 H 2 - α 2 ∇p(t, •) 2 H + nM C(a, b) 2 p(t, •) 2 L 2 2 + ∇p(t, •) 2 H 2M ≤ 1 + nM C(a, b) 2 p(t, •) 2 H 2 + f (t, •) 2 H 2 - α 4 ∇p(t, •) 2 H ,
where M > 2/α is some large enough constant which stems from applying Young's inequality, whereas C(a, b) > 0 is a constant which stems from Theorem 2.1 and continuously depends on the L ∞ norms of b and of the derivatives of a uniquely, and a routine application of Gronwäll's inequality together with Theorem 2.1 yield (7.4). Finally, assume f = 0 and that p is a non-negative density in L 2 (R n , R). Thanks to the fact that u + , u -∈ H 1 (R n , R) for every u ∈ H 1 (R n , R), with ∇(u + ) = ∇u1 {u>0} and ∇(u -) = ∇u1 {u<0} , and that u -= (-u) + , leveraging the notation we introduced previously and applying [START_REF] Chipot | Elements of Nonlinear Analysis[END_REF]Lemma 11.2]

with u = -p yield d dt p(t, •) -2 H 2 = - ∂p ∂t (t, •), p(t, •) - V * = - n i,j=1 R n 1 2 a α ij (t, y) ∂(p + -p -) ∂y j (t, y) ∂(-p -) ∂y i (t, y) dy + n i=1 R n   n j=1 1 2 ∂a α ij ∂y j (t, y) -b i (t, y)   ∂(p + -p -) ∂y i (t, y)p(t, y) -dy ≤ - α 2 ∇(p -)(t, •) 2 H + C(a, b) n i=1 R n ∂p - ∂y i (t, y)p(t, y) -dy ≤ - α 2 ∇(p -)(t, •) 2 H + nM C(a, b) 2 p(t, •) -2 H 2 + ∇(p -)(t, •) 2 H 2M ≤ nM C(a, b) 2 p(t, •) -2 H 2 ,
and thanks to a routine application of Gronwäll's inequality, the fact that p(x) ≥ 0 almost everywhere yields p(t, x) ≥ 0 for every t ∈ [0, T ] and almost every x ∈ R n . At this step, for every k

∈ N choose a cut-off function ϕ k ∈ C ∞ c (R n , [0, 1]) such that ϕ k (x) = 1 for every x ∈ B R n k (0), supp(ϕ k ) ⊆ B R n 2k (0)
, and whose first and second derivatives are uniformly bounded. For every t ∈ [0, T ], from the definition of FPE f and the monotone and dominated convergence theorems (here, we leverage both Theorem 2.1 and the fact that p(t, •) ∈ L 2 (R n , R), t ∈ [0, T ]) we may compute To conclude, it is clear that we just need to prove that the curve µ : [0, T ] → P(R n ) defined in the statement of the theorem is narrowly continuous. For this, let t ∈ [0, T ] and (

t k ) k∈N ⊆ [0, T ] such that t k → t for k → ∞. Note that p ∈ C(0, T ; H) implies that p(t n , •)-p(t, •) L 2 → 0 for k → ∞. In particular, for any function ϕ ∈ C c (R n , R), as soon as k → ∞ we infer that R n ϕ(y)p(t n , y) dy - R n ϕ(y)p(t, y) dy ≤ ϕ L 2 p(t n , •) -p(t, •) L 2 → 0.
We conclude from the fact that the narrow and weak* topologies coincide in P(R n ).

Proof of Theorem 7.11. Given that the proof is substantially long, for the sake of clarity we divide it in several step. Below, we adopt the notation we introduced and used in the proof of Theorem 7.7.

1) A second parabolic estimate. In this section, we provide computations by rather considering SFPE f with 0

= f ∈ L 2 ([0, T ] × R n , R). Let (v i ) i∈N be a countable basis of H 2(m+1) (R n , R) (and in turn of H ℓ (R n , R), for ℓ = 0, . . . , 2m + 1), such that (v i ) i∈N is orthonormal in H 1 (R n , R). Therefore, there is (x i ) i∈N ⊆ R such that (A.2) lim j→∞ j i=1 xi v i = p, in H 1 (R n , R).
Moreover, by revisiting the proof of [START_REF] Chipot | Elements of Nonlinear Analysis[END_REF]Theorem 11.7], one easily see that for every k ∈ N there exists x k ∈ AC([0, T ], R k ) such that the function

p k (t, •) k i=1 x k i (t)v i ∈ C(0, T ; L 2 (R n , R)) ∩ L 2 (0, T ; H 1 (R n , R))
is the unique solution to the variational problem

(A.3)      d dt (p k (t, •), v i ) L 2 + A α,a,b (t; p k (t, •), v i ) = (f (t, •), v i ) H , i = 1, . . . , k, x k i (0) = xi , i = 1, . . . , k,
which in addition satisfies (see also [START_REF] Chipot | Elements of Nonlinear Analysis[END_REF]Page 197])

(A.4)                ∂p k ∂t (t, •) = k i=1 d dt x k i (t)v i ∈ L 2 (0, T ; H 1 (R n , R)), ∂p k ∂y j (t, •) = k i=1 x k i (t) ∂v i ∂y j ∈ AC(0, T ; L 2 (R n , R)), j = 1, . . . , n, p k → p strongly in L 2 (0, T ; H 1 (R n , R)).
Moreover, from the proof of Theorem 7.7, it is clear that each p k satisfies (7.4).

At this step, by multiplying each equation in (A.3) by d dt x k i (t) and summing those over i = 1, . . . , k, thanks to (A.4) for almost every t ∈ [0, T ] we may compute

∂p k ∂t (t, •) 2 L 2 = k i=1 d dt x k i (t) d ds s=t (p k (s, •), v i ) L 2 = k i=1 d dt x k i (t) (f (t, •), v i ) L 2 -A α,a,b (t; p k (t, •), v i ) ≤ f (t, •) L 2 ∂p k ∂t (t, •) L 2 - n i,j=1 R n 1 2 a α ij (t, y) ∂p k ∂y j (t, y) d dt ∂p k ∂y i (t, y) dy (A.5) - n i=1 R n   n j=1 1 2 ∂a α ij ∂y j (t, y) -b i (t, y)   ∂ ∂y i ∂p k ∂t (t, y) p k (t, y) dy.
We are going to bound the last two terms in (A.5). Specifically, for the first to the last term, thanks to Theorem 2.1 and (A.4) for every t ∈ [0, T ] we obtain that

- t 0 R n n i,j=1 1
- α 2 ∇p k (t, •) 2 L 2 + C α, (a, b) Hm p 2 H 1 + t 0 f (s, •) 2 L 2 ds , (A.6)
where C α, (a, b) Hm > 0 is a constant which continuously depends on α and (a, b) Hm uniquely. Below, we will implicitly overload the constant C α, (a, b) Hm . We now focus on the last term in (A.5). For this, first thanks to Theorem 2.1 one obtains that, for every

i = 1, . . . , n, ϕ ∈ C ∞ c (R n , R), and almost every t ∈ [0, T ], - R n n j=1 1 2 ∂a α ij ∂y j (t, y) -b i (t, y) ∂ϕ ∂y i (y)p k (t, y) dy = = R n ∂ ∂y i     n j=1 1 2 ∂a α ij ∂y j -b i   p k   (t, y) ϕ(y) dy.
Since from (A.4) we infer the existence of a sequence (ϕ

ℓ ) ℓ∈N ⊆ C ∞ c (R n , R) such that ∂p k ∂t (t, •) -ϕ ℓ H 1 → 0, for ℓ → ∞,
∂s (s, •) 2 L 2 ds ≤ C α, (a, b) H + m p 2 H 1 + t 0 f (s, •) 2 L 2 ds . (A.7)
2) First-order-in-time and second-order-in-space regularity. In this section, we provide computations by rather considering SFPE f with 0 = f ∈ L 2 ([0, T ]×R n , R).

We first show the inclusion

∂p ∂t ∈ L 2 (0, T ; L 2 (R n , R)) ∼ = L 2 ([0, T ] × R n , R
), this latter identification being true since L 2 (R n , R) is a separable Hilbert space. For this, we show the existence of ξ ∈ L 2 (0, T ; L 2 (R n , R)) such that the following holds in L 2 (R n , R):

(A.8) T 0 ξ(t, •)ψ(t) dt = - T 0 p(t, •) dψ dt (t) dt, for all ψ ∈ C ∞ c ([0, T ], R).
For this, we first note that, thanks to (A.7) there exists ξ ∈ L 2 (0, T ;

L 2 (R n , R)) such that ∂p k ∂t k∈N ⊆ L 2 (0, T ; L 2 (R n , R))
weakly converges to ξ, up to a subsequence. Now, for any ψ ∈ C ∞ c ([0, T ], R) and any ϕ ∈ L 2 (R n , R), thanks to the last (convergence) property in (A.4) we may compute At this step, from our choice for the countable basis (v i ) i∈N ⊆ H 2(m+1) (R n , R), for almost every t ∈ [0, T ] and every i ∈ N, in particular there exists z i (t) ∈ R such that (A.12) p(t, •) = lim x k i (t)v i (•), in H 1 (R n , R),

T 0 ξ(t, •)ψ(t) dt, ϕ L 2 = T 0 ξ(t, •), ψ(t)ϕ L 2 dt = = lim k→∞ T 0 ∂p k ∂t (t,
for almost every t ∈ [0, T ], and from the orthonormality of (v i ) i∈N in H 1 (R n , R) we obtain that z i (t) = x k i (t), for every k ∈ N, i = 1, . . . , k and almost every t ∈ [0, T ]. Hence, (A.12) yields that the mapping p : [0, T ] → H 2 (R n , R) is strongly Bochner measurable. In addition, combining the elliptic estimate (A.11) together with the estimates (7.4) and (A.7) finally provides that p ∈ L 2 (0, T ; H 2 (R n , R)) with and SFPE 0 follows from the previous regularity properties and a density argument.

3) Second-order-in-time and fourth-order-in-space regularity. We now turn to the original setting SFPE 0 , i.e., SFPE f with f = 0. The first step to further improve the regularity of p consists of formally differentiating (A.1) with respect to time and studying solutions q : [0, T ] × R n → R to the new variational problem In particular, from such properties we readily deduce that, for every j = 1, . . . , n, Clearly, p = w is solution to (A.16). In particular, if so we would obtain that ∂p ∂t ∈ L 2 (0, T ; H 2 (R n , R)) and ∂ 2 p ∂t 2 ∈ L 2 ([0, T ]×R n , R), and thanks to the choice of the basis (v i ) i∈N ⊆ H 2(m+1) (R n , R) and to a higher order elliptic regularity argument applied to (A.9)-(A.10) with f = 0, we would also obtain that p ∈ L 2 (0, T ; H 4 (R n , R)), that is the sought second-order-in-time and fourth-order-in-space regularity.

Hence, to conclude we need to prove that the integro-differential variational problem (A.16) may have one solution at most. For this, let z 1 , z 2 ∈ L 2 (0, T ; H 2 (R n , R)), Therefore, we are in the setting of Theorem 7.7 and of our previous computations. In particular, we may combine (A.13) with (A.17) to obtain that, for every t ∈ [0, T ], .14). In particular, it is clear how to extend properties (A.15) and define updated integro-differential variational problems (A.16) by induction, which, thanks to iteratively higher order elliptic regularity arguments applied to (A.9)-(A.10) and the elliptic estimate (A.13), provide that ∂ ℓ p ∂t ℓ ∈ L 2 (0, T ; H 2(m+1-ℓ) (R n , R)), for ℓ = 1, . . . , m + 1, and p ∈ L 2 (0, T ; H 2(m+1) (R n , R)). The conclusion follows.

  3) ∂p ∂t (t, y) = (L a,b t ) * p(t, y), (t, y) ∈ [0, T ] × R n ,

Theorem 5 . 7 . 2 .δε ε 2 ,

 5722 Let the coefficients ( a, b) ∈ H + m,R * be the unique solution to LP with λ = log 1 δε ε With probability at least 1 -δ, it holds that: E( a, b) ≤ C(a * , b * ) log 1 where the constant C(a * , b * ) > 0 depends on a * and b * uniquely.

δε ε 2 ,

 2 and therefore, thanks to Theorem 3.4, we may apply the estimate (3.3) to ρ, which in combination with Lemma 5.6 with the choice λ = log 1 δε ε 2 in particular yields sup t∈[0,T ] p a, b (t, •) -p(t, •) 2 L 2 ≤ C ( a, b) Hm C(a * , b * ) log 1 where the constant C(a * , b * ) > 0 depends on a * and b * uniquely, whereas the constant C ( a, b) Hm > 0 continuously depends on ( a, b) Hm uniquely. Up to overloading these constants, combined with (5.4) this latter inequality readily yields (5.6)

Theorem 5 . 8 .

 58 Denote the fill distance between X D and D with h Q sup (t,x)∈D min ℓ=1,...,Q (t k , x k ) -(t, x) .

  and (a * , b * ) ∈ H + m,R * satisfy Assumptions (A) and (B), and denote by X and p respectively the (unique) solutions to SDE and SFPE with coefficients (a * , b * ) ∈ H + m,R * . There exists a constant C(a * , b * ) > 0 which only depends on a * and b * , such that by choosing the following learning parameters for the fixed precision parameters 0 < ε, δ < 1:

  t)v i (•), in H 2 (R n , R),for almost every t ∈ [0, T ]. But, up to extracting a subsequence, (A.4) yields p(t, •) = lim k→∞ k i=1

2 L 2

 22 ds , t ∈ [0, T ]. (A.13)Before moving on with additional regularity properties, we note that, thanks to Theorem 2.1 and the fact that p ∈ L 2 (0, T ; H 2 (R n , R)), integrating by parts the variational problem (A.1) with f = 0 readily yieldsT 0 R n ∂p ∂t (t, y) -(L a,b t ) * p(t, y) ϕ(y) dy dt = 0, ϕ ∈ C ∞ c (R n , R),

  t, •), ϕ) L 2 + A α,a,b (t; q(t, •), ϕ)+ +A 0, ȧ, ḃ(t; p(t, •), ϕ) = 0, ϕ ∈ H 1 (R n , R), q(0,•) = (L a,b 0 ) * p(•). Problem (A.14) is well-posed. Indeed, since p ∈ L 2 (0, T ; H 2 (R n , R)), thanks to Theorem 2.1 one easily shows thatA 0, ȧ, ḃ(t; p(t, •), ϕ) = -R n (L 0, ȧ, ḃ t ) * p(t, y)ϕ(y) dy, for every ϕ ∈ H 1 (R n , R) and almost every t ∈ [0, T ]. Therefore, since (L 0, ȧ, ḃ t ) * p ∈ L 2 ([0, T ] × R n , R) and (L a,b 0 ) * p ∈ H 1 (R n , R), problem (A.14) fits the setting of Theorem 7.7, thus from our previous computations there exists a unique solutionq ∈ L 2 (0, T ; H 2 (R n , R)) to (A.14), which additionally satisfies ∂q ∂t ∈ L 2 ([0, T ]×R n , R).At this step, we define the functionw : [0, T ] × R n → R (t, y) → p(y) + t 0 q(s, •) ds (y) = p(y) + t 0 q(s, y) ds,where the Lebesgue integral in the last equality stems from the properties of the Bochner integral for functions in L 2 ([0, T ]× R n , R). Clearly, w ∈ C 0 (0, T ; H 2 (R n , R)), and Fubini theorem yields, for everyϕ 1 ∈ L 2 (R n , R) and ψ ∈ C ∞ c ([0, T ], R), , y)ψ(t) dt ϕ 1 (y) dy = 0,and, for almost every t ∈ [0, T ] and every ϕ 2 ∈ C ∞ c (R n , R), y) ds ϕ 2 (y) dy, j = 1, . . . , n.

(A. 15 ) 0 A 0 A 0 A( 0 A

 150000 ∂w ∂t = q ∈ L 2 (0, T ; H 2 (R n , R)),Thanks to Theorem 2.1 and the properties listed in (A.15), for everyϕ ∈ C ∞ c (R n , R)and almost every t ∈ [0, T ], we may computed dt (w(t, •), ϕ) L 2 = (q(t, •), ϕ) L 2 = (L a,b 0 ) * p, ϕ L 2 -t α,a,b (s; q(s, •), ϕ) ds -t 0, ȧ, ḃ(s; p(s, •) ± w(s, •), ϕ) ds = R n (L a,b 0 ) * p(y)ϕ(y) dyy) -ḃi (s, y) w(s, y) y) -b i (s, y) ∂w ∂s (s, y) ds ∂ϕ ∂y i (y) dy t 0, ȧ, ḃ(s; (p -w)(s, •), ϕ) ds.Therefore, from[START_REF] Chipot | Elements of Nonlinear Analysis[END_REF] Theorem 11.5] and the fact thatR n (L a,b 0 ) * p(y)ϕ(y) dy = -A α,a,b (0; p, ϕ),we conclude that w solves the following integro-differential variational problem t, •), ϕ) L 2 + A α,a,b (t; w(t, •), ϕ)+ + t 0, ȧ, ḃ(s; (p -w)(s, •), ϕ) ds = 0, ϕ ∈ H 1 (R n , R), w(0, •) = p(•).

L 2 (s 0 (z 1 - 1 -

 2011 [0, T ] × R n , R), satisfy (A.[START_REF] Genon-Catalot | Asymptotic Equivalence of Nonparametric Diffusion and Euler Scheme Experiments[END_REF]. By defining the mappingh : [0, T ] → L 2 (R n , R) t → t 0 (L 0, ȧ, ḃ s ) * (z 1 -z 2 )(s, •) ds, it is readily checked that h ∈ L 2 ([0, T ] × R n , R), in particular satisfying t 0 h(s, •) 2 L 2 ds ≤ ≤ C α, (a, b) Hm t 0 z 2 )(r, •) 2 H 2 dr ds, t ∈ [0, z 2 )(t, •), ϕ) L 2 + A α,a,b (t; (z 1 -z 2 )(t, •), ϕ) = = (h(t, •), ϕ) L 2 , ϕ ∈ H 1 (R n , R), (z 1 -z 2 )(0, •) = 0.

t 0 (z 1 -s 0 (z 1 -

 0101 z 2 )(s, •) 2 H 2 ds ≤ C α, (a, b) Hm t 0 h(s, •) 2 L 2 ds ≤ C α, (a, b) Hm t 0 z 2 )(r, •) 2 H 2 dr ds,and a routine application of Gronwäll's inequality allows us to conclude that z 1 = z 2 .4) Conclusion.Given Theorem 2.1, the regularity of the initial condition p ∈ H 2m+1 (R n , R), and the choice of the basis (v i ) i∈N ⊆ H 2(m+1) (R n , R), one may easily iterate the previous computations by induction to define successive time derivatives of p via iteratively considering formal differentiation-in-time of the variational problem (A

  for almost every t ∈ [0, T ].

					Hence,
	the definition of O a,b t (•) and Theorem 7.5 yield
	(7.8)	O a,b t	g(t, •) = lim k→∞	R n	g k (t, y)p a,b (t, y) dy, for almost every t ∈ [0, T ].
	On the one hand, combined with Theorem 7.7 and Pettis' theorem, (7.8) yields the
	Bochner-measurability of the mapping t ∈ [0, T ] → O a,b t (•) ∈ L 2 (R n , R) * , and there-fore from (7.7) we deduce that O a,b •

  = 1, . . . , n and almost every t ∈ [0, T ]. Therefore, for every ϕ ∈ C ∞ c (R n ; R), i, j = 1, . . . , n, and almost every t ∈ [0, T ], it holds that

										with	∂(a α ij p) ∂y j	(t, •) = p(t, •)	∂a α ij ∂y j	(t, •) + a α ij (t, •)	∂p ∂y j	(t, •),
	for every i, j R n a α ij (t, y)p(t, y)	∂ 2 ϕ ∂y j ∂y i	(y) dy =
							= -	R n	a α ij (t, y)	∂p ∂y j	(t, y) + p(t, y)	∂a α ij ∂y j	(t, y)	∂ϕ ∂y i	(y) dy,
	showing that p solves FPE f with coefficients (a, b) ∈ H + m , α > 0. Next, since p ∈ L 2 (0, T ; V ) and ∂p ∂t ∈ L 2 (0, T ; V * ), from (A.1) we may compute
		d dt	p(t, •) 2 H 2	=	∂p ∂t	(t, •), p(t, •)	V *	= (f (t, •), p(t, •)) H -A α,a,b (t; p(t, •), p(t, •)),
	and therefore Young's inequality yields
	d dt	p(t, •) 2 H 2	≤ f (t, •) H p(t, •) H -	α 2	H ∇p(t, •) 2 + C(a, b)	n i=1 R n	∂p(t, y) ∂y i	p(t, y) dy

  again Theorem 2.1 and (7.4) yield

	-	0	t	n i=1 R n = t 0 i=1 R n   n j=1 n + t 0 n i=1 R n 1 2  ∂a α ij ∂y j  n j=1   n (s, y) -b i (s, y)   ∂ ∂y i 1 2 ∂ 2 a α ij ∂y i ∂y j (s, y) -∂b i ∂y i (s, y) ∂p k ∂s   ∂p k (s, y) p k (s, y) dy ds = (s, y)p k (s, y) dy ds ∂s  j=1 1 2 ∂a α ij ∂y j (s, y) -b i (s, y)  ∂p k ∂y i (s, y) ∂p k ∂s (s, y) dy ds
				≤ C α, (a, b) Hm	0	t	R n	|p k (s, y)| +	n i=1	∂p k ∂y i	(s, y)	∂p k ∂s	(s, y) dy ds
				≤	1 2	0	t	∂p k ∂s	(s, •)	2 L 2	ds + C α, (a, b) Hm	p 2 H 1 +

t 0 f (s, •) 2 L 2 ds , for every t ∈ [0, T ]. By summing up this latter inequality with (A.6) and (A.5), via a routine Granwäll's inequality argument we infer that, for k ∈ N and t ∈ [0, T ], t 0 ∂p k

  •), ψ(t)ϕ , and the equivalence (A.8) readily follows.Next, we prove that p ∈ L 2 (0, T ; H 2 (R n , R)). For this, we first note that, from Theorem 2.1 and the variational problem (A.1), thanks to our previous computations the following variational equality holds true for almost every t ∈ [0, T ]:([0, T ] × R n , R).Therefore, Theorem 2.1 and the classical elliptic regularity theory (see, e.g.,[10, Section 6.3]) imply, for almost every t ∈ [0, T ], both that p(t, •) ∈ H 2 (R n , R) and that (A.11) p(t, •) 2 H 2 ≤ C α, (a, b) Hm p(t, •) 2 L 2 + g(t, •) 2 L 2 .

	(A.9)	R n + i=1 R n n 1 2 i,j=1 n  a α ij (t, y)  n j=1 1 2 ∂a α ∂p ∂y j ij ∂y j	(t, y) (t, y) -b i (t, y) ∂ϕ ∂y i (y) dy   ∂ϕ ∂y i	for ϕ ∈ H 1 (R n , R) R n (y)p(t, y) dy = g(t, y)ϕ(y) dy,
	where
	(A.10)							g f -	∂p ∂t	∈ L 2
															L 2	dt = lim k→∞	0	T	∂p k ∂t	(t, •)ψ(t) dt, ϕ	L 2
		= -lim k→∞	0	T	p k (t, •)	dψ dt	(t) dt, ϕ	L 2	= -lim k→∞	0	T	p k (t, •),	dψ dt	(t)ϕ	L 2	dt
		= -	0	T	p(t, •),	dψ dt	(t)ϕ	L 2	dt = -	0	T	p(t, •)	dψ dt	(t) dt, ϕ	L 2

Appendix A. Proofs of Sections 2 and 7.

A.1. Proofs of Section 2.

Proof of Theorem 2.1. The first and second statements in Theorem 2.1 stem from Morrey theorem as soon as we show the existence of a linear and bounded operator:

for which there exists a constant C > 0 such that: