Diffusion limit for a stochastic kinetic problem

We study the limit of a kinetic evolution equation involving a small parameter and perturbed by a smooth random term which also involves the small parameter. Generalizing the classical method of perturbed test functions, we show the convergence to the solution of a stochastic diffusion equation.


Introduction
Our aim in this work is to develop new tools to study the limit of kinetic equations to fluid models in the presence of randomness. Without noise, this is a thoroughly studied field in the literature. Indeed, kinetic models with small parameters appear in various situations and it is important to understand the limiting equations which are in general much easier to simulate numerically. In this article, we consider the following model problem with initial condition where L is a linear operator (see (3) below). We will study the behavior in the limit ε → 0 of its solution f ε .
In the deterministic case m ε = 0, such a problem occurs in various physical situations: we refer to [DGP00] and references therein. The unknown f ε (t, x, v) is interpreted as a distribution function of particles, having position x and degrees of freedom v at time t. The variable v belongs to a measure space (V, µ) where µ is a non-negative finite measure. The actual velocity is a(v), where a ∈ L ∞ (V ; R d ). The operator L expresses the particle interactions. Here, we consider the most basic interaction operator, given by Note that L is dissipative since In the absence of randomness, the density ρ ε = V f ε dµ converges to the solution of the linear parabolic equation (see section 2.3 for a precise statement): is assumed to be positive definite. We thus have a diffusion limit in the partial differential equation (PDE) sense.
When a random term with the scaling considered here is added to a differential equation, it is classical that, at the limit ε → 0, a stochastic differential equation with time white noise is obtained. This is also called a diffusion limit in the probabilistic language, since the solution of such a stochastic differential equation is generally called a diffusion. Such convergence has been proved initially by Khasminskii [Has66a,Has66b] and then, using the martingale approach and perturbed test functions, in the classical article [PSV77] (see also [EK86], [FGPS07], [Kus84]).
The goal of the present article is twofold. First, we generalize the perturbed test function method to the context of a PDE and develop some tools for that. We believe that they will be of interest for future article dealing with more complex PDEs. Second, we simultaneously take the diffusion limit in the PDE and in the probabilistic sense. This is certainly relevant in a situation where a noise with a correlation in time of the same order as a typical length of the deterministic mechanism is taken. Our main result states that under some assumptions on the random term m, in particular that it satisfies some mixing properties, the density ρ ε = V f ε dµ converges to the solution of the stochastic partial differential equation where K is as above, W is a Wiener process in L 2 (T d ) and the covariance operator Q can be written in terms of m. As is usual in the context of diffusion limit, the stochastic equation involves a Stratonovitch product.
As already mentioned, we use the concept of solution in the martingale sense. This means that the distribution of the process satisfies an equation written in terms of the generator (see section 3.2 for instance). This generator acts on test functions and the perturbed test function method is a clever way to choose the test functions such that one can identify the generator of the limiting equation.
Instead of expanding the solution of the random PDE f ε as is done in a Hilbert development in the PDE theory, we work on the test functions acting on the distributions of the solutions.
In section 2, we set some notations, describe precisely the random driving term, recall the deterministic result and finally state our main result. Section 3 studies the kinetic equation for ε fixed. In section 4, we build the correctors involved in the perturbed test function method and identify the limit generator. Finally, in section 5, we prove our result. We first show a uniform bound on the L 2 norm of the solutions, prove tightness of the distributions of the solutions and pass to the limit in the martingale formulation.
We are not aware of any result on probabilistic diffusion limit using perturbed test functions in the context of PDE. A diffusion limit is obtained for the nonlinear Schrödinger equation in [Mar06], [dBD10], [DT10] but there the driving noise is one dimensional and the solution of the PDE depends continuously on the noise so that in this case an easier argument can be used.

Preliminary and main result 2.1 Notations
We work with PDEs on the torus T d , this means that the space variable x ∈ [0, 1] d and periodic boundary conditions are considered. The variable v belongs to a measure space (V, µ) where µ is a probability measure. We shall write for simplicity L 2 x,v instead of L 2 (T d ×V, dx⊗dµ), its scalar product being denoted by (·, ·). We use the same notation for the scalar product of L 2 (T d ); note that this is consistent since µ(V ) = 1. Similarly, we denote by u L 2 the norm (u, u) 1/2 , wether u ∈ L 2 x,v or L 2 (T d ). We use the Sobolev spaces on the torus H γ (T d ). For γ ∈ N, they consist of periodic function which are in L 2 (T d ) as well as their derivatives up to order γ. For general γ ≥ 0, they are easily defined by Fourier series for instance. For γ < 0, H γ (T d ) is the dual of H −γ (T d ). Classically, for γ 1 > γ 2 , the injection of H γ1 (T d ) in H γ2 (T d ) is compact. We use also L ∞ (T d ) and W 1,∞ (T d ), the subspace of L ∞ (T d ) of functions with derivatives in L ∞ (T d ). Finally, L 2 (V ; H 1 (T d )) is the space of functions f of v and x such that all derivatives with respect to x are in L 2 (T d ) and the square of the norm is finite.

The driving random term
The random term m ε has the scaling where m is a stationary process on a probability space (Ω, F , P) and is adapted to a filtration (F t ) t∈R . Note that m ε is adapted to the filtration (F ε t ) t∈R , with Our basic assumption is that, considered as a random process with values in a space of spatially dependent functions, m is a stationary homogeneous Markov process taking values in a subset E of W 1,∞ (T d ). We assume that m is stochastically continuous. Note that m is supposed not to depend on the variable v.
The law ν of m(t) is supposed to be centered: In fact, we also assume that m is uniformly bounded in W 1,∞ (T d ) so that E is included in a ball of W 1,∞ (T d ). We denote by (P t ) t≥0 a transition semigroup on E associated to m and by M its infinitesimal generator. As is usual in the context of diffusion limit, we use the notion of solution of the martingale problem and need mixing properties on m. We assume that there is a subset D M of C b (E), the space of bounded continuous functions on E, such that, for every ψ ∈ D M , M ψ is well defined and is a continuous and integrable martingale. Moreover, we suppose that m is ergodic and satisfies some mixing properties in the sense that there exists a subset P M of C b (E) such that for any θ ∈ P M the Poisson equation has a unique solution ϕ ∈ D M satisfying E ϕdν = 0 provided E θdν = 0.
We denote by M −1 θ this solution and assume that it is given by: In particular, we assume that the above integral is well defined. We need that P M contains sufficiently many functions. In particular, we assume that for each x ∈ T d , the evaluation function ψ x defined by ψ x (n) = n(x), n ∈ E, is in P M . Also, we assume that, for any f, g ∈ L 2 x,v , the function θ f,g : n → (f, ng) is in We need that M −1 I takes values in a ball of W 1,∞ (T d ) and take C n large enough so that for all n ∈ E. It is natural to require the following compatibility assumption, which would follow from continuity properties of M −1 : Note that by (5), θ f,g and ψ x satisfy the centering condition (7). We will also assume that for any f, g ∈ L 2 x,v , the function is in P M .
To describe the limit equation, we remark that since m(0) has law ν, Note that, m being stationary, so that k is symmetric. Let Q be the linear operator on L 2 (T d ) associated to the kernel k: Then Q is a positive operator. Indeed, for f ∈ L 2 (T d ), denoting by ψ f the map and, by the mixing properties of m, Besides, Q is also self-adjoint and compact since k is symmetric and bounded. We can therefore define its square root Q 1/2 . Note that Q 1/2 is Hilbert-Schmidt on L 2 (T d ) and, denoting by Q 1/2 L2 its Hilbert-Schmidt norm, we have We will not analyze here in detail which kind of processes satisfies our assumptions. The requirement (8) that m and M −1 m are a.s. bounded in W ∞ (T d ) are quite strong. An example of process we may consider is where the processes (m j ) j∈N are independent real valued centered stationary, satisfying the bound |m j (t)| ≤ C, a.s., t ∈ R, for a given C > 0. We are then reduced to analysis on a product space. The invariant measure of m is then easily constructed from the invariant measures of the m j 's. Also, the Poisson equation can be solved provided each Poisson equation associated to m j can be solved. This can easily be seen by working first on functions ψ depending only on a finite number of j.
The precise description of the sets D M and P M depends on the specific processes m j , j ∈ N. For instance, if m j are Poisson processes taking values in finite sets S j , then D M and P M can be taken as the set of bounded functions on i∈N S j . More general Poisson processes could be considered (see [FGPS07]). Actually, the hypothesis (8) can be slightly relaxed. The boundedness assumption is used two times. First, in the proof of (13) and (14), but there it would be sufficient to know that m has finite exponential moments. It is used in a more essential way in Proposition 8. There, we need that the square of the norm of m and M −1 m have some exponential moments. However, (under suitable assumptions on the variance of the processes for example), we may consider driving random terms given by Gaussian processes, or more generally diffusion processes.

The deterministic equation
There are also some structure hypotheses on the first and second moments of and suppose that the following symmetric matrix is definite positive: An example of (V, µ, a) satisfying the hypotheses above is given by In the deterministic case m = 0, the limit problem when ε → 0 is a diffusion equation, as asserted in the following theorem.
Assume (9)-(10). Then the density ρ ε := V f ε dµ converges in weak-L 2 t,x to the solution ρ to the diffusion equation This result is a contained is [DGP00] where a more general diffusive limit is analyzed. Note that, actually, strong convergence of (ρ ε ) can be proved by using compensated compactness, see [DGP00] also.

Main result
In our context, the limit of the Problem (1)-(2) is a stochastic diffusion equation.
Theorem 2 (Diffusion Limit in the stochastic case). Assume that (f ε 0 ) is bounded in L 2 x,v and that Assume (5) with initial condition: It is not difficult to see that formally, (11) is the Itô form of the Stratonovitch equation Theorem 2 remains true in the slightly more general situation where the coefficient in factor of the noise in (1) whereσ is a smooth, sublinear function.
3 Resolution of the kinetic Cauchy Problem
Since A is closed and densely defined, by the Hille-Yosida Theorem [CH98], it defines a unitary group e tA on L 2 ). The proof of this result is not difficult and left to the reader. The last statement is easily obtained since A commutes with derivatives with respect to x. Energy estimates can be obtained. Indeed, for smooth integrable solutions f ε to (1)-(2), we have the a priori estimate By (4) and (8), this gives the bound hence, by Gronwall's Lemma, the following bound (depending on ε): Similarly, we have It is sufficient to assume f ε 0 ∈ L 2 (V ; H 1 (T d )) (resp. f ε 0 ∈ L 2 (V ; H 2 (T d ))) to prove (13) (resp. (14)). By density, the inequality holds true for

Generator
The process f ε is not Markov but the couple (f ε , m ε ) is. Its infinitesimal generator is given by : These are differential operators with respect to the variables f ∈ L 2 x,v , n ∈ E. Here and in the following, D denotes differentiation with respect to f and we identify the differential with the gradient. For a C 2 function on L 2 x,v , we also use the second differential D 2 ϕ of a function ϕ, it is a bilinear form and we sometimes identify it with with a bilinear operator on L 2 x,v , by the formula: Let us define a set of test functions for the martingale problem associated to the generator L ε .
Definition 4. We say that Ψ is a good test function if x,v and maps bounded sets onto bounded sets is continuous from L 2 (V ; H 1 (T d ))×E to R and maps bounded sets onto bounded sets of R • for any f ∈ L 2 (V ; H 1 (T d )), Ψ(f, ·) ∈ D M .
We have the following result.
Proposition 5. Let Ψ be a good test function.Then for any f ε 0 ∈ L 2 (V ; H 1 (T d )), is a continuous and integrable (F ε t ) martingale. Proof: Let s, t ≥ 0 and let s = t 1 < · · · < t n = t be a subdivision of [s, t] such that max i |t i+1 − t i | = δ. We have for any F ε s measurable and bounded g hal-00576610, version 1 -14 Mar 2011 We write Since f ε 0 ∈ L 2 (V ; H 1 (T d )), we deduce from (14) and the assumption on Ψ that a δ is uniformly integrable with respect to (s, ω). Also f ε is almost surely continuous and m ε is stochastically continuous. It follows that DΨ(f ε (s), m ε (t i+1 )) − DΨ(f ε (s), m ε (s)) converges to 0 in probability when δ goes to zero for any s. By uniform integrability, we deduce that A converges to 0. Similarly, we have and, by the same argument, B converges to zero when δ goes to zero. The result follows.

The limit generator
To prove the convergence of (ρ ε ), we use the method of the perturbed testfunction [PSV77]. The method of [PSV77] has two steps: first construct a corrector ϕ ε to ϕ so that L ε ϕ ε is controlled, then, in a second step, use this with particular test-functions to show the tightness of (ρ ε ). In the first step, we are led to identify the limit generator acting on ϕ.

Correctors
In this section, we try to understand the limit equation at ε → 0. To that purpose, we investigate the limit of the generator L ε by the method of perturbed test-function.
We restrict our study to smooth test functions and introduce the following class of functions. Let ϕ ∈ C 3 (L 2 x,v ). We say that ϕ is regularizing and subquadratic if there exists a constant C ϕ ≥ 0 such that for all f, h, k, l ∈ L 2 x,v , for all m, m i ∈ {0, · · · , 3}, i = {1, 2, 3}. Note that regularizing and subquadratic functions define good test functions (depending on f only).
Given ϕ regularizing and subquadratic, we want to construct ϕ 1 , ϕ 2 good test functions, such that The limit generator L is to be determined. By the decomposition (15), this is equivalent to the system of equations (17d)

Order ε −2
Equation (17a) constrains ϕ to depends onf = V f dµ uniquely: and imposes that the limit generator L acts on ϕ(f ) uniquely, as expected in the diffusive limit, in which we obtain an equation on the unknown V f dµ. Indeed, since ϕ is independent on n, (17a) reads (Lf, Dϕ(f )) = 0.

Order ε −1
Let us now solve the second equation (17b). To that purpose, we need to invert L L * . Let us work formally in a first step to derive a solution. Assume that m(t, n) is a Markov process with generator M and define the Markov process (g(t, f ), m(t, n)). Its generator is precisely L L * . Denote by (Q t ) t≥0 its transition semigroup. Since both g and m satisfy mixing properties, the couple (g, m) also. In particular, we have and it is expected that, under the necessary condition L A * ϕ = 0, a solution to (17b) is given by Let us now compute L A * ϕ. By (18), we have for h ∈ L 2 x,v , (h, Dϕ(f )) = (h, Dϕ(f )), where as above the upper bar denotes the average with respect to v. Hence L A * ϕ = −(Af , Dϕ(f )) + (f n, Dϕ(f )).
Since the first moments of a(v) and m(t) vanish, hence L A * ϕ = 0. We then write f ), m(t, n))) dt.
Note that g is deterministic andḡ =f , so that Furthermore, regarding the term Ag(t, f ), we have Ag(t, f ) = e −t Af .

By (18), this is equivalent to
This computation is formal but it is now easy to define ϕ 1 by (22) and to check that it satisfies (17b). It is also clear that ϕ 1 is a good test function.
With respect to ϕ, L A * ϕ 1 can be decomposed as the sum of a first-order and a second-order operator: with (l, Dϕ(f )) = ((−A + M −1 I(n))(−Af + f n), Dϕ(f )), Alternatively, using (18), we have (l, Dϕ(f )) = ((−A + M −1 I(n))(−Af + f n), Dϕ(f )), By the cancellation properties (9) and (5), using the same notation as above, we obtain where, we recall, the average ψ is defined by (21). Let A denote the unbounded operator where K is defined in (10). We define the generator L , acting on functions of ρ ∈ L 2 (T d ), as follows L ψ(ρ) =(Aρ, Dψ(ρ)) Observe that, by definition of K, A 2 ρ = Aρ for all ρ ∈ H 2 (T d ), therefore L A * ϕ 1 (f ) = L ϕ(f ). Then, (17c) is formally satisfied if we set We now develop (27) to obtain an explicit candidate for ϕ 2 . We first check that Using Ag(t, f ) = e −t Af , t ≥ 0, and the decomposition (24), we obtain ϕ 2 (f, n) as the sum of two terms: and Then we observe that, for ψ satisfying ψ = 0, Therefore, recalling (f M −1 I(n), g) = M −1 (f n, g), (28) and (29) are simplified into and respectively. By (8), and (16), we prove that ϕ 2 (f, n) grows at most quadratically with respect to f L 2 . It is easy to check that ϕ 2 is a good test function and to control L A * ϕ 2 . We obtain the following result.

Limit equation
Let us develop the expression (32) obtained for the limit generator L . We have: To recognize the part containing D 2 ϕ, we identify D 2 ϕ with its Hessian and first assume that it is associated to a kernel Φ. Then, we write: Denote by q the kernel of Q 1/2 , then which gives E D 2 ϕ(ρ) · (ρM −1 n, ρn)dν(n) By approximation, this formula holds for all C 2 function ϕ. We conclude that L is the generator associated to the stochastic PDE dρ = div(K∇ρ)dt + 1 2 where W is a cylindrical Wiener process.

Diffusive limit
Our aim now is to prove the convergence in law of ρ ε = V f ε dµ to ρ, solution to (12), or equivalently Equation (33). To that purpose, we use again the perturbed test function method to get a bound on the solutions in L 2 x,v then we prove that ρ ε is tight in C([0, T ]; H −η ), η > 0.

Bound in L 2
x,v Proposition 8 (Uniform L 2 x,v bound). Assume (8). Then, for all T > 0, there exists a constant C depending on T and sup ε>0 f ε 0 L 2 only such that, for all t ∈ [0, T ], E f ε (t) L 2 ≤ C.
Proof: Set ϕ(f ) := 1 2 f 2 L 2 . We seek for one corrector ϕ 1 ∈ C 2 (L 2 x,v × E) such that, for the modified test-function ϕ ε := ϕ + εϕ 1 , the term L ε ϕ ε (f ε , m ε ) can be accurately controlled. We compute, for f ∈ L 2 (V ; H 1 (T d )), n ∈ E, Since M ϕ(f, n) = 0 (ϕ being independent on n), and since Dϕ(f, n) = f , the first term in (34) is which has a favorable sign. Since A is skew-symmetric, L A * ϕ = (f n, f ). This term is difficult to control and we choose ϕ 1 to compensate it. We set so that M ϕ 1 = −(f n, f ) and the second term in (34) is By (8), we obtain The remainder L A * ϕ 1 satisfies the following bounds By (8), (35), (36), we obtain for a given constant depending on C n but not on ε. Note that Ψ = ϕ ε is a good test function so that by Proposition 5 we have Since ϕ 1 (f, n) = (f M −1 I(n), f ) and ϕ ε = ϕ + εϕ 1 , we deduce, for ε small enough, where C is a (possibly different) constant depending also on C n . By Gronwall's Lemma, we obtain a bound independent on ε on E f ε (t) 2 L 2 . By Proposition 5, Proposition 6 and Proposition 7, we deduce: x,v ) be a regularizing and subquadratic function satisfying (18). There exist two good test functions ϕ 1 , ϕ 2 such that, defining ϕ ε = ϕ + εϕ 1 + ε 2 ϕ 2 , for all f ∈ L 2 x,v , n ∈ E, where C is a constant depending on the constant C n in (8) and C ϕ . Moreover is a continuous integrable martingale for the filtration (F ε t ) generated by m ε . Finally, for 0 ≤ s 1 ≤ · · · ≤ s n ≤ s ≤ t and ψ ∈ C b ((L 2 x,v ) n ), with another constant C depending on the constant C n in (8), on C ϕ and on the supremum of ψ.
Proof: Everything has already been proved except for the last statement. It suffices to write: Then, we multiply by ψ(ρ ε (s 1 ), . . . , ρ ε (s n )) take the expectation, use the bounds written above and finally Proposition 8 to conclude.
where Id is the identity on L 2 (T d ). Since Let 0 ≤ s ≤ t. To get an estimate on the H −γ -norm we expand the square to obtain Let {p j ; j ≥ 1} be a complete orthonormal system in L 2 (T d ). Since ρ ε (s) is F s -measurable, we have By (38), we obtain where ψ(ρ) := 1 2 J γ ρ 2 L 2 , l j (ρ) := (p j , J γ ρ). It is clear that ψ and l j are subquadratic and regularizing as in (16). For example, for all m 1 , m 2 , m 3 ∈ {0, . . . , 3} since a ∈ L ∞ (V, µ) and J 3 ∇ 3 is an operator of order lower than 0. We can therefore use Corollary 9 to get a correction on L ε ϕ(f ε , m ε ) for ϕ = ψ or l j . In particular, where the constant C may depends on T , C n , C ψ (the constant in (16) for ψ) and sup ε>0 f ε 0 L 2 . Similarly, where C is as above and also independent on j, as easily checked. By (32) and the analysis of L in Section 4.2, we have where C is as above and independent on j. By (39) and the choice of γ, we obtain (37).
By Prokhorov's Theorem [Bil99], there is a subsequence still denoted by (ρ ε ) and a probability measure µ on C([0, T ]; H −η (T d )) such that the law µ ε of ρ ε converges to µ weakly on C([0, T ]; H −η (T d )). We then show that µ is a solution of the martingale problem, with a set of test functions precised below, associated to the limit equation (11).
Thanks to Proposition 8, we can approximate every subquadratic and regularizing functions by functions in C b (H −η (T d )) which are subquadratic and regularizing with a uniform constant in (16) and which converge pointwise, and prove that (42) holds true for every subquadratic and regularizing functions.
We deduce that is a martingale with quadratic variation t 0 ρ(s)Q 1/2 ρ(s)Q 1/2 * ds. Thanks to martingale representation (see for instance [DPZ92]), up to a change of probability space, there exist a cylindrical Wiener process W such that It is well known that this equation has a unique solution with paths in the space C([0, T ]; H −η (R d )). This can be shown for instance by energy estimates using Itô formula after a suitable regularization argument. Moreover pathwise uniqueness implies uniqueness in law and we deduce that µ is the law of this solution and is uniquely determined. Finally, the whole sequence (µ ε ) converges to µ weakly in the space of probability measures on C([0, T ]; H −η (R d )).