Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilitistic interpretation

ESAIM: Probability and Statistics

January 2004, Vol. 8, p. 36–55 DOI: 10.1051/ps:2003018

POINTWISE CONVERGENCE OF BOLTZMANN SOLUTIONS FOR GRAZING COLLISIONS IN A MAXWELL GAS VIA A PROBABILITISTIC

INTERPRETATION

H´ el` ene Gu´ erin

Abstract. Using probabilistic tools, this work states a pointwise convergence of function solutions of the 2-dimensional Boltzmann equation to the function solution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results of Fournier (2000) on the Malliavin calculus for the Boltzmann equation. Moreover, using the particle system introduced by Gu´erin and M´el´eard (2003), some simulations of the solution of the Landau equation will be given.

This result is original and has not been obtained for the moment by analytical methods.

Mathematics Subject Classification. 60J75, 60H10, 60H07, 82C40.

Received June 4, 2002. Revised November 27, 2002.

1. Introduction

The Boltzmann equation [5, 6] describes the behaviour of particles in a rarefied gas. More precisely, it describes in dimension 2 the behaviour of the density f(t, v, x) of particles having the velocityv ∈R² at time t ≥0 and at point x ∈ R². We consider in this work the spatially homogenous case, which means that the density does not depend on the position x of particles. In 1936, Landau [19] derived from the Boltzmann equation a new equation called the Fokker-Planck-Landau equation, usually considered as an approximation of the homogeneous Boltzmann equation in the limit of grazing collisions. These equations take the form

∂f

∂t =Q(f, f) (1.1)

where Qis a quadratic operator depending on the nature of the collisions. In this paper, we consider the case of a Maxwell gas in dimension 2. Then the Boltzmann equation writes

∂f

∂t =Q_B(f, f) (BE)

Keywords and phrases.Boltzmann equation without cutoff for a Maxwell gas, Landau equation for a Maxwell gas, nonlinear stochastic differential equations, Malliavin calculus.

1 Universit´e Paris 10, UFR SEGMI, Modal’X, 200 avenue de la R´epublique, 92000 Nanterre, France;

e-mail:[email protected]

c EDP Sciences, SMAI 2004

with a collision operatorQ_B given by Q_B(f, f)(t, v) =

v∗∈R²

_π

θ=−π(f(t, v)f(t, v_∗)−f(t, v)f(t, v_∗))β(θ) dθdv_∗

wherev, v_∗ are the pre-collisional velocities andv, v_∗the post-collisional velocities and where the cross-section β is an even positive function from [−π, π]\{0}toR⁺ such that_π

−πθ²β(θ)dθ <∞.

The relation between the post-collisional velocities and the pre-collisional velocities in dimension 2 is the following

v=v+A(θ)(v−v_∗) ; v_∗ =v−A(θ)(v−v_∗) with

A(θ) = 1 2

cosθ−1 −sinθ sinθ cosθ−1

.

We are interested in cases for which the molecules in the gas interact according to an inverse power law in _d¹s

withs≥2, wheredis the distance between particles. Consequently, the functionβ has a singularity in 0 of the formβ(θ) ∼

θ→0Cθ^−s−1s+1, withC a positive constant. We assume that

Assumption (A): β is an even positive function on [−π, π]\{0}of the formβ=β₀+β₁ such that 1) β₁is an even and positive function on [−π, π];

2) there existk₀>0,θ₀∈(0, π) andr∈(1,3) such thatβ₀(θ) = k₀

|θ|^rI_[−θ0,θ0](θ).

The second equation we consider is the Landau equation:

∂f

∂t =Q_L(f, f) (LE)

with the collision operatorQ_Ldefined by Q_L(f, f) =1

2 2 i,j=1

∂

∂v_i

R²dv_∗a_ij(v−v_∗)

f(t, v_∗) ∂f

∂v_j (t, v)−f(t, v) ∂f

∂v_∗j(t, v_∗) witha= (a_ij)_1≤i,j≤2 a nonnegative symmetric matrix of the form in the Maxwell case

a(z) = Λ|z|²Π (z) (1.2)

where Π (z) is the orthogonal projection on (z)^⊥ and Λ is a positive constant precised below.

Many authors have been interested in proving rigorously the convergence of Boltzmann to Landau, in differ- ent cases of scattering cross-section and initial data. Firstly Arsen’ev and Buryak [2] proved the convergence of solutions of the Boltzmann equation towards solutions of the Landau equation under very restrictive assump- tions. Desvillettes [8] gave a mathematical framework for more physical situations, but excluding the case of Coulomb potential which has been studied by Degond and Lucquin [7]. Degond and Lucquin stated an asymp- totic development of the Boltzmann kernel when the collisions become grazing. Then, Goudon [12] and Villani [23] proved in two independent works the existence of a solution of the Landau equation for soft potentials using the asymptotic of grazing collisions, with a bounded entropy and energy function as initial data. More recently, Gu´erin and M´el´eard [16] proved the convergence of solutions of the Boltzmann equation to a solution of the Landau equation for ‘moderately soft’ potentials with a probabilistic representation when the initial data is a probability measure with a finite fourth-order moment. All those works prove an L¹-weak convergence of the solutions. Alexandre and Villani [1] stated in a recent work a strong convergence inL^p for some soft potentials including the case of a Coulomb gas.

The aim of this paper is to prove a pointwise convergence of function-solutions of the Boltzmann equation to the function-solution of the Landau equation onR²for a Maxwell gas, which is unknown by analytical methods.

We recall that in the case of Maxwell molecules, there is uniqueness of the solution of the Landau equation (see for example [15], Cor. 7). Fournier [10] and Gu´erin [15] proved respectively from probability measure solutions the existence of weak function solutions of the Boltzmann equation and of the Landau equation when the initial data is not a Dirac measure. To this aim, they used an efficient probabilistic tool: the Malliavin calculus for processes with jumps in [10] and the Malliavin calculus for white noises in [15]. From the result of Gu´erin and M´el´eard in [16] on the convergence of the probability measure solutions following the asymptotic of grazing collisions, it seems to be natural to study the convergence of function solutions.

In the asymptotics of grazing collisions, we only consider collisions with an infinitesimal angle of deviation.

To this aim, we renormalize the cross-section β of the Boltzmann equation to concentrate on such collisions.

We use the approximation introduced by Desvillettes [8]: for any ε > 0, let β^ε be the function defined on [−επ, επ]\{0}by

β^ε(θ) = 1 ε³β

θ ε

(1.3) We notice that the mass of the functionβ^εconcentrates on the values ofθnear 0 whenεtends to 0,i.e. when the collisions become grazing, in the following sense:

for anyθ₀>0,β^ε(θ)−→

ε→00 uniformly onθ≥θ₀ (1.4)

and _επ

−επsin θ

2 ₂

β^ε(θ) dθ−→

ε→0Λ (1.5)

where Λ = ¹_2π

0 θ²β(θ)dθ > 0 is the constant appearing in the expression (1.2) of the matrix a. This asymp- totic (1.3) is a particular case of the one introduced by Villani in [23], and used by Gu´erin and M´el´eard in [16].

We prove here the following theorem:

Theorem 1.1. Let β be an even function on [−π, π]\{0} satisfying Assumption (A). Assume that the initial dataP₀ is a probability measure with finite moments of all orders andP₀ is not a Dirac mass.

We define β^ε(θ) = ε⁻³β(θ/ε) and we denote by f^ε the function-solution of the Boltzmann equation (BE) associated with the cross-sectionβ^ε (obtained by Fournier in [10]). The functionf^ε is of class C^∞ onR² ([10], Th. 3.2).

Then the sequence (f^ε(t, .))_ε>0 is pointwise convergent on R² asε tends to0 for any t >0 and the limiting functionf is the function-solution of the Landau equation. Moreover,f(t, .)is of classC^∞and there is pointwise convergence of derivatives of any orders.

This theorem states a strong convergence result of solutions of the Boltzmann equation to the solution of Landau equation for a Maxwell gas when the collisions become grazing. Goudon [12] and Villani [23] proved L¹ -weak convergence, but in the more general case of soft potentials and in dimension 3. It seems that their methods can not give a stronger result.

Theorem 1.1 gives a new proof of the existence of regular function-solution for the Landau equation via a probabilistic approach.

We have to restrict our study to the dimension 2 because of the nonregularity of the Boltzmann coefficients in R³ (see [11], Lem. 2.6). Fournier [10] built the functionsf^εusing the Fourier transforms of the probability measure solutions. Consequently, since the Boltzmann measure-solutions converge, it suffices to prove that their Fourier transforms are uniformly bounded by integrable functions on R², when the collisions become grazing to obtain the convergence of the function-solutions. The proof is based upon a careful study of the results of Fournier [10] (the details of the proof are given in Sect. 4).

In the last part of this paper, we use the Monte-Carlo algorithm following the asymptotic of grazing collisions developed by Gu´erin and M´el´eard in [16]. We firstly simulate the convergence of solutions of the Boltzmann

equation to the solution of the Landau equation for a degenerate initial distribution, and then we observe the behaviour in time of the solution of the Landau equation and of its entropy.

Notations

- DT will denote the Skorohod spaceD([0, T],R²) of cadlag functions from [0, T] intoR². - C_b²(R²) is the space of real bounded functions of classC²with bounded derivatives.

- M₂(R) is the set of matrices of order 2×2. The matrix A^∗ is the adjoint of the matrix A and the matrixIdenotes the identity matrix inM₂(R).

- The bracket., .denotes the scalar product inR².

2. Some definitions

Letβ be defined by Assumption (A) and β^εbe defined by (1.3). We define the Boltzmann equation (BE^ε) associated with the cross-sectionβ^ε:

∂f

∂t =Q_Bε(f, f) (BE^ε)

with

Q_B^ε(f, f)(t, v) =

v∗∈R²

_π

θ=−π(f(t, v)f(t, v_∗)−f(t, v)f(t, v_∗))β^ε(θ) dθdv_∗.

The collision operators of the Boltzmann and the Landau equations preserve momentum and kinetic energy.

Equations of the form (1.1) have to be understood in a weak sense, i.e. f is a solution of the equation if for any test functionsφ,

d dt

R²φ(v)f(t, v)dv =

R²φ(v)Q(f, f)(t, v)dv.

As detailed for example in [10], a standard integration by parts and a compensation due to the bad integrability behaviour of β^εyield to the definition of a function-solution of the Boltzmann equation:

Definition 2.1. Letε >0 be fixed. A function-solution of (BE^ε) is a functionf^εsatisfying for anyφ∈C_b²(R²) the equation

d dt

R²f^ε(t, v)φ(v)dv=

R²×R²K_β^φε(v, v_∗)f^ε(t, v)dvf^ε(t, v_∗)dv_∗ (2.1) whereK_β^φε is defined by

K_β^φε(v, v_∗) = −b^ε∇φ(v).(v−v_∗) +

_επ

−επ

φ(v+A(θ) (v−v_∗))−φ(v)−A(θ) (v−v_∗).∇φ(v)

β^ε(θ) dθ (2.2) withb^ε= ¹_2επ

−επ(1−cosθ)β^ε(θ)dθ.

Using the conservation of the mass in (2.1), we introduce a definition of probability measure solutions of (BE^ε):

Definition 2.2. Letε >0 be fixed. LetP₀be a probability measure with a finite 2-order moment. A measure family (P_t^ε)_t≥0 is a measure-solution of (BE^ε) if it satisfies for anyφ∈ C_b²

R²

R²φ(v)P_t^ε(dv) =

R²φ(v)P₀(dv) + _t

0

R²×R²K_β^φε(v, v_∗)P_s^ε(dv)P_s^ε(dv_∗) ds. (2.3)

In the same way, we give the following definition of a function-solution for the Landau equation:

Definition 2.3. A functionf is a function-solution of (LE) iff satisfies for eachφ∈C_b²(R²) d

dt

R²f(t, v)φ(v) dv=

R²×R²L^φ(v, v_∗)f(t, v)dvf(t, v_∗) dv (2.4) whereL^φis the Landau kernel defined on R²×R² by:

L^φ(v, v_∗) =1 2

2 i,j=1

∂_ij²φ(v)a_ij(v−v_∗) + 2 i=1

∂_iφ(v)b_i(v−v_∗)

withb_i(z) = 2 j=1

∂_ja_ij(z) =−Λzi.

We also state a definition of measure-solutions of (LE) as in Definition 2.2.

We notice that the Boltzmann kernelK_β^φε is pointwise convergent onR²×R²to the Landau kernelL^φwhen εtends to 0 for anyφ∈C_b²

R²

(see for example [12] or [23]).

3. The convergence of the function-solutions

We give in this section the main idea of the proof of Theorem 1.1.

In all the following, P₀ is assumed to be a probability measure with a finite two-order moment and β a positive even function on [−π, π]\{0}satisfying Assumption (A).

In the probabilistic study of the Boltzmann equation, we consider in fact (2.3) as the evolution equation of the family of the time marginals of a jump process. The distribution of this process will be solution of the following nonlinear martingale problem:

Definition 3.1. Let ε > 0 be fixed. We say that a probability measure P^ε on DT solves the nonlinear martingale problem (M P^ε) starting atP₀ if forX the canonical process underP^ε, the law ofX₀ isP₀ and for anyφ∈ C²_b

R² ,

φ(X_t)−φ(X₀)− _t

0

R²K_β^φε(X_s, v_∗)P_s^ε(dv_∗) ds (3.1) is a square-integrable martingale, whereP_s^εis the marginal ofP^εat time s.

Taking expectation in (3.1), we notice that ifP^εis a solution of (M P^ε),then (P_t^ε)_t≥0 is a measure-solution of (BE^ε).

Fournier proved in [10] the existence of a solutionP^εof (M P^ε) for anyε >0. Moreover, Gu´erin and M´el´eard in [16] stated the tightness of the sequence (P^ε)_ε>0 when the collisions become grazing (ε →0) in the more general case of soft potentials and in dimension 3 (using the same arguments, the convergence theorem is still true in dimension 2). In the particular case of Maxwellian molecules, there is convergence of the sequence (P^ε)_ε>0 to the measure-solution of the Landau equation (LE) thanks to the uniqueness of this solution (see [15], Cor. 7). We will use those results under the following form:

Theorem 3.2. Letβ^ε=ε⁻³β(θ/ε). For anyε >0, there exists a solutionP^εof the martingale problem(M P^ε).

Moreover, the sequence (P_t^ε)_ε>0 converges as εgoes to 0 to a distribution P_t which is the measure-solution of the Landau equation.

Let us remark that to obtain a function-solution from a measure-solution (P_t^ε)_t≥0, it suffices to prove that

∀t >0P_t^εadmits a densityf^ε(t, .) with respect to the Lebesgue measure onR². Then the functionf^εsatisfies Definition 2.1. Fournier [10] stated the following theorem using the Malliavin calculus for processes with jumps:

Theorem 3.3. Let ε∈(0,1) be fixed. Assume thatP₀ is not a Dirac measure.

1) The Boltzmann equation(BE^ε)admits a function-solution f^ε with initial dataP₀.

2) If P₀ belongs toL^p for any p≥1, then for any t >0, for any couple α= (α₁, α₂)∈ N², there exists a constant C_t,α^ε such that the following inequality holds for all ϕ∈ C^∞

R²

with compact support

R²∂_αϕ(v)P_t^ε(dv)

≤C_t,α^ε ϕ_∞ (3.2)

where ∂_α denotes the partial derivative _∂α^∂1^αx¹⁺₁∂^αα22x₂. Consequently, the function-solutionf^ε is infinitely differ- entiable on R² and is given by:

f^ε(t, v) = 1 4π²

R²

Pˆ_t^ε(x) e−i<v,x>dx wherePˆ_t^ε is the Fourier transform ofP_t^ε.

We want to state the convergence of the function-solutionsf^ε of the Boltzmann equation (BE^ε) when the grazing collisions prevail.

Thanks to the convergence of measure-solutions (P_t^ε)_t≥0of the Boltzmann equation to the measure-solution (P_t)_t≥0of the Landau equation (see Th. 3.2), the sequence

Pˆ_t^ε

ε>0is pointwise convergent onR²to the Fourier transform ˆP_t ofP_t, for anyt≥0.

Approximating the functions ϕ(v) = ei<v,x> withx= (x₁, x₂)∈R², by compact support functions of class C^∞, we obtain from inequality (3.2) that∀x∈R² and∀α₁, α₂≥2

Pˆ_t^ε(x)≤inf

1, C_t,(α^ε _1,α2)

|x₁|^α1|x₂|^α2 ·

Thus if we prove that the constantsC_t,α^ε are uniformly bounded inεby a constantC_t,αfor anyα∈N², using the Lebesgue theorem, we easily deduce that the function-solutionsf^ε(t, v) (and its derivatives of any orders) of the Boltzmann equation converge asεgoes to 0 to the function-solutionf(t, v) =

R²Pˆ_t(x) ei<v,x>dx(respectively, its derivatives) of the Landau equation (obtained in [15]) for anyv∈R²andt >0. Consequently the theorem will be proved.

4. The proof of Theorem 1.1

We assume from now without restriction thatε∈(0,1/2].

To state that the constantsC_t,α^ε appearing in (3.2) are uniformly bounded inε, we have to study the proof of Theorem 3.3. Fournier [10] proved the existence of function-solutions by the mean of a nonlinear stochastic differential equation giving a pathwise version of the probabilistic interpretation.

4.1. The Pathwise approach

Letε >0 be fixed,P₀be a probability measure with a finite 2-order moment andβ satisfy Assumption (A).

Let us consider two probability spaces to highlight the nonlinearity of the equation: the first one is the abstract space (Ω,F,{Ft}_t∈[0,T], P) and the second one is ([0,1],B([0,1]),dα). The processes on ([0,1],B([0,1]),dα) will be calledα-processes, the expectation underdαwill be denoted byE_α and the laws byLα.

On (Ω,F, P) we consider a Poisson measureN^ε(dθ,dα,dt) on [−π, π]×[0,1]×[0, T] with intensity measure ν^ε(dθ,dα,dt) =β^ε(θ) dθdαdt and with compensated measure ˜N^ε(dθ,dα,dt).

Theorem 4.1. (see [10], Th. 2.8) Let V₀ be a random variable with distribution P₀. There exists a couple of processes (V^ε, W^ε)on Ω×[0,1]satisfying the nonlinear stochastic differential equation (SDE^ε):

V_t^ε = V₀+ _t

0

₁

0

_επ

−επA(θ)(V_s−^ε −W_s−^ε (α)) ˜N^ε(ds,dα,dθ)−b^ε _t

0

₁

0

V_s−^ε −W_s−^ε (α) dαds with L(V^ε) =Lα(W^ε) =P^ε.

Moreover E[ sup

0≤t≤T|V_t^ε|²] =E_α[ sup

0≤t≤T|W_t^ε|²]<∞. There is uniqueness in law of P^ε.

Corollary 4.2. Thanks to Itˆo’s formula, the measureP^ε is also a solution of the martingale problem (M P^ε).

Consequently,(P_t^ε)_t≥0 is a measure-solution of the Boltzmann equation for Maxwellian molecules.

Moreover we easily prove (see [16], Sect. 3.3):

Lemma 4.3. Assume that V₀ is a random vector in R² belonging to L^p for any p ≥ 1. Then for any T >

0, p≥1, there exists a constantK_p independent of εsuch that E[ sup

0≤t≤T|V_t^ε|^p] =E_α[ sup

0≤t≤T|W_t^ε|^p]≤K_p. (4.1)

Using the Malliavin calculus for a stochastic differential equation driven by a Poisson process, Fournier [10]

proved that each time-marginalP_t^εsatisfies (3.2) for anyt >0 and the coefficientsC_t,α^ε depend on the Malliavin derivatives of V^ε. Consequently, to controlC_t,α^ε we have to estimate the Malliavin’s derivatives.

4.2. Some recalls on the Malliavin calculus

The Malliavin calculus in the case of a stochastic differential equation driven by a Poisson process, also called the stochastic calculus of variations, has been adapted to the case of the Boltzmann equation by Graham and M´el´eard [13] and Fournier [10] from the arguments of Bichteler, Gravereaux and Jacod in [3] and [4].

Let us consider a fixed time interval [0, T],T >0. Letε∈ 0,¹₂

be fixed.

Let us explain the main idea of this framework. We build a perturbation replacing θ with θ+ < λ, v^ε >

in order to obtain a new family of random measures N_λ^ε (forλ ∈Λ, Λ being a neighborhood of 0 in R² and v^ε a well-chosen predictable function from Ω×[0, T]×[−εθ₀, εθ₀]×[0,1] toR²). Then, we build a family of probability measuresP_λ^ε=G^ε_λ,T.P^εon Ω such thatL((V₀, N_λ^ε)|P_λ^ε) =L((V₀, N^ε)|P^ε). By this way, we obtain a perturbed processV_λ^ε satisfyingL

V_λ,t^ε |P_λ^ε

=L(V_t^ε|P^ε), and thus E

ϕ

V_λ,t^ε

G^ε_λ,t

=E[ϕ(V_t^ε)], for any Borel bounded functionϕ on R². Differentiating this equality atλ= 0, using an L²-differentiate of V_λ,t^ε and G^ε_λ,t, we finally obtain an equality of the form

E[ϕ(V_t^ε).DV_t^ε] =−E[ϕ(V_t^ε)DG^ε_t] which is the first step to satisfy inequality (3.2) of Theorem 3.3.

Consequently, the constant C_t,α^ε appearing in (3.2) depends on the moments of the derivatives of V_t^ε, of det⁻¹(DV_t^ε) and of the derivatives of DG^ε_t. Under some assumptions on the initial data P₀, Fournier [10]

obtained estimates of those moments. Consequently, we still have to state that those moments are uniformly bounded inεto prove Theorem 1.1. The derivatives of V_t^ε andDG^ε_t depend strongly on the random function v^ε introduced in the perturbation. The functionv^εused by Fournier in [10] does not allow to obtain uniform bounds of the moments inε∈(0,1/2] (see Rem. 4.3). So, we consider another perturbation which we describe now.

4.3. The perturbation and the Malliavin derivatives

δ^ε(θ) =c ε^1−r|θ|^r+1

1− |θ| εθ₀

(4.2) withc a constant independent ofεsuch thatc≤

θ^r₀

θ₀+r+ 2 +r2^r−1₋₁

. We notice that δ^ε(θ) +(δ^ε)(θ)<1.

Letg^εbe aR²-valued predictable function such that for anyω, t, α, ε, the mapθ→g^ε(ω, t, θ, α) is of class C¹ withg^ε_∞+g^ε_∞≤1 whereg^ε is the derivative ofg^ε with respect toθ.

We then define the random functionv^εon Ω×[0, T]×[−εθ0, εθ₀]×[0,1] by

v^ε(ω, t, θ, α) =g^ε(ω, t, θ, α)δ^ε(θ). (4.3) We denote byv^ε the derivative of v^εwith respect toθ.

Let Λ⊂B(0,1) be a neighbourhood of 0 inR². Forλ∈Λ, we consider the following perturbation γ^ε,λ(ω, t, θ, α) =θ+λ, v^ε(ω, t, θ, α) ·

We notice that the mapθ−→γ^ε,λ(ω, t, θ, α) is an increasing bijection from [−εθ₀, εθ₀] into itself (for anyε≤¹₂ and|θ| ≤εθ₀,|v^ε(θ)|<1 thanks to the choice ofc).

Recalling thatβ =β₁+β₀, the Poisson measureN split into N₀+N₁, where N₀ and N₁ are independent Poisson measures on [0, T]×[0,1]×[−π, π] with intensitiesν₀(dθ,dα,ds) =β₀(θ)dθdαds andν₁(dθ,dα,ds) = β₁(θ)dθdαdsrespectively. We denote by ˜N₀ and ˜N₁the associated compensated measures.

For λ∈ Λ, we defineN₀^ε,λ = γ^ε,λ(N₀^ε) the image measure of N₀^ε by the mapγ^ε,λ: if A ⊂[0, T]×[0,1]× [−εθ₀, εθ₀] is a Borel set,

N₀^ε,λ(ω, A) = _T

0

₁

0

_εθ0

−εθ0

IA

s, γ^ε,λ(ω, s, θ, α), α

N₀^ε(ω,dθ,dα,ds). We consider the shift S^ε,λ defined by

V₀◦S^ε,λ(ω) =V₀(ω), N₀^ε◦S^ε,λ(ω) =N₀^ε,λ(ω), andN₁^ε◦S^ε,λ(ω) =N₁^ε(ω) . Proposition 4.4. Let G^ε,λ be the Dol´eans-Dade martingale:

G^ε,λ_t = 1 + _t

0

₁

0

_εθ0

−εθ0

G^ε,λ_s−

Y^ε,λ(s, θ, α)−1N˜₀^ε(dθ,dα,ds) whereY^ε,λ is the following predictable real valued function onΩ×[0, T]×[−εθ₀, εθ₀]×[0,1]

Y^ε,λ(ω, s, θ, α) = (1 +λ, v^ε(ω, t, θ, α))β₀^ε

γ^ε,λ(ω, t, θ, α) β₀^ε(θ) · Then G^ε,λ_t is positive for any t∈[0, T].

Proof. Let us notice that

Y^ε,λ(s, θ, α)−1≤ |λ|d^ε(θ)

with d^ε(θ) = δ^ε(θ) +|δ^ε(θ)|+r2^r−1δε_|θ|^(θ). According to Appendix (Lem. 6.2), d^ε ∈ ∩

p≥2L^p(β₀^ε(θ) dθ) with moments uniformly bounded inε. Consequently, G^ε,λ is well defined and if

M_t^ε,λ= 1 + _t

0

₁

0

_επ

−επ

Y^ε,λ(s, θ, α)−1N˜₀^ε(dθ,dα,ds)

then (see Jacod and Shiryaev [18], p. 59),

G^ε,λ_t = e^Mtε,λ

s≤t

1 + ∆M_s^ε,λ

e^{−∆Msε,λ}.

Moreover, since ε≤1/2, for|θ| ≤εθ₀

Y^ε,λ(s, θ, α)−1 ≤ d^ε(θ)≤ 1 2cθ₀^r

θ₀+r+ 2 +r2^r−1

≤ 1/2

thanks to the choice ofc (see (4.2)). Thus, the jumps of M^ε,λ are greater than −1/2 which implies thatG^ε,λ_t is positive.

Let P^ε,λ be the probability measure defined by P^ε,λ =G^ε,λ_T .P^ε. Using the Girsanov theorem for random measures, we notice that P^ε,λ◦

S^ε,λ₋₁

= P^ε (for more details see [10], Prop. 3.7). We consider now the perturbed process V^ε,λ = V^ε◦S^ε,λ. Following Fournier [10], Section 3, and Appendix (Lem. 6.2), we notice thatV^ε,λandG^ε,λ belong toL^pfor anyp≥1 with bounded moments inε, and they are differentiable atλ= 0.

We give the expressions of their derivatives:

- the derivative ofG^ε,λ atλ= 0 is the following random vector in R² DG^ε_t=

_t

0

₁

0

_εθ0

−εθ0

v^ε(s, θ, α)−rv^ε(s, θ, α) θ

N˜₀^ε(dθ,dα,ds) ; - the derivative ofV_t^εis a 2×2 matrix which satisfies the equation

DV_t^ε = −b^ε 2

_t

0 DV_s^εds+ _t

0

₁

0

_επ

−επA(θ)DV_s^ε−N˜^ε(dθ,dα,ds) +

_t

0

₁

0

_εθ0

−εθ0

A(θ)(V_s− −W_s−(α))(v^ε(s, θ, α))^∗N₀^ε(dθ,dα,ds) (4.4) which can be also written

DV_t^ε=M_t^ε.H_t^ε (4.5)

whereM^εis the following invertible Dol´eans-Dade martingale M_t^ε=I−b^ε

2 _t

0 M_s^εds+ _t

0

₁

0

_επ

−επA(θ)M_s^ε−N˜^ε(dθ,dα,ds) (4.6) and

H_t^ε= _t

0

₁

0

_εθ0

−εθ0

(M_s^ε−)⁻¹(I+A(θ))⁻¹A(θ) (V_s^ε−−W_s^ε−(α)) (v^ε(s, θ, α))^∗N₀^ε(dθ,dα,ds). (4.7)

We want to state that the moments of the derivatives ofV_t^ε, of det⁻¹(DV_t^ε) and of the derivatives ofDG^ε_t are uniformly bounded inε. We will just give here a detailed proof of the term det⁻¹(DV_t^ε). We easily obtain the bounds for the two other terms studying the construction ofDG^ε_t and ofDV_t^ε, using the definition (4.3) ofv^ε and the bounds given in Appendix (Lem. 6.2).

The derivatives of V_t^ε and ofDG^ε_t depend strongly on v^ε. The choice ofv^ε is important. The moments of DG^ε_t are uniformly bounded inε, if there exists a positive constantK₁independent ofεsuch that

_εθ0

0

δ^ε(θ) +|δ^ε(θ)|+rδ^ε(θ) θ

₂

β₀^ε(θ)dθ≤K₁.

The moments ofDV_t^ε are uniformly bounded inε, if there exists a positive constantK₂ independent ofεsuch

that _εθ0

0 δ^ε(θ)β₀^ε(θ)dθ≤K₂. Nevertheless, the integral_εθ0

0 δ^ε(θ)β₀^ε(θ)dθmust not tend to 0 asεgoes to 0. If not, the variableDV_t^εconverges to 0 in L² as εtends to 0 (see Expression (4.4) ofDV_t^ε) , and we have no hope to obtain uniform bounds for the term det⁻¹(DV_t^ε).

In the sequel, we will consider more precisely the perturbationv^ε defined by v^ε(t, θ, α) = ¯g(V_s^ε−−W_s^ε−(α), M_s^ε−, θ)δ^ε(θ) with for anyx∈R²,y∈ M₂(R)

¯

g(x, y, θ) = (A(θ)x)^∗

(I+A(θ))⁻¹ _∗

y⁻¹_∗

ζ(x, y, θ) ζ(x, y, θ) = h(A(θ)x)k(I+A(θ))k(y)

whereδ^εis defined by (4.2) and the functionshandksatisfy the following assumptions:

-his the function fromR² to (0,1] defined byh(x) =

1 +|x|²₋₁

;

-kis a function fromM2(R) to [0,1] such thatk(y) = 0 if and only if dety= 0 and such that the map y−→ y⁻¹_∗

k(y) if dety= 0

0 if dety= 0 (4.8)

is of classC_b^∞ fromM₂(R) to itself.

Consequently, the processH^ε introduced in (4.5) writes H_t^ε =

_t

0

₁

0

_επ

−επ(M_s^ε−)⁻¹Γ (V_s^ε−−W_s^ε−(α), θ)

(M_s^ε−)⁻¹ _∗

×ζ(V_s^ε−−W_s^ε−(α), M_s^ε−, θ)δ^ε(θ)N₀^ε(dθ,dα,ds) with for anyx∈R²,

Γ (x, θ) = (I+A(θ))⁻¹(A(θ)x) (A(θ)x)^∗

(I+A(θ))⁻¹ _∗

.

4.4. Study of det

(DV

)

Since the derivative ofV_t^εcan be written asDV_t^ε=M_t^ε.H_t^εfor any t≥0, we study independently the term M_t^ε and the termH_t^ε.

Theorem 4.5. Assume (A) and P₀ ∈ ∩p<∞L^p. For every t ≥ 0, (detM_t^ε)⁻¹ admits moments of all orders uniformly in ε.

Proof. By [10] (Th. 3.20),M_t^εis invertible and its inverse (M_t^ε)⁻¹ satisfies the equation (M_t^ε)⁻¹ = I−b^ε

2 _t

0 (M_s^ε)⁻¹ds− _t

0

₁

0

_επ

−επ(M_s^ε−)⁻¹(I+A(θ))⁻¹A(θ) ˜N^ε(dθ,dα,ds) +

_t

0

₁

0

_επ

−επ(M_s^ε−)⁻¹A(θ) (I+A(θ))⁻¹A(θ)β^ε(θ) dθdαds (4.9) with

(I+A(θ))⁻¹A(θ) = sinθ cosθ+ 1

0 −1

1 0

and

A(θ) (I+A(θ))⁻¹A(θ) =1 2

sinθ cosθ+ 1

−sinθ 1−cosθ cosθ−1 −sinθ

. Since_π

0 θ²β(θ) dθ <∞, the sequence (b^ε)_ε>0 is bounded.

We notice that, _επ

−επ

|sinθ|

cosθ+ 1 _p

β^ε(θ) dθ=ε⁻² _π

−π

|sinεθ|

cosεθ+ 1 _p

β(θ) dθ.

For any ε ∈ (0,¹₂], the function θ −→ _cos^|sin_εθ+1^εθ| β(θ) is continuous on [−π, π]\ {0} and for ε small enough,

|sinεθ|

cosεθ+1 ≤εθ.

Consequently, the sequence _cos^|sinθ|_{θ+1 p}

β^ε(θ) dθ

ε∈(0,1/2] is bounded for anyp≥2.

Using the same arguments, we notice that the integrals _επ

−επ

sin²θ+|sinθ(1−cosθ)|

cosθ+ 1

p

β^ε(θ) dθ=ε⁻² _π

−π

sin²εθ+|sinεθ(1−cosεθ)|

cosεθ+ 1

p

β(θ) dθ are uniformly bounded inε,ε∈(0,¹₂], for anyp≥1.

Then, using usual estimates, Gronwall’s lemma in (4.9), we easily deduce that for anyp≥1, there exists a constantK_p (independent ofε) such that∀ε∈(0,¹₂],

E

(M_t^ε)^−p

≤K_p.

Thus (detM_t^ε)⁻¹is uniformly bounded inεinL^pfor any t≥0.

Theorem 4.6. Assume that (A) is satisfied and V₀∈ ∩

p≥1L^p. For everyt≥0 (detH_t^ε)⁻¹ admits moments of all orders uniformly in ε.

Lemma 4.7. The map (ε, t, Y)−→ L(V_t^ε, Y) is weakly continuous on 0,¹₂

×[0, T]×

Y ∈R²:|Y|= 1 whereP_t⁰=L

V_t⁰

is the measure-solution of the Landau equation at time t.

Proof. Let (ε_n, t_n, Y_n) be a sequence such that (ε_n, t_n, Y_n) →

n→∞(ε, t, Y) in [0,¹₂]×[0, T]×

Y ∈R²:|Y|= 1 . Letψ∈ C_b²(R) and we defineψ_Y onR²of classC_b² byv −→ψ_Y (v) =ψ(v, Y). We consider the sequence

d_n=E

ψ_Y (V_t^ε)−ψ_Yn V_t^ε_nⁿ

. We want to state thatd_n−→0 asngoes to +∞.