Fractional diffusion limit for a stochastic kinetic equation
aa r X i v : . [ m a t h . A P ] N ov FRACTIONAL DIFFUSION LIMIT FOR A STOCHASTICKINETIC EQUATION
Sylvain De Moor
ENS Cachan Bretagne - IRMAR, Université Rennes 1,Avenue Robert Schuman, F-35170 Bruz, FranceEmail: [email protected]
Abstract
We study the stochastic fractional diffusive limit of a kinetic equation involv-ing a small parameter and perturbed by a smooth random term. Generalizing themethod of perturbed test functions, under an appropriate scaling for the small pa-rameter, and with the moment method used in the deterministic case, we show theconvergence in law to a stochastic fluid limit involving a fractional Laplacian.
Keywords : Kinetic equations, diffusion limit, stochastic partial differential equa-tions, perturbed test functions, fractional diffusion.
In this paper, we consider the following equation ∂ t f ε + 1 ε α − v · ∇ x f ε = 1 ε α Lf ε + 1 ε α m ε f ε in R + t × R dx × R dv , (1.1)with initial condition f ε (0) = f ε in R dx × R dv , (1.2)where < α < , L is a linear operator (see (1 . below) and m ε a random processdepending on ( t, x ) ∈ R + × R d (see Section 2.2). We will study the behaviour inthe limit ε → of its solution f ε .The solution f ε ( t, x, v ) to this kinetic equation can be interpreted as a distributionfunction of particles having position x and degrees of freedom v at time t . Thevariable v belongs to the velocity space R d that we denote by V .The collision operator L models diffusive and mass-preserving interactions of theparticles with the surrounding medium; it is given by Lf = Z V f d v F − f, (1.3)where F is a velocity equilibrium function such that F ∈ L ∞ , F ( − v ) = F ( v ) , F > a.e., R V F ( v ) d v = 1 and which is a power tail distribution F ( v ) ∼ | v |→∞ κ | v | d + α . (1.4)Note that F ∈ ker( L ) . Power tail distribution functions arise in various contexts,such as astrophysical plasmas or in the study of granular media. For more details1n the subject, we refer to [11].In this paper, we derive a stochastic diffusive limit to the random kinetic model (1 . − (1 . , using the method of perturbed test functions. This method providesan elegant way of deriving stochastic diffusive limit from random kinetic systems;it was first introduced by Papanicolaou, Stroock and Varadhan [12]. The book ofFouque, Garnier, Papanicolaou and Solna [7] presents many applications to thismethod. A generalisation in infinite dimension of the perturbed test functionsmethod arose in recent papers of Debussche and Vovelle [5] and de Bouard andGazeau [8].For the random kinetic model (1 . − (1 . , the case α = 2 and v replaced by a ( v ) where a is bounded is derived in the paper of Debussche and Vovelle [5]. Here westudy a different scaling parametrized by < α < and we relax the boundednesshypothesis on a since we study the case a ( v ) = v . Note that, in our case, in orderto get a non-trivial limiting equation as ε goes to , we exactly must have a ( v ) unbounded; furthermore, we can easily extend the result to velocities of the form a ( v ) where a is a C -diffeomorphism from V onto V . In the deterministic case, i.e. m ε ≡ , Mellet derived in [10] and [11] with Mouhot and Mischler a diffusion limitto this kinetic equation involving a fractional Laplacian. As a consequence, for therandom kinetic problem (1 . − (1 . , we expect a limiting stochastic equation witha fractional Laplacian.As in the deterministic case, the fact that the equilibrium F have an appropriategrowth when | v | goes to + ∞ , namely of order | v | − d − α , is essential to derive a non-trivial limiting equation when ε goes to .To derive a stochastic diffusive limit to the random kinetic model (1 . − (1 . , weuse a generalisation in infinite dimension of the perturbed test functions method.Nevertheless, the fact that the velocities are not bounded gives rise to non-trivialdifficulties to control the transport term v ·∇ x . As a result, we also use the momentmethod applied in [10] in the deterministic case. The moment method consists inworking on weak formulations and in introducing new auxiliary problems, namelyin the deterministic case χ ε − εv · ∇ x χ ε = ϕ, where ϕ is some smooth function; thus we introduce in the sequel several additionalauxiliary problems to deal with the stochastic part of the kinetic equation. Solvingthese problems is based on the inversion of the operator L − εA + M where M isthe infinitesimal generator of the driving process m . Finally, we have to combineappropriately the moment and the perturbed test functions methods.We also point out similar works using a more probabilistic approach of Basile andBovier [1] and Jara, Komorowski and Olla [9].2 Preliminaries and main result
In the sequel, L F − denotes the F − weighted L ( R d × V ) space equipped with thenorm k f k := Z R d Z V | f ( x, v ) | F ( v ) d v d x. We denote its scalar product by ( ., . ) . We also need to work in the space L ( R d ) , or L x for short. The scalar product in L x will be denoted by ( ., . ) x . When f ∈ L F − ,we denote by ρ the first moment of f over V i.e. ρ = R V f d v . We often use thefollowing inequality k ρ k L x ≤ k f k , which is just Cauchy-Schwarz inequality and the fact that R V F ( v ) d v = 1 . Finally, S ( R d ) stands for the Schwartz space on R d , and S ′ ( R d ) for the space of tempereddistributions on R d .We recall that the operator L is defined by (1 . . It can easily be seen that L isa bounded operator from L F − to L F − . Note also that L is dissipative since, for f ∈ L F − , ( Lf, f ) = −k Lf k . (2.1)In the sequel, we denote by g ( t, · ) the semi-group generated by the operator L on L F − . It satisfies, for f ∈ L F − , d d t g ( t, f ) = Lg ( t, f ) ,g (0 , f ) = f, and it is given by g ( t, f ) = Z V f d v F (1 − e − t ) + f e − t , t ≥ , f ∈ L F − , so that g ( t, · ) is a contraction, that is, for f ∈ L F − , k g ( t, f ) k ≤ k f k , t ≥ . (2.2)We now introduce the following spaces S γ for γ ∈ R . First, we define the followingoperator on L ( R d ) J := − ∆ x + | x | , with domain D( J ) := (cid:8) f ∈ L ( R d ) , ∆ x f, | x | f ∈ L ( R d ) (cid:9) . Let ( p j ) j ∈ N d be the Hermite functions, defined as p j ( x , ..., x d ) := H j ( x ) · · · H j d ( x d ) e − | x | , j = ( j , ..., j d ) ∈ N d and H i stands for the i − th Hermite’s polynomial on R . The functions ( p j ) j ∈ N d are the eigenvectors of J with associated eigenvalues ( µ j ) j ∈ N d := (2 | j | + 1) j ∈ N d where | j | := | j | + · · · + | j d | . Furthermore, one can checkthat J is invertible from D( J ) to L ( R d ) , and that it is self-adjoint. As a result,we can define J γ for any γ ∈ R .Then, for γ ∈ R , we can also view J γ as an operator on S ′ ( R d ) . Let u ∈ S ′ ( R d ) ,we define J γ u ∈ S ′ ( R d ) by setting, for all ϕ ∈ S ( R d ) , h J γ u, ϕ i := h u, J γ ϕ i . Finally, we introduce, for γ ∈ R , S γ ( R d ) := { u ∈ S ′ ( R d ) , J γ u ∈ L ( R d ) } , equipped with the norm k u k S γ ( R d ) = k J γ u k L ( R d ) . In the sequel, we need to know the asymptotic behaviour of the quantities k p j k L x , k∇ x p j k L x , k D p j k L x and k ( − ∆) α p j k L x as | j | → ∞ . In fact, classical propertiesof the Hermite functions give the following bounds k p j k L x = 1 , k∇ x p j k L x ≤ µ j , k D p j k L x ≤ µ j , k ( − ∆) α p j k L x ≤ µ j . (2.3)We finally recall the definition of the fractional power of the Laplacian. It can beintroduced using the Fourier transform in S ′ ( R d ) by setting, for u ∈ S ′ ( R d ) , F (( − ∆) α u )( ξ ) = | ξ | α F ( u )( ξ ) . Alternatively, we have the following singular integral representation, see [13], − ( − ∆) α u ( x ) = c d,α PV Z R d [ u ( x + h ) − u ( x )] d h | h | d + α , for some constant c d,α which only depends on d and α . The random term m ε is defined by m ε ( t, x ) := m (cid:18) tε α , x (cid:19) , where m is a stationary process on a probability space (Ω , F , P ) and is adapted to afiltration ( F t ) t ≥ . Note that m ε is adapted to the filtration ( F εt ) t ≥ = ( F ε − α t ) t ≥ . We assume that, considered as a random process with values in a space of spatiallydependent functions, m is a stationary homogeneous Markov process taking valuesin a subset E of L ( R d ) ∩ W , ∞ ( R d ) . In the sequel, E will be endowed with the norm k · k ∞ of L ∞ ( R d ) . Besides, we denote by B ( E, X ) (or B ( E ) when X = R ) the set of4ounded functions from E to X endowed with the norm k g k ∞ := sup n ∈ E k g ( n ) k X for g ∈ B ( E, X ) .We assume that m is stochastically continuous. Note that m is supposed not todepend on the variable v . For all t ≥ , the law ν of m t is supposed to be centered E [ m t ] = Z E n d ν ( n ) = 0 . The subset E has the following properties. We fix a family ( η i ) i ∈ N of functions in W , ∞ ( R d ) such that S := X i ∈ N k η i k W , ∞ < ∞ , and we assume that every n ∈ E can be uniquely written as n = X i ∈ N n i ( n ) η i , (2.4)with | n i ( n ) | ≤ K for all i ∈ N and all n ∈ E . Note that the preceding seriesconverges absolutely and that E is included in the ball B(0 , KS ) of W , ∞ ( R d ) .Finally, since m is centered, we also suppose that for all i ∈ N , Z E n i ( n ) d ν ( n ) = 0 . (2.5)We denote by e tM a transition semi-group on E associated to m . We supposethat the transition semi-group is Feller i.e. e tM maps continuous functions of n oncontinuous functions of n for all t ≥ . In the sequel we also need to consider e tM as a transition semi-group on the space B ( E, L F − ) and not only on B ( E ) . Thus,if g ∈ B ( E, L F − ) , e tM acts on g pointwise, that is, [ g e tM g ]( x, v ) = e tM [ g ( x, v )] , ( x, v ) ∈ R d × V. In both cases, we denote by M the infinitesimal generator associated to the transi-tion semi-group. Note that we do not distinguish on which space B ( E, X ) , X = R or L F − , the operators are acting since it will always be clear from the context.Then, for X = R or X = L F − , D( M ) stands for the domain of M ; it is defined asfollows: D( M ) := (cid:26) u ∈ B ( E, X ) , lim h → e hM − Ih u exists in B ( E, X ) (cid:27) , and if u ∈ D( M ) , we set M u := lim h → e hM − Ih u in B ( E, X ) . We suppose that there exists µ > such that for all g ∈ B ( E ) verifying thecondition R E g ( n ) d ν ( n ) = 0 , k e tM g k ∞ ≤ e − µt k g k ∞ , t ≥ . (2.6)5oreover, we suppose that m is ergodic and satisfies some mixing properties in thesense that there exists a subspace P M of B ( E ) such that for any g ∈ P M , thePoisson equation M ψ = g − Z E g ( n ) d ν ( n ) =: b g, has a unique solution ψ ∈ D( M ) satisfying R E ψ ( n ) d ν ( n ) = 0 . We denote by M − b g this unique solution, and assume that it is given by M − b g ( n ) = Z ∞ e tM b g ( n ) d t, n ∈ E. (2.7)Thanks to (2 . , the above integral is well defined. In particular, it implies thatfor all n ∈ E , lim t →∞ e tM b g ( n ) = 0 . We assume that for all i ∈ N , n n i ( n ) is in P M and that for all n ∈ E , | M − n i ( n ) | ≤ K . As a consequence, we simply define M − I by M − I ( n ) := X i ∈ N M − n i ( n ) η i , n ∈ E. We also suppose that for all f ∈ L ( R d ) , the functions g f : n ∈ E ( f, n ) x and n ∈ E M − g f ( n ) are in P M .We will suppose that for all t ≥ , E k m t k L x < ∞ , E k M − I ( m t ) k L x < ∞ . (2.8)To describe the limiting stochastic PDE, we then set k ( x, y ) = E Z R m ( y ) m t ( x ) d t, x, y ∈ R d . The kernel k is, thanks to (2 . , the fact that m is stationary and Cauchy-Schwarzinequality, in L ( R d × R d ) and such that Z R d k ( x, x ) d x < ∞ . Furthermore, we can check (see [5]), since m is stationary, that k is symmetric. Asa result, we introduce the operator Q on L ( R d ) associated to the kernel kQf ( x ) = Z R d k ( x, y ) f ( y ) d y, x ∈ R d , which is self-adjoint and trace class. Furthermore, since we assumed that the func-tions g f : n ∈ E ( f, n ) x and n ∈ E M − g f ( n ) are in P M , we can show, see[5, Lemma 1], that Q is non-negative, that is ( Qf, f ) x ≥ for all f ∈ L ( R d ) . Asa result, we can define the square root Q which is Hilbert-Schmidt on L ( R d ) .6t remains to make some hypothesis on M . We set, for all n ∈ E , θ ( n ) = Z E nM − I ( n ) d ν ( n ) − nM − I ( n ) , (2.9)and, for i, j ∈ N , θ i,j = R E n i M − n j d ν − n i M − n j , so that θ = X i,j ∈ N θ i,j η i η j . We suppose that for all i, j, k, l ∈ N and s, t ≥ , n n i ( n ) ,n θ i ( n ) ,n e tM n i ( n ) e sM n j ( n ) ,n e tM θ i,j ( n ) e sM n k ( n ) ,n e tM θ i,j ( n ) e sM θ k,l ( n ) , (2.10)are in D( M ) , with k n i k ∞ + k θ i,j k ∞ + k M n i k ∞ + k M θ i,j k ∞ ≤ K, (2.11) k M [ e tM n i e sM n j ] k ∞ + k M [ e tM θ i,j e sM n j ] k ∞ + k M [ e tM θ i,j e sM θ k,l ] k ∞ ≤ Ke − µ ( s + t ) . (2.12) Remark
The above assumptions (2 . − (2 . on the process m are verified, forinstance, when m is a Poisson process taking values in E .We now state two lemmas which will be very useful in the following. Lemma 2.1
Let p ∈ B ( E ) be a function in D( M ) such that k M p k ∞ ≤ K . Thenwe have, for all h > , (cid:13)(cid:13)(cid:13)(cid:13) e hM − Ih p − M p (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ K. Proof
We just write, for all n ∈ E , (cid:12)(cid:12)(cid:12)(cid:12) e hM − Ih p ( n ) − M p ( n ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h Z h M e sM p ( n ) d s − M p ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h Z h e sM M p ( n ) d s − M p ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ K, where we used the contraction property of the semigroup e tM . This concludes theproof. (cid:3) Remark
The proof is still valid if p ∈ B ( E, L F − ) ; we just have to replace theabsolute values by the L F − -norm. 7 emma 2.2 For all i, j, k, l ∈ N and s, t ≥ , the functions n n i ( n ) ,n θ i,j ( n ) ,n e tM n i e sM n j ( n ) ,n e tM θ i,j e sM n k ( n ) ,n e tM θ i,j e sM θ k,l ( n ) , n M n i ( n ) ,n M θ i,j ( n ) ,n M [ e tM n i e sM n j ]( n ) ,n M [ e tM θ i,j e sM n k ]( n ) ,n M [ e tM θ i,j e sM θ k,l ]( n ) , (2.13) are continuous. Proof
We fix i, j, k, l ∈ N and s, t ≥ . First of all, n n i ( n ) is obviouslycontinuous since it is linear. We recall that θ i,j = R E n i M − n j d ν − n i M − n j .With (2 . and (2 . , we have M − n j = Z ∞ e tM n j d t, which is continuous with respect to n ∈ E by (2 . , (2 . , (2 . and the domi-nated convergence theorem. As a result, n n i ( n ) M − n j ( n ) is continuous; andalso n R E n i ( n ) M − n j ( n ) d ν ( n ) by the dominated convergence theorem. Hence n θ i,j ( n ) is continuous. The continuity of n i and θ i,j now immediately gives thecontinuity of the three last functions of the left group of the lemma by the Fellerproperty of the semigroup e tM .For the remaining functions, just remark that if p ∈ B ( E ) is in D( M ) and contin-uous, then M p is the uniform limit on E when h → of the functions e hM − Ih p, which are continuous by the Feller property of the semigroup. Hence
M p is con-tinuous. This ends the proof. (cid:3)
In this section, we solve the linear evolution problem (1 . − (1 . thanks to asemigroup approach. We thus introduce the linear operator A := − v · ∇ x on L F − with domain D( A ) := { f ∈ L F − , v · ∇ x f ∈ L F − } . The operator A has dense domain and, since it is skew-adjoint, it is m -dissipative.Consequently A generates a contraction semigroup ( T ( t )) t ≥ , see [3]. We recallthat D( A ) is endowed with the norm k · k D( A ) := k · k + k A · k , and that it is aBanach space. Proposition 2.3
Let
T > and f ε ∈ L F − . Then there exists a unique mildsolution of (1 . − (1 . on [0 , T ] in L ∞ (Ω) , that is there exists a unique f ε ∈ L ∞ (Ω , C ([0 , T ] , L F − )) such that P − a.s. f εt = T (cid:18) tε α − (cid:19) f ε + Z t T (cid:18) t − sε α − (cid:19) (cid:18) ε α Lf εs + 1 ε α m εs f εs (cid:19) ds, t ∈ [0 , T ] . ssume further that f ε ∈ D( A ) , then there exists a unique strong solution f ε ∈ L ∞ (Ω , C ([0 , T ] , L F − )) ∩ L ∞ (Ω , C ([0 , T ] , D( A ))) of (1 . − (1 . . Proof
Subsections 4.3.1 and 4.3.3 in [3] gives that P − a.s. there exists a uniquemild solution f ε ∈ C ([0 , T ] , L F − ) and it is not difficult to slightly modify the proofto obtain that in fact f ε ∈ L ∞ (Ω , C ([0 , T ] , L F − )) (we intensively use that for all t ≥ and ε > , k m εt k ∞ ≤ K ).Similarly, subsections 4.3.1 and 4.3.3 in [3] gives us P − a.s. a strong solution f ε ∈ C ([0 , T ] , L F − ) ∩ C ([0 , T ] , D( A )) of (1 . − (1 . and once again one can easily getthat in fact f ε ∈ L ∞ (Ω , C ([0 , T ] , L F − )) ∩ L ∞ (Ω , C ([0 , T ] , D( A ))) . (cid:3) We are now ready to state our main result:
Theorem 2.4
Assume that ( f ε ) ε> is bounded in L F − and that ρ ε := Z V f ε d v −→ ε → ρ in L ( R d ) . Then, for all η > and T > , ρ ε := R V f ε d v converges in law in C ([0 , T ] , S − η ) to the solution ρ to the stochastic diffusion equation dρ = − κ ( − ∆) α ρ d t + 12 Hρ + ρQ dW t , in R + t × R dx , (2.14) with initial condition ρ (0) = ρ in L ( R d ) , and where W is a cylindrical Wienerprocess on L ( R d ) , κ := κ c d,α Z ∞ | t | α e − t d t, (2.15) and H := Z E nM − I ( n ) d ν ( n ) ∈ W , ∞ . (2.16) Remark
The limiting equation (2 . can also be written in Stratonovitch form dρ = − κ ( − ∆) α ρ d t + ρ ◦ Q dW t . Notation
In the sequel, we will note . the inequalities which are valid up to con-stants of the problem, namely K , S , µ , d , α , k L k , sup ε> k f ε k and real constants.Nevertheless, when we need to emphasize the dependence of a constant on a pa-rameter, we index the constant C by the parameter; for instance the constant C ϕ depends on ϕ . The process f ε is not Markov (indeed, by (1 . , we need m ε to know the incrementsof f ε ) but the couple ( f ε , m ε ) is. From now on, we denote by L ε its infinitesimalgenerator, it is given by L ε Ψ( f, n ) = 1 ε α ( Lf + εAf, D Ψ( f, n )) + 1 ε α ( f n, D Ψ( f, n )) + 1 ε α M Ψ( f, n ) , Ψ : L F − × E → R is enough regular to be in the domain of L ε . Thuswe begin this section by introducing a special set of functions which lie in thedomain of L ε and satisfy the associated martingale problem. In the following, if Ψ : L F − → R is differentiable with respect to f ∈ L F − , we denote by D Ψ( f ) itsdifferential at a point f and we identify the differential with the gradient. Definition 3.1
We say that
Ψ : L F − × E → R is a good test function if ( i ) ( f, n ) Ψ( f, n ) is differentiable with respect to f ; ( ii ) ( f, n ) D Ψ( f, n ) is continuous from L F − × E to L F − and maps boundedsets onto bounded sets; ( iii ) for any f ∈ L F − , Ψ( f, · ) ∈ D M ; ( iv ) ( f, n ) M Ψ( f, n ) is continuous from L F − × E to R and maps bounded setsonto bounded sets. Proposition 3.1
Let Ψ be a good test function. If f ε ∈ D( A ) , M ε Ψ ( t ) := Ψ( f εt , m εt ) − Ψ( f ε , m ε ) − Z t L ε Ψ( f εs , m εs ) d s is a continuous and integrable ( F εt ) t ≥ martingale, and if | Ψ | is a good test func-tion, its quadratic variation is given by h M ε Ψ i t = Z t ( L ε | Ψ | − L ε Ψ)( f εs , m εs ) d s. Proof
This is classical, we use the same kind of ideas and follow the proof of [5,Proposition 6] and [7, Appendix 6.9].
In this section, we study the limit of the generator L ε when ε → . The limitgenerator L will characterize the limit stochastic fluid equation. To derive the diffusive limiting equation, one has to study the limit as ε goes to of quantities of the form L ε Ψ where Ψ is a good test function. From now on, wechoose a specific form for the test functions that we keep thorough the paper. Wetake ϕ in the Schwartz space S ( R d ) and we set Ψ( f, n ) := ( f, ϕF ) (4.1)It is clear that Ψ is a good test function. Remember that, when ε → , we willobtain a fluid limit equation verified by the macroscopic quantity ρF ; the testfunction Ψ takes this point in consideration since Ψ( f, n ) = Ψ( f ) = Ψ( ρF ) . In thesequel, we will show that the knowledge of the limits L ε Ψ and L ε | Ψ | as ε goes10o where Ψ is defined as (4 . is sufficient to obtain our result. Nevertheless, wenow have to correct Ψ and | Ψ | so as to obtain non-singular limits. Here, we showformally how we correct Ψ (the formal work on | Ψ | is similar).We search the correction Ψ ε of Ψ . First of all, to correct the deterministic part,we follow the moment method presented in [10] so we set Ψ ε ( f, n ) = ( f, χ ε F ) where χ ε solves the auxiliary problem χ ε − εv · ∇ x χ ε = ϕ. Now, to correct the stochastic part, we try an Hilbert expansion method (adaptedto our scaling) coupled with the idea of auxiliary equation brought in the momentmethod so that we complete the definition of Ψ ε as Ψ ε ( f, n ) = ( f, χ ε F ) + ε α ( f, δ ε F ) + ε α ( f, θ ε F ) , where δ ε and θ ε are to be defined. We then compute, since the first term in theexpansion of Ψ ε does not depend on n ∈ E , L ε Ψ ε ( f, n ) = 1 ε α ( Lf + εAf, χ ε F ) (4.2) + 1 ε α ( f n, χ ε F ) + 1 ε α ( Lf + εAf, δ ε F ) + 1 ε α ( f, M δ ε F ) (4.3) + ( f n, δ ε F ) + ( Lf + εAf, θ ε F ) + ( f, M θ ε F ) + ε α ( f n, θ ε F ) . (4.4)The first term (4 . above converges as ε goes to to ( − κ ( − ∆) α f, ϕF ) , see [10],that is to the infinitesimal generator of the fractional Laplacian applied to Ψ : weget the deterministic term of the limiting equation.Since L is auto-adjoint and A is skew-adjoint, the three following terms (4 . canbe rewritten as ε α ( f, nχ ε F ) + 1 ε α ( f, ( L − εA + M )( δ ε F )) . Then we cancel these singular term by choosing δ ε such that ( L − εA + M )( δ ε F ) = − nχ ε F. Formally, this equation can be solved with the resolvent operator of L − εA + M so that we have δ ε ( x, v, n ) F ( v ) = Z + ∞ e tM g ( t, nχ ε F )( x + εvt, v ) d t. With this expression of δ ε F and since χ ε → ϕ as ε → , see [10], we have that δ ε F converges to − M − I ( n ) ϕF when ε → . So, neglecting an error term, we cansuppose that (4 . writes ( f, − nM − I ( n ) ϕF ) + ( Lf + εAf, θ ε F ) + ( f, M θ ε F ) + ε α ( f n, θ ε F ) . L ε Ψ ε as ε goes to does depend on n ∈ E .Since the expected limit is L Ψ where Ψ does not depend on n , we have to correctonce again the remaining terms to break the dependence with respect to n of thelimit. The right way to do so, given the mixing properties of the operator M , is tosubtract the mean value: we write (4 . as ( f, − HϕF ) + ( f, θ ( n ) ϕF ) + ( Lf + εAf, θ ε F ) + ( f, M θ ε F ) + ε α ( f n, θ ε F ) , where H and θ are respectively defined in (2 . and (2 . . Now, we choose θ ε sothat ( L − εA + M )( θ ε F ) = − θ ( n ) ϕF, so that (4 . becomes ( f, − HϕF ) + ε α ( f n, θ ε F ); it allows us to conclude that L ε Ψ ε converges to L Ψ as ε → where L is the in-finitesimal generator of the equation (2 . (note that D Ψ ≡ so that no stochas-tic appears here).As we said previously, the same kind of work can be done to correct | Ψ | . In thefollowing subsections, we define rigorously the corrections of Ψ and | Ψ | . As it is said above, we use the moment method presented in [10] to correct thedeterministic part of the equation (1 . . Let χ ε be the solution of the auxiliaryproblem χ ε − εv · ∇ x χ ε = ϕ. (4.5)We recall, see [10], that the solution of (4 . is given by χ ε ( x, v ) = Z + ∞ e − t ϕ ( x + εvt ) d t, x ∈ R d , v ∈ V. (4.6)We now detail few results on χ ε . Proposition 4.1
The function χ ε F is in L F − with k χ ε F k ≤ k ϕ k L x . (4.7) Furthermore, for any λ > , we have the following estimate: k ( χ ε − ϕ ) F k . C λ ε k∇ x ϕ k L x + k ϕ k L x λ . (4.8) Proof
See Appendix A. (cid:3)
In the two following lemmas, we study in detail the convergence to the fractionalLaplace operator. We recall that κ has been defined by (2 . . Lemma 4.2
For any λ > , we have the following estimate: (cid:13)(cid:13)(cid:13)(cid:13) ε − α Z V [ χ ε ( · , v ) − ϕ ( · )] F ( v ) d v + κ ( − ∆) α ϕ (cid:13)(cid:13)(cid:13)(cid:13) L x . (Λ( ε ) + λ )( k ϕ k L x + k D ϕ k L x ) , (4.9) for a certain function Λ , which only depends on ε , such that Λ( ε ) → when ε → . roof See Appendix A. (cid:3)
Lemma 4.3
For any λ > , we have the following estimate: (cid:12)(cid:12) ε − α ( εAf + Lf, χ ε F ) + ( κ ( − ∆) α f, ϕF ) (cid:12)(cid:12) . (Λ( ε ) + λ ) k f k ( k ϕ k L x + k D ϕ k L x ) , (4.10) for a certain function Λ , which only depends on ε , such that Λ( ε ) → when ε → . Proof
See Appendix A. (cid:3) δ ε We recall that g ( t, · ) denotes the semi-group generated by the operator L on L F − and that the function χ ε has been defined in (4 . . Then, we define the function δ ε : R d × V × E → R by δ ε ( x, v, n ) F ( v ) := Z + ∞ e tM g ( t, nχ ε F )( x + εvt, v ) d t, and we give here some properties of δ ε . We recall that the test function ϕ has beenfixed in Section . . Proposition 4.4
The function δ ε F belongs to B ( E, L F − ) with k δ ε F k B ( E,L F − ) . k ϕ k L x . (4.11) It satisfies ( L − εA + M )( δ ε F ) = − nχ ε F, (4.12) with k M δ ε F k B ( E,L F − ) . k ϕ k L x . (4.13) Furthermore, for any λ > , we have the two following estimates: k δ ε F + M − I ( n ) ϕF k B ( E,L F − ) . C λ k∇ x ϕ k L x ε + k ϕ k L x λ, (4.14) k M δ ε F + nχ ε F k B ( E,L F − ) . C λ k∇ x ϕ k L x ε + k ϕ k L x λ. (4.15) Proof
Proof of (4.11). The definition of δ ε F can be rewritten, thanks to (2 . , as δ ε ( x, v, n ) F ( v ) = + ∞ X i =0 Z + ∞ e tM n i ( n ) g ( t, η i χ ε F )( x + εvt, v ) d t =: + ∞ X i =0 α i ( x, v, n ) . Then we fix i ∈ N and n ∈ E . We have k α i ( · , · , n ) k = Z R d Z V (cid:18)Z + ∞ e tM n i ( n ) g ( t, η i χ ε F )( x + εvt, v ) d t (cid:19) d vF ( v ) d x ≤ Z R d Z V (cid:18)Z + ∞ Ke − µt | g ( t, η i χ ε F ) | ( x + εvt, v ) d t (cid:19) d vF ( v ) d x ≤ K µ Z R d Z V Z + ∞ e − µt | g ( t, η i χ ε F ) | ( x + εvt, v ) d t d vF ( v ) d x = K µ Z ∞ e − µt k g ( t, η i χ ε F ) k d t ≤ K µ k η i χ ε F k ≤ K µ k η i k W , ∞ k ϕF k , (2 . , (2 . , (2 . , Cauchy-Schwarz inequality, the contraction prop-erty of the semigroup g ( t, · ) (2 . and finally (4 . . We thus get k α i k B ( E,L F − ) ≤ Kµ k η i k W , ∞ k ϕF k . Since S = P i ∈ N k η i k W , ∞ < ∞ , we finally deduce that the series defining δ ε F converges absolutely in B ( E, L F − ) and that k δ ε F k B ( E,L F − ) . k ϕF k = k ϕ k L x . Proof of (4.12) . Fix i ∈ N , α i maps E into L F − . We claim that α i ∈ D( M ) with,for all n ∈ E , M α i ( x, v, n ) = Z + ∞ M e tM n i ( n ) g ( t, η i χ ε F )( x + εvt, v ) d t =: β i ( x, v, n ) in L F − . Indeed, for n ∈ E , we have Z R d Z V (cid:18) e hM α i ( x, v, n ) − α i ( x, v, n ) h − β i ( x, v, n ) (cid:19) d vF ( v ) d x = Z R d Z V (cid:18)Z ∞ (cid:20) e ( t + h ) M − e tM h − M e tM (cid:21) n i ( n ) g ( t, η i χ ε F )( x + εvt, v ) d t (cid:19) d vF ( v ) d x ≤ Z R d Z V (cid:18)Z ∞ e − µt (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) e hM − Ih − M (cid:21) n i (cid:13)(cid:13)(cid:13)(cid:13) ∞ | g ( t, η i χ ε F ) | ( x + εvt, v ) d t (cid:19) d vF ( v ) d x ≤ µ (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) e hM − Ih − M (cid:21) n i (cid:13)(cid:13)(cid:13)(cid:13) ∞ k η i k W , ∞ k ϕF k . Since by (2 . , n n i ( n ) ∈ D( M ) we deduce that (cid:13)(cid:13)(cid:13)(cid:13) e hM α i − α i h − β i (cid:13)(cid:13)(cid:13)(cid:13) B ( E,L F − ) ≤ µ (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) e hM − Ih − M (cid:21) n i (cid:13)(cid:13)(cid:13)(cid:13) ∞ k η i k W , ∞ k ϕF k −→ h → , which is just what we needed. Now, with (2 . , we apply Lemma 2.1 so that wededuce, with the fact that P i ∈ N k η i k W , ∞ < ∞ and the dominated convergencetheorem, that δ ε F ∈ D( M ) with M [ δ ε F ]( x, v, n ) = ∞ X i =0 β i ( x, v, n ) , where the series converges absolutely in B ( E, L F − ) . We fix i ∈ N , n ∈ E and v ∈ V . We recall that η i is in W , ∞ ( R d ) and that χ ε is defined by (4 . where ϕ is in theSchwartz space S ( R d ) . Then it is easily seen that η i χ ε F and η i χ ε F are in W , ( R d ) with respect to x . Therefore, since g ( t, η i χ ε F ) = η i χ ε F F (1 − e − t ) + η i χ ε F e − t , weobtain that h := t ∈ (0 , ∞ ) g ( t, η i χ ε F )( x + εvt, v ) is in W , ∞ ((0 , ∞ ) , L x ) with h ′ ( t )( x, v ) = Lg ( t, η i χ ε F )( x + εvt, v ) + εv · ∇ x g ( t, η i χ ε F )( x + εvt, v ) , L x . Furthermore, with (2 . , h := t ∈ (0 , ∞ ) e tM n i ( n ) is clearly in W , ((0 , ∞ ) , R ) with h ′ ( t ) = M e tM n i ( n ) . We now get by integration by parts β i ( x, v, n ) = Z + ∞ M e tM n i ( n ) g ( t, η i χ ε F )( x + εvt, v ) d t = (cid:2) e tM n i ( n ) g ( t, η i χ ε F )( x + εvt, v ) (cid:3) ∞ − Z + ∞ e tM n i ( n ) d. d t g ( t, η i χ ε F )( x + εvt, v ) d t = − n i ( n ) η i χ ε F ( x, v ) − Z + ∞ e tM n i ( n ) Lg ( t, η i χ ε F )( x + εvt, v ) d t − εv · Z + ∞ e tM n i ( n ) ∇ x g ( t, η i χ ε F )( x + εvt, v ) d t, where all the equalities have to be understood in L x . We easily see that the lasttwo terms of the preceding equality are respectively equal in L x to − Lα i ( x, v, n ) and εAα i ( x, v, n ) . As a result, we just proved that for all i ∈ N and n ∈ E , wehave the following equality for almost all x ∈ R d and v ∈ V : ( L − εA + M ) α i ( x, v, n ) = − n i ( n ) η i χ ε F ( x, v ) . (4.16)Now, the right hand term of the last equality is clearly in L F − . Since α i is in L F − , Lα i ∈ L F − ; and we proved above that M α i ∈ L F − . As a consequence Aα i is in L F − and the preceding equality is valid in L F − . We want to sum over i ∈ N . We previously proved that we have, in B ( E, L F − ) , P + ∞ i =0 M α i = P + ∞ i =0 β i = M [ δ ε F ] . Since the series P i ∈ N α i converges absolutely in B ( E, L F − ) and since L is a bounded operator on L F − , we also deduce that we have, in B ( E, L F − ) , P + ∞ i =0 Lα i = L [ δ ε F ] . Since P i ∈ N n i η i converges absolutely in W , ∞ ( R d ) to n , weobtain that P i ∈ N n i η i χ ε F converges absolutely in B ( E, L F − ) to nχ ε F . Finally,with (4 . and the fact that A is a closed operator, we also have, in B ( E, L F − ) , P + ∞ i =0 Aα i = A [ δ ε F ] . Summing (4 . over i ∈ N now gives ( L − εA + M )( δ ε F ) = − nχ ε F . Proof of (4.13) We just proved that
M δ ε F = P + ∞ i =0 β i , with β i ( x, v, n ) = Z + ∞ M e tM n i ( n ) g ( t, η i χ ε F )( x + εvt, v ) d t = Z + ∞ e tM M n i ( n ) g ( t, η i χ ε F )( x + εvt, v ) d t, so that we immediately deduce (4 . thanks to (2 . . Proof of (4 . . Let λ > . First of all, we point out that g ( t, η i ϕF ) = η i ϕF sothat − M − n i ( n ) η i ϕF ( x, v ) = Z ∞ e tM n i ( n ) g ( t, η i ϕF )( x, v ) d t.
15e can then write, for i ∈ N and n ∈ E , k α i ( · , · , n ) + M − n i ( n ) η i ϕF k ≤ Z R d Z V (cid:18)Z + ∞ e tM n i ( n ) g ( t, η i ( χ ε − ϕ ) F )( x + εvt, v ) d t (cid:19) d vF ( v ) d x + Z R d Z V (cid:18)Z + ∞ e tM n i ( n ) [ g ( t, η i ϕF )( x + εvt, v ) − g ( t, η i ϕF )( x, v )] d t (cid:19) d vF ( v ) d x. Similarly as the very beginning of the proof, we can bound the first term by K µ k η i k W , ∞ k ( χ ε − ϕ ) F k , and we recall that we have, with (4 . , k ( χ ε − ϕ ) F k ≤ C λ ε k∇ x ϕ k L x + 4 k ϕ k L x λ . For the second term, B say, we write B = Z R d Z V (cid:18)Z + ∞ e tM n i ( n ) [ η i ϕF ( x + εvt, v ) − η i ϕF ( x, v )] d t (cid:19) d vF ( v ) d x. ≤ K µ k η i k W , ∞ Z R d Z V Z + ∞ e − µt [ ϕ ( x + εvt ) − ϕ ( x )] d tF ( v ) d v d x. We can then mimic the proof of Proposition . to get the following bound Z R d Z V Z + ∞ e − µt [ ϕ ( x + εvt ) − ϕ ( x )] d tF ( v ) d v d x ≤ C λ µ ε k∇ x ϕ k L x + 4 µ k ϕ k L x λ . To sum up, we just obtained, for i ∈ N and n ∈ E , k α i ( · , · , n ) + M − n i ( n ) η i ϕF k . k η i k W , ∞ (cid:16) C λ k∇ x ϕ k L x ε + k ϕ k L x λ (cid:17) . k η i k W , ∞ (cid:0) C λ k∇ x ϕ k L x ε + k ϕ k L x λ (cid:1) . We can now sum over i ∈ N to obtain, k δ ε F + M − I ( n ) ϕF k B ( E,L F − ) . C λ k∇ x ϕ k L x ε + k ϕ k L x λ, which is the bound expected. Proof of (4 . . We recall that
M δ ε F = P + ∞ i =0 β i , with β i defined above. Notethat nχ ε F ( x, v ) = + ∞ X i =0 Z + ∞ e tM M n i ( n ) η i χ ε F ( x, v ) d t, so that we decompose M δ ε ( x, v, n ) F ( v ) + nχ ε F ( x, v ) into two terms + ∞ X i =0 Z + ∞ e tM M n i ( n ) [ g ( t, η i χ ε F )( x + εvt, v ) − g ( t, η i ϕF )( x, v )] d t + + ∞ X i =0 Z + ∞ e tM M n i ( n ) [ η i ϕF )( x, v ) − η i χ ε F ( x, v )] d t.
16s we have done previously, we can show that the first term is, in B ( E, L F − ) , . (cid:0) C λ k∇ x ϕ k L x ε + k ϕ k L x λ (cid:1) . We bound the second term in B ( E, L F − ) as . k ( χ ε − ϕ ) F k , that is, thanks to (4 . , . (cid:0) C λ k∇ x ϕ k L x ε + k ϕ k L x λ (cid:1) . It finally gives thebound expected. This concludes the proof. (cid:3) θ ε We recall that, for all n ∈ E , θ ( n ) = Z E nM − I ( n ) d ν ( n ) − nM − I ( n ) , and that, for i, j ∈ N , θ i,j = R E n i M − n j d ν − n i M − n j . Then we define thefunction θ ε : R d × V × E → R by θ ε ( x, v, n ) F ( v ) := Z + ∞ e tM g ( t, θ ( n ) ϕF )( x + εvt, v ) d t, that is, θ ε ( x, v, n ) F ( v ) := + ∞ X i,j =0 Z + ∞ e tM θ i,j ( n ) g ( t, η i η j ϕF )( x + εvt, v ) d t, and, similarly as Proposition 4.4, we obtain the Proposition 4.5
The function θ ε F belongs to B ( E, L F − ) with k θ ε F k B ( E,L F − ) . k ϕ k L x . (4.17) It satisfies ( L − εA + M )( θ ε F ) = − θ ( n ) ϕF, (4.18) with k M θ ε F k B ( E,L F − ) . k ϕ k L x . (4.19) ζ ε We set, for all ( f, n ) ∈ L F − × E , ξ ε ( f, n ) = ( f, δ ε F ) n − Z E ( f, δ ε F ) n d ν ( n ) , and, for i ∈ N , ξ εi = ( f, δ ε F ) n i . We then define the function ζ ε : R d × V × L F − × E → R by ζ ε ( x, v, f, n ) F ( v ) := Z + ∞ e tM g ( t, ξ ε ( f, n ) ϕF )( x + εvt, v ) d t. Similarly as Proposition . , we have the17 roposition 4.6 Let f ∈ L F − be fixed. The function ζ ε F ( f ) belongs to B ( E, L F − ) with k ζ ε F ( f ) k B ( E,L F − ) . k f kk ϕ k L x . (4.20) It satisfies ( L − εA + M )( ζ ε F ( f )) = − ξ ε ( f, n ) ϕF, (4.21) with k M ζ ε F ( f ) k B ( E,L F − ) . k f kk ϕ k L x . (4.22) Note that f ζ ε F ( f ) is linear. Furthermore, we have for all f ∈ D( A ) , k ζ ε ( Lf + εAf, · ) F k B ( E,L F − ) . k f kk ϕ k L x (cid:0) C λ k∇ x ϕ k L x ε + k ϕ k L x λ (cid:1) . (4.23) Proof
We will only prove (4 . and (4 . . For the former, we write for i ∈ N and ( f, n ) ∈ L F − × E , M ξ εi ( f, n ) = M ( f, δ ε ( n ) F ) n i ( n ) − Z E M ( f, δ ε ( n ) F ) n i ( n ) d ν ( n )= + ∞ X j =0 Z + ∞ M n i ( n ) e tM n j ( n )( f, g ( t, η j χ ε F ) F ) d t − Z E + ∞ X j =0 Z + ∞ M n i ( n ) e tM n j ( n )( f, g ( t, η j χ ε F ) F ) d t d ν ( n ) , so that, with (2 . , we have | M ξ εi ( f, n ) | . k f kk ϕ k L x . With the definition of ζ ε ,it is now easy to obtain (4 . .For (4 . , we fix i ∈ N and focus on ξ εi ( f, n ) . We have for all ( f, n ) ∈ D( A ) × E , ξ εi ( Lf + εAf, n ) = ( Lf + εAf, δ ε ( n ) F ) n i − Z E ( Lf + εAf, δ ε ( n ) F ) n i d ν ( n )= ( f, ( L − εA )[ δ ε ( n ) F ]) n i − Z E ( f, ( L − εA )[ δ ε ( n ) F ]) n i d ν ( n )= − ( f, M δ ε ( n ) F + nχ ε F ) n i + Z E ( f, M δ ε ( n ) F + nχ ε F ) n i d ν ( n ) , where we used (4 . . Thanks to (4 . , we thus obtain that, for all ( f, n ) ∈ D( A ) × E , | ξ εi ( Lf + εAf, n ) | . k f k (cid:0) C λ k∇ x ϕ k L x ε + k ϕ k L x λ (cid:1) . With the expression of ζ ε , it is now easy to get the required estimate. This con-cludes the proof. (cid:3) In this section, we precisely define the corrections of the two test functions Ψ and | Ψ | that we derived in a formal way in Subsection 4.1.First, we define a deterministic correction by Ψ ε ∗ ( f, n ) := ( f, χ ε F ) , f ∈ L F − , n ∈ E. Ψ are defined by, for ( f, n ) ∈ L F − × E , ( ϕ ε ( f, n ) := ( f, δ ε ( n ) F ) ,ϕ ε ( f, n ) := ( f, θ ε ( n ) F ) . The stochastic corrections for | Ψ | are defined by, for ( f, n ) ∈ L F − × E , ( φ ε ( f, n ) := 2( f, χ ε F )( f, δ ε ( n ) F ) ,φ ε ( f, n ) := 2( f, ζ ε ( f, n ) F ) + 2( f, χ ε F )( f, θ ε ( n ) F ) . Finally, the corrections Ψ ε, and Ψ ε, of Ψ and | Ψ | are defined by ( Ψ ε, ( f, n ) := Ψ ε ∗ + ε α ϕ ε + ε α ϕ ε , Ψ ε, ( f, n ) := | Ψ ε ∗ | + ε α φ ε + ε α φ ε . Proposition 4.7
For i = 1 , and ( f, n ) ∈ L F − × E , we have the following esti-mates: ϕ εi ( f, n ) . k f kk ϕ k L x , φ εi ( f, n ) . k f k k ϕ k L x , (4.24) M ϕ εi ( f, n ) . k f kk ϕ k L x , M φ εi ( f, n ) . k f k k ϕ k L x . (4.25) Furthermore, the functions Ψ ε ∗ , | Ψ ε ∗ | , ϕ ε , ϕ ε , φ ε and φ ε are good test functions.Besides, for ( f, n ) ∈ L F − × E , | ( f, Dφ ε ( f, n )) | . k f k k ϕ k L x . (4.26) Proof
Estimates (4 . and (4 . are justified by Cauchy Schwarz inequality and (4 . , (4 . , (4 . , (4 . , (4 . and (4 . .Concerning the fact that all the functions cited above are good test functions, wefirst note that the case of Ψ ε ∗ and | Ψ ε ∗ | is easy to prove.Let us deal with the case of ϕ ε . Conditions ( i ) and ( iii ) of Definition 3.1 areobviously verified. For condition ( ii ) , we have to prove that Dϕ ε ( f, n ) ≡ δ ε ( n ) F iscontinuous with respect to ( f, n ) ∈ L F − × E , i.e. that n δ ε ( n ) F is continuous.We recall that δ ε ( x, v, n ) F ( v ) = P + ∞ i =0 α i ( x, v, n ) in B ( E, L F − ) where α i ( x, v, n ) := Z + ∞ e tM n i ( n ) g ( t, η i χ ε F )( x + εvt, v ) d t. Now, n n i ( n ) is continuous with Lemma 2.2, and we thus have thanks to (2 . , (2 . , (2 . and the dominated convergence theorem that n α i ( n ) is continuous.Since the series of the α i defining δ ε F converges in B ( E, L F − ) , we obtain thecontinuity of n δ ε ( n ) F . Furthermore, we can show that ( f, n ) Dϕ ε ( f, n ) maps bounded sets onto bounded sets thanks to (4 . . So condition ( ii ) is verified.Similarly, by the continuity of n M n i ( n ) (Lemma 2.2) and by (4 . , we provethat condition ( iv ) is verified.Similarly, we can prove that ϕ ε , φ ε and φ ε are good test functions.Finally, since ζ ε ( f, n ) is linear in f , for ( f, n ) ∈ L F − × E , Dφ ε ( f, n )( f ) = 4( f, ζ ε ( f, n ) F ) + 4( f, χ ε F )( f, θ ε ( n ) F ) , so that (4 . , (4 . and (4 . gives (4 . . (cid:3) roposition 4.8 The function ( f, n )
7→ | Ψ ε, | ( f, n ) is a good test function. Fur-thermore, we have, for all ( f, n ) ∈ L F − × E , the following bounds: | M | ϕ ε | ( f, n ) | . k f k k ϕ k L x , | M [ ϕ ε ϕ ε ]( f, n ) | . k f k k ϕ k L x , | M | ϕ ε | ( f, n ) | . k f k k ϕ k L x , (4.27) and ε − α | M | Ψ ε, | − ε, M Ψ ε, | . k f k k ϕ k L x . (4.28) Proof
In the expression of | Ψ ε, | , since Ψ ε ∗ , ϕ ε and ϕ ε are good test functions byProposition 4.7, it is easy to prove that | Ψ ε ∗ | , Ψ ε ∗ ϕ ε and Ψ ε ∗ ϕ ε are also good testfunctions. It remains to focus on the cases of | ϕ ε | , ϕ ε ϕ ε and | ϕ ε | . We only showthe case of | ϕ ε | since the others are proved similarly.First, note that point ( i ) of Definition 3.1 is clearly verified by | ϕ ε | with D | ϕ ε | ( f, n )( h ) =2( f, δ ε ( n ) F )( h, δ ε ( n ) F ) and this function of ( f, n ) maps bounded sets onto boundedsets (thanks to (4 . ) and is continuous (is it linear in f and continuous in n since n δ ε ( n ) F is continuous, see the proof of Proposition 4.7). Then we write | ϕ ε | ( f, n ) = ( f, δ ε ( n ) F ) = + ∞ X i =0 Z + ∞ e tM n i ( n )( f, g ( t, η i χ ε F ) F ) d t ! = X i,j Z ∞ Z ∞ e tM n i ( n ) e sM n j ( n )( f, g ( t, η i χ ε F ) F )( f, g ( s, η j χ ε F ) F ) d t d s, so that, with (2 . , (2 . and Lemma 2.1, we can mimic the proof of Proposition4.4 to show that | ϕ ε | ∈ D( M ) with M | ϕ ε | ( f, n ) = X i,j Z ∞ Z ∞ M [ e tM n i e sM n j ]( n )( f, g ( t, η i χ ε F ) F )( f, g ( s, η j χ ε F ) F ) d t d s. Furthermore, with (2 . , ( f, n ) M | ϕ ε | ( f, n ) maps bounded sets onto boundedsets (it gives the first bound of (4 . ); with (2 . , (2 . and the dominated con-vergence theorem, it is continuous with respect to n . Since it is linear in f andmaps bounded sets onto bounded sets, it is continuous with respect to ( f, n ) .To sum up, we proved that | ϕ ε | ( f, n ) verifies points ( ii ) , ( iii ) and ( iv ) of Definition3.1. Finally, we obtain (4 . thanks to (4 . , (4 . and (4 . . (cid:3) We first define the limit generator L . For ψ = Ψ or ψ = | Ψ | , and all ρ ∈ L ( R d ) ,we set L ψ ( ρ ) := ( ρF, − κ ( − ∆) α Dψ ( ρF )) − Z E ( ρF nM − I ( n ) , Dψ ( ρF )) d ν ( n ) − Z E D ψ ( ρF )( ρF n, ρF M − I ( n )) d ν ( n ) , Proposition 4.9 If ( f, n ) ∈ D( A ) × E , for any λ > , we have the followingestimate: (cid:12)(cid:12) L ε Ψ ε, ( f, n ) − L Ψ( ρ ) (cid:12)(cid:12) . k f k (cid:2) Λ( ε )( k ϕ k L x + k D ϕ k L x ) + C λ k∇ x ϕ k L x ε + k ϕ k L x ε α + ( k ϕ k L x + k D ϕ k L x ) λ (cid:3) . (4.29) We can also write the right-hand side of the previous bound as k f k (Λ( ε ) C ϕ,λ + C ϕ λ ) , (4.30) where in each case Λ stands for a function which only depends on ε such that Λ( ε ) → when ε → . Proof
We recall that, thanks to Proposition 4.7, Ψ ε ∗ , ϕ ε and ϕ ε are good testfunctions. Then, we compute: L ε Ψ ε ∗ ( f, n ) = 1 ε α ( Lf + εAf, χ ε F ) + 1 ε α ( f n, χ ε F ) , where we used the fact that M Ψ ε ∗ ( f, n ) = 0 since Ψ ε ∗ does not depend on n . Wealso have ε α L ε ϕ ε ( f, n ) = 1 ε α ( Lf + εAf, δ ε ( n ) F ) + ( f n, δ ε ( n ) F ) + 1 ε α ( f, M δ ε ( n ) F )= 1 ε α ( f, ( L − εA + M )[ δ ε ( n ) F ]) + ( f n, δ ε ( n ) F ) , where we used the fact that L (resp. A ) is auto-adjoint (resp. skew-adjoint) anddue to the equation verified by δ ε F (4 . , we are led to ε α L ε ϕ ε ( f, n ) = − ε α ( f n, χ ε F ) + ( f n, δ ε ( n ) F ) . Furthermore, we have ε α L ε ϕ ε ( f, n ) = ( f, ( L − εA + M )[ θ ε ( n ) F ]) + ε α ( f n, θ ε ( n ) F ) , that we rewrite, thanks to the equation verified by θ ε F (4 . , as ε α L ε ϕ ε ( f, n ) = − ( f, θ ( n ) ϕF ) + ε α ( f n, θ ε ( n ) F ) . To sum up, L ε Ψ ε, ( f, n ) = L ε Ψ ε ∗ ( f, n ) + ε α L ε ϕ ε ( f, n ) + ε α L ε ϕ ε ( f, n ) , hence L ε Ψ ε, ( f, n ) = 1 ε α ( Lf + εAf, χ ε F ) + ( f n, δ ε ( n ) F ) − ( f, θ ( n ) ϕF ) + ε α ( f n, θ ε ( n ) F )= 1 ε α ( εAf + Lf, χ ε F ) − Z E ( f nM − I ( n ) , ϕF ) d ν ( n )+ ( f n, ( δ ε ( n ) F + M − I ( n ) ϕF )) + ε α ( f n, θ ε ( n ) F ) .
21e point out that D Ψ( f ) ≡ and ( f ψ , ψ F ) = ( ρF ψ , ψ F ) if ψ and ψ do notdepend on v ∈ V so that we have | L ε Ψ ε ( f, n ) − L Ψ( ρ ) | ≤ | ε − α ( εAf + Lf, χ ε F ) + ( κ ( − ∆) α f, ϕF ) | + | ( f n, ( δ ε ( n ) F + M − I ( n ) ϕF )) | + ε α | ( f n, θ ε ( n ) F ) | . We recall that, for all n ∈ E , k n k W , ∞ . so that ( | ( f n, ( δ ε ( n ) F + M − I ( n ) ϕF )) | . k f kk δ ε F + M − IϕF k B ( E,L F − ) , | ( f n, θ ε ( n ) F ) | . k f kk θ ε F k B ( E,L F − ) . Then the bounds (4 . , (4 . and (4 . immediately give the result; this con-cludes the proof. (cid:3) Proposition 4.10 If ( f, n ) ∈ D( A ) × E , for any λ > , we have the followingestimate: | L ε Ψ ε, ( f, n ) − L | Ψ | ( ρ ) | . Λ( ε ) C ϕ,λ k f k + C ϕ k f k λ, for a certain function Λ , which only depends on ε , such that Λ( ε ) → when ε → . Proof
We recall that, thanks to Proposition 4.7, | Ψ ε ∗ | , φ ε and φ ε are good testfunctions. Then, we compute: L ε | Ψ ε ∗ | ( f, n ) = 2 ε α ( L + εAf, χ ε F )( f, χ ε F ) + 2 ε α ( f n, χ ε F )( f, χ ε F ) , where we used the fact that M | Ψ ε ∗ | ( f, n ) = 0 since Ψ ε ∗ does not depend on n . Wealso have, with the fact that Dϕ ( f )( h ) = 2( h, χ ε F )( f, δ ε ( n ) F )+2( h, δ ε ( n ) F )( f, χ ε F ) , ε α L ε φ ε ( f, n ) = 2 ε α ( L + εAf, χ ε F )( f, δ ε ( n ) F ) + 2 ε α ( L + εAf, δ ε ( n ) F )( f, χ ε F )+ 2( f n, χ ε F )( f, δ ε ( n ) F ) + 2( f n, δ ε ( n ) F )( f, χ ε F ) + 2 ε α ( f, M δ ε ( n ) F )( f, χ ε F )= 2 ε α ( L + εAf, χ ε F )( f, δ ε ( n ) F ) + 2 ε α ( f, ( L − εA + M )[ δ ε ( n ) F ])( f, χ ε F )+ 2( f n, χ ε F )( f, δ ε ( n ) F ) + 2( f n, δ ε ( n ) F )( f, χ ε F ) . Thanks to the equation satisfied by δ ε F (4 . , we finally get ε α L ε φ ε ( f, n ) = 2 ε α ( L + εAf, χ ε F )( f, δ ε ( n ) F ) − ε α ( f n, χ ε F )( f, χ ε F )+ 2( f n, χ ε F )( f, δ ε ( n ) F ) + 2( f n, δ ε ( n ) F )( f, χ ε F ) . Besides, we have ε α L ε φ ε ( f, n ) = 2( f, ( L − εA + M )[ ζ ε ( f, n ) F ])+2( f, ( L − εA + M )[ θ ε ( n ) F ])( f, χ ε F )+ 2( Lf + εAf, χ ε F )( f, θ ε ( n ) F ) + 2( f, ζ ε ( Lf + εAf, n ) F ) + ε α ( f n, Dφ ε ( f, n )) , θ ε F and ζ ε F (4 . and (4 . , ε α L ε φ ε ( f, n ) = − f, ξ ε ϕF ) − f, θ ( n ) ϕF )( f, χ ε F )+ 2( Lf + εAf, χ ε F )( f, θ ε ( n ) F ) + 2( f, ζ ε ( Lf + εAf, n ) F ) + ε α ( f n, Dφ ε ( f, n )) . To sum up, L ε Ψ ε, ( f, n ) = L ε | Ψ ε ∗ | ( f, n ) + ε α L ε φ ε ( f, n ) + ε α L ε φ ε ( f, n ) , hence L ε Ψ ε, ( f, n ) = 2 ε α ( L + εAf, χ ε F )( f, χ ε F ) + 2 ε α ( L + εAf, χ ε F )( f, δ ε ( n ) F )+ 2( f n, χ ε F )( f, δ ε ( n ) F ) + 2( f n, δ ε ( n ) F )( f, χ ε F ) − f, ξ ε ϕF ) − f, θ ( n ) ϕF )( f, χ ε F ) + 2( Lf + εAf, χ ε F )( f, θ ε ( n ) F ) + 2( f, ζ ε ( Lf + εAf, n ) F )+ ε α ( f n, Dφ ε ( f, n )) . Now, with the definitions of θ , ξ and the limit generator L , we write the followingdecomposition L ε Ψ ε, ( f, n ) − L | Ψ | ( ρ ) = P i =1 τ i ( f, n ) , where τ : = 2 ε α ( L + εAf, χ ε F )( f, χ ε F ) − − κ ( − ∆) α f, ϕF )( f, ϕF ) ,τ : = − Z E ( f, nM − I ( n ) ϕF )( f, ( χ ε − ϕ ) F ) d ν ( n ) ,τ : = 2 Z E ( f, ( δ ε ( n ) F + M − I ( n ) ϕF ))( f n, ϕF ) d ν ( n ) ,τ : = 2( f n, ( δ ε ( n ) F + M − I ( n ) ϕF ))( f, χ ε F ) , τ := 2( f, δ ε ( n ) F )( f, ( χ ε − ϕ ) F ) ,τ : = 2 ε α ( Lf + εAf, χ ε F )( f, δ ε ( n ) F ) , τ := 2( Lf + εAf, χ ε F )( f, θ ε ( n ) F ) ,τ : = 2( f, ζ ε ( Lf + εAf, n ) F ) , τ := ε α ( f n, Dφ ε ( f, n )) . To conclude the proof, we are now about to bound every τ i . For τ , we write τ = 2 ε α ( L + εAf, χ ε F )( f, χ ε F ) − − κ ( − ∆) α f, ϕF )( f, χ ε F )+ 2( f, − κ ( − ∆) α ϕF )( f, ( χ ε − ϕ ) F ) , so that, with (4 . , | τ | . k f kk ϕ k L x (cid:12)(cid:12)(cid:12)(cid:12) ε α ( L + εAf, χ ε F ) + ( κ ( − ∆) α f, ϕF ) (cid:12)(cid:12)(cid:12)(cid:12) + 2 k f k k κ ( − ∆) α ϕ k L x k ( χ ε − ϕ ) F k , and we use (4 . and (4 . . Similarly, we bound τ thanks to (4 . , τ thanks to (4 . , τ thanks to (4 . and (4 . , τ thanks to (4 . and (4 . . For τ , we write τ = 2 ε α (cid:18) ε α ( Lf + εAf, χ ε F ) − ( − κ ( − ∆) α f, ϕF ) (cid:19) ( f, δ ε ( n ) F )+ 2 ε α ( f, − κ ( − ∆) α ϕF )( f, δ ε ( n ) F ) ,
23o that, with (4 . , | τ | . ε α k f kk ϕ k L x (cid:12)(cid:12)(cid:12)(cid:12) ε α ( L + εAf, χ ε F ) + ( κ ( − ∆) α f, ϕF ) (cid:12)(cid:12)(cid:12)(cid:12) + ε α k f k k κ ( − ∆) α ϕ k L x k ϕ k L x , and we use (4 . . We handle the case of τ similarly. We bound τ thanks to (4 . , and τ thanks to (4 . .Finally, the combination of the bounds on the τ i exactly yields the required result.This concludes the proof. (cid:3) L F − In this section, we prove a uniform estimate of the L F − norm of the solution f ε with respect to ε . To do so, we will again use the perturbed test functions method.Thus, let us begin by defining a correction function. Namely, we introduce thefunction ι ε : R d × V × E → R with ι ε ( x, v, n ) := + ∞ X i =0 Z + ∞ e tM n i ( n ) η i ( x + εvt ) d t. Similarly as Proposition 4.4, we can prove the
Proposition 5.1
The function ι ε is in L ∞ ( R d × V × E ) with k ι ε k L ∞ ( R d × V × E ) . . (5.1) It satisfies ( M − εA )( ι ε ) = − n. (5.2) Proposition 5.2
For all p ≥ and f ∈ D( A ) , we have the following bound E sup t ∈ [0 ,T ] k f εt k p . . (5.3) Proof
We set, for all f ∈ L F − , Θ( f ) := k f k , which is easily seen to be a goodtest function. Then„ with the fact that A is skew-adjoint, (2 . , and the fact that Θ does not depend on n ∈ E , we get for f ∈ D( A ) and n ∈ E , L ε Θ( f, n ) = 1 ε α ( Lf + εAf, f ) + 1 ε α ( f n, f ) + 1 ε α M Θ( f, n )= − ε α k Lf k + 1 ε α ( f n, f ) . The first term has a favourable sign to obtain our bound. The second term is moredifficult to control, and we correct Θ as follows. We set φ ε ( f, n ) = ( f, ι ε ( n ) f ) and Θ ε ( f, n ) := Θ( f, n ) + ε α φ ε ( f, n ) . We can show, with the same method as in the24roof of Proposition 4.7, that φ ε is a good test function. We then use integrationsby parts and (5 . to discover ε α L ε φ ε ( f, n ) = 2 ε α ( Lf, ι ε ( n ) f ) + 2 ε α ( εAf, ι ε ( n ) f ) + 2( f n, ι ε ( n ) f ) + 1 ε α ( f, M ι ε ( n ) f )= 2 ε α ( Lf, ι ε ( n ) f ) + 1 ε α ( f, ( M − εA )[ ι ε ( n )] f ) + 2( f n, ι ε ( n ) f )= 2 ε α ( Lf, ι ε ( n ) f ) − ε α ( f n, f ) + 2( f n, ι ε ( n ) f ) . To sum up, since L ε Θ ε ( f, n ) = L ε Θ( f, n ) + ε α L ε φ ε ( f, n ) , we have L ε Θ ε ( f, n ) = − ε α k Lf k + 2 ε α ( Lf, ι ε ( n ) f ) + 2( f n, ι ε ( n ) f ) . We use (5 . to bound the second term: ε α ( Lf, ι ε ( n ) f ) . ε α k Lf kk f k≤ k Lf k ε α + 12 k f k . k Lf k ε α + k f k . Besides, note that with (5 . the third term is . k f k . Finally we just proved that | L ε Θ ε ( f, n ) | . k f k . (5.4)As in Proposition 3.1, since Θ ε is a good test function, we now set, M ε Θ ε ( t ) := Θ ε ( f εt , m εt ) − Θ ε ( f ε , m ε ) − Z t L ε Θ ε ( f εs , m εs ) d s, which is a continuous and integrable ( F εt ) t ≥ martingale. By definition of Θ , Θ ε and M ε , k f εt k = 12 k f ε k − ε α ( φ ε ( f εt , m εt ) − φ ε ( f ε , m ε )) + Z t L ε Θ ε ( f εs , m εs ) d s + M ε Θ ε ( t ) . Since with (5 . we have | φ ε ( f, n ) | . k f k , we can write, with (5 . , k f εt k . k f ε k + ε α k f εt k + Z t k f εs k d s + sup t ∈ [0 ,T ] | M ε Θ ε ( t ) | , that is, for ε sufficiently small and by Gronwall Lemma, k f εt k . k f ε k + sup t ∈ [0 ,T ] | M ε Θ ε ( t ) | . (5.5)Furthermore, similarly as Proposition 4.8, we can show that | Θ ε | is a good testfunction, and that | L ε | Θ ε | − ε L ε Θ ε | = ε − α | M | Θ ε | − ε M Θ ε | . k f k (1 + Λ( ε )) , Λ which only depends on ε and such that Λ( ε ) → as ε → . Inparticular, for ε small enough, | L ε | Θ ε | − ε L ε Θ ε | . k f k . Besides, with Proposition 3.1, the quadratic variation of M ε Θ ε ( t ) is given by h M ε Θ ε i t = Z t ( L ε | Θ ε | − ε L ε Θ ε )( f εs , m εs ) d s. As a result, with Burkholder-Davis-Gundy and Hölder inequalities, we get E [ sup t ∈ [0 ,T ] | M ε Θ ε | p ] . E [ |h M ε Θ ε i T | p ] . Z T E [ k f εs k p ] d s. (5.6)By (5 . , we have E [ k f εt k p ] . E [ k f ε k p ] + E [ sup t ∈ [0 ,T ] | M ε Θ ε ( t ) | p ] , so that we get E [ k f εT k p ] . E [ k f ε k p ] + Z T E [ k f εs k p ] d s, that is, by Gronwall lemma, E [ k f εT k p ] . E [ k f ε k p ] . This actually holds true for any t ∈ [0 , T ] . Thus, using (5 . and then (5 . givesfinally the result. (cid:3) In this section we state the following proposition which sums up all the resultsobtained above. This will be convenient to handle the tightness and convergencesteps. We recall that the corrections Ψ ε,i , i = 1 , are defined in Section 4.4. Proposition 6.1
Let f ε ∈ D( A ) . For i = 1 , , M εi ( t ) := Ψ ε,i ( f εt , m εt ) − Ψ ε,i ( f ε , m ε ) − Z t L ε Ψ ε,i ( f εs , m εs ) d s, t ∈ [0 , T ] , is a continuous and integrable martingale for the filtration ( F εt ) t ≥ generated by ( m εt , t ≥ . The quadratic variation of M ε is given by h M ε i t = Z t ( L ε | Ψ ε, | − ε, L ε Ψ ε, )( f εs , m εs ) d s, t ∈ [0 , T ] and we have, for all t ∈ [0 , T ] , | L ε | Ψ ε, | − ε, L ε Ψ ε, | ( f εt , m εt ) . sup t ∈ [0 ,T ] k f εt k k ϕ k L x . (6.1)26 urthermore, for any λ > , ≤ s ≤ · · · ≤ s n ≤ s ≤ t and G ∈ C b (( L ( R d )) n ) , (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20)(cid:18) Ψ( ρ εt F ) − Ψ( ρ εs F ) − Z ts L Ψ( ρ εσ ) d σ (cid:19) G ( ρ εs , ..., ρ εs n ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) . Λ( ε ) C ϕ,λ + C ϕ λ, (6.2) (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20)(cid:18) | Ψ | ( ρ εt F ) − | Ψ | ( ρ εs F ) − Z ts L | Ψ | ( ρ εσ ) d σ (cid:19) G ( ρ εs , ..., ρ εs n ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) . Λ( ε ) C ϕ,λ + C ϕ λ, (6.3) for some function Λ , which only depends on ε , such that Λ( ε ) → when ε → .Finally, for all t ∈ [0 , T ] , we have the following estimate: | L ε Ψ ε, | ( f εt , m εt ) . sup t ∈ [0 ,T ] k f εt k ( k ϕ k L x + k∇ x ϕ k L x + k D ϕ k L x + k ( − ∆) α ϕ k L x ) . (6.4) Proof
For i = 1 , , Proposition 4.7 gives that Ψ ε,i is a good test function, and itimplies, with Proposition 3.1, that M εi is a continuous and integrable martingale.Besides, with Proposition 4.8, | Ψ ε, | is a good test function, hence the formula forthe quadratic variation of M ε .Note that L ε | Ψ ε, | − ε, L ε Ψ ε, = ε − α ( M | Ψ ε, | − ε, M Ψ ε, ) from whichwe deduce (6.1) due to (4.28).We continue with the proof of (6 . . Observe that Ψ = Ψ ε, +(Ψ − Ψ ε ∗ ) − ε α ϕ ε − ε α ϕ ε so that we can write Ψ( f εt ) − Ψ( f εs ) − Z ts L Ψ( ρ εσ ) d σ = M ε ( t ) − M ε ( s )+ (Ψ − Ψ ε ∗ )( f εt ) − (Ψ − Ψ ε ∗ )( f εs ) − ε α ϕ ε ( f εt ) − ε α ϕ ε ( f εt )+ ε α ϕ ε ( f εs ) + ε α ϕ ε ( f εs ) + Z ts L ε Ψ ε, ( f εσ , m εσ ) − L Ψ( ρ εσ ) d σ. Then, we multiply by G ( ρ εs , ..., ρ εs n ) and take the expectation. Note that, since M ε is a martingale for the filtration ( F εt ) t ≥ generated by ( m εt , t ≥ , we have E [( M ε ( t ) − M ε ( s )) G ( J − ηr ρ εs , ..., J − ηr ρ εs n )] = 0 . Then, it suffices to use (4 . , (4 . , (4 . , the uniform L F − bound (5 . and Ψ( f ) = Ψ( ρF ) to obtain (6 . . A similar work can be done to obtain (6 . .It remains to prove (6 . . We simply write, for ( f, n ) ∈ D( A ) × E , | L ε Ψ ε, ( f, n ) | ≤ | L ε Ψ ε, ( f, n ) − L Ψ( f, n ) | + | L Ψ( f, n ) | . We apply (4 . with ε ≤ and λ = 1 so that | L ε Ψ ε, ( f, n ) − L Ψ( f, n ) | . k f k ( k ϕ k L x + k∇ x ϕ k L x + k D ϕ k L x ) . Since, clearly, | L Ψ( f, n ) | . k f k ( k κ ( − ∆) α ϕ k L x + k ϕ k L x ) , the proof is complete. (cid:3) Tightness
In this section, in order to be able to take the limit ε → in law of the fam-ily of processes ( ρ ε ) ε> , we prove its tightness in an appropriate space, namely C ([0 , T ] , S − η ( R d )) . Precisely, the result is the following. Proposition 7.1
Let η > . Then the family ( ρ ε ) ε> is tight in C ([0 , T ] , S − η ( R d )) . Proof
We will here specialize the test function ϕ ∈ S ( R d ) into the functions ( p j ) j ∈ N d , which are defined in Section 2.1. So we set, for j ∈ N d and f ∈ L F − , Ψ j ( f ) := ( f, p j F ) , and we index by j ∈ N d all the corrections defined in Section 4.4. Thanks toProposition 6.1, we consider the continuous martingales M ε ,j ( t ) := Ψ ε ,j ( f εt , m εt ) − Ψ ε ,j ( f ε , m ε ) − Z t L ε Ψ ε ,j ( f εs , m εs ) d s. We also define, for j ∈ N d and t ∈ [0 , T ] , θ εj ( t ) := Ψ j ( f ε ) + Z t L ε Ψ ε ,j ( f εs , m εs ) d s + M ε ,j ( t ) . Note that θ εj ( t ) = Ψ j ( f ε ) + Ψ ε ,j ( f εt , m εt ) − Ψ ε ,j ( f ε , m ε ) , so that, with Cauchy-Schwarz inequality and (4 . , | θ εj ( t ) | . sup t ∈ [0 ,T ] k f ε ( t ) kk p j k L x = sup t ∈ [0 ,T ] k f ε ( t ) k . Hence, by the uniform L F − bound (5 . , E sup t ∈ [0 ,T ] (cid:12)(cid:12) θ εj ( t ) (cid:12)(cid:12) . . (7.1)We now observe that, for t ∈ [0 , T ] , Ψ j ( f εt ) − θ εj ( t ) = (cid:2) (Ψ j − Ψ ε ∗ ,j ) − ε α ϕ ε ,j − ε α ϕ ε ,j (cid:3) ( f εt , m εt ) − (cid:2) (Ψ j − Ψ ε ∗ ,j ) − ε α ϕ ε ,j − ε α ϕ ε ,j (cid:3) ( f ε , m ε ) , and it gives, with Cauchy-Schwarz inequality, (4 . , (4 . ), and (2 . , (cid:12)(cid:12) Ψ j ( f εt ) − θ εj ( t ) (cid:12)(cid:12) . sup t ∈ [0 ,T ] k f εt kk ( χ εj − p j ) F k + ( ε α + ε α ) k f εt kk p j k L x ≤ sup t ∈ [0 ,T ] k f εt k ( C λ ε k∇ x p j k L x + k p j k L x λ + ( ε α + ε α ) k p j k L x ) ≤ sup t ∈ [0 ,T ] k f εt k ( C λ εµ j + λ + ε α + ε α ) . (7.2)28rom now on, we fix γ > d/ . Observe that, by (7 . , a.s. and for all t ∈ [0 , T ] , the series defined by u εt := P j ∈ N d θ εj ( t ) J − γ p j converges in L ( R d ) , which isembedded in S ′ ( R d ) . We then set θ εt := J γ X j ∈ N d θ εj ( t ) J − γ p j , which exists a.s. and for all t ∈ [0 , T ] in S ′ ( R d ) . In fact, we see that a.s. and forall t ∈ [0 , T ] , θ εt is in S − γ ( R d ) . Indeed, k θ εt k S − γ ( R d ) = k J γ u εt k S − γ ( R d ) = k u εt k L x < ∞ . We point out that Ψ j ( f εt ) = ( ρ εt F, p j F ) = ( ρ εt , p j ) x so that h ρ ε ( t ) − θ ε ( t ) , p j i = Ψ j ( f εt ) − h J γ u εt , p j i = Ψ j ( f εt ) − h u εt , J γ p j i = Ψ j ( f εt ) − h u εt , p j i µ γj = Ψ j ( f εt ) − θ εj ( t ) . By (7 . , it permits to write, for t ∈ [0 , T ] , k ρ ε ( t ) − θ ε ( t ) k S − γ ( R d ) . X j ∈ N d µ − γj sup t ∈ [0 ,T ] k f εt k ( C λ ε µ j + λ + ε α + ε α ) . sup t ∈ [0 ,T ] k f εt k ( C λ ε + ε α + ε α + λ ) where the second bound comes from our choice γ > d/ (we recall, see Section2.1, that µ j = 2 | j | + 1 ). Thanks to the uniform L F − bound (5 . , it finally leadsto the following estimate: E sup t ∈ [0 ,T ] k ρ ε ( t ) − θ ε ( t ) k S − γ ( R d ) . C λ ε + ε α + ε α + λ. (7.3)We now fix η > . For any δ > , let w ( ρ, δ ) := sup | t − s | <δ k ρ ( t ) − ρ ( s ) k S − η ( R d ) denote the modulus of continuity of a function ρ ∈ C ([0 , T ] , S − η ( R d )) . Since theinjection L ( R d ) ⊂ S − η ( R d ) is compact, and by Ascoli’s theorem, the set K R := ( ρ ∈ C ([0 , T ] , S − η ( R d )) , sup t ∈ [0 ,T ] k ρ k L ( R d ) ≤ R, w ( ρ, δ ) < ε ( δ ) ) , where R > and ε ( δ ) → when δ → , is compact in C ([0 , T ] , S − η ( R d )) . To provethe tightness of ( ρ ε ) ε> in C ([0 , T ] , S − η ( R d )) , it thus suffices, see [2], to prove thatfor all σ > , there exists R > such that P ( sup t ∈ [0 ,T ] k ρ ε k L ( R d ) > R ) < σ, (7.4)and lim δ → lim sup ε → P ( w ( ρ ε , δ ) > σ ) = 0 . (7.5)29y the continuous embedding L ( R d ) ⊂ S − η ( R d ) and Markov’s inequality, we have P ( sup t ∈ [0 ,T ] k ρ ε k L ( R d ) > R ) ≤ P ( sup t ∈ [0 ,T ] k f ε k L F − > R ) ≤ R E [ sup t ∈ [0 ,T ] k f ε k L F − ] , and it gives (7 . thanks to the uniform L F − bound (5 . .Similarly, we will deduce (7 . by Markov’s inequality and a bound on E [ w ( ρ ε , δ )] for δ > . Actually, by interpolation, the continuous embedding L ( R d ) ⊂ S − η ( R d ) and the uniform L F − bound (5 . , we have E sup | t − s | <δ k ρ ( t ) − ρ ( s ) k S − η♭ ≤ E sup | t − s | <δ k ρ ( t ) − ρ ( s ) k υS − η♯ for a certain υ > if η ♯ > η ♭ > . As a result, it is indeed sufficient to work with η = γ . In view of (7 . , we first want to obtain an estimate of the increments of θ ε . We have, for j ∈ N d and ≤ s ≤ t ≤ T , θ εj ( t ) − θ εj ( s ) = Z ts L ε Ψ ε ,j ( f εσ , m εσ ) d σ + M ε ,j ( t ) − M ε ,j ( s ) . By (6 . and the uniform L F − bound (5 . , we have E (cid:12)(cid:12)(cid:12)(cid:12)Z ts L ε Ψ ε ,j ( f εσ , m εσ ) d σ (cid:12)(cid:12)(cid:12)(cid:12) . C j | t − s | , where C j := ( k p j k L x + k∇ x p j k L x + k D p j k L x + k ( − ∆) α p j k L x ) . Furthermore, using Burkholder-Davis-Gundy inequality, E | M ε ,j ( t ) − M ε ,j ( s ) | . E |h M ε ,j i t − h M ε ,j i s | , and thanks to (6 . , the uniform L F − bound (5 . and the fact that k p j k L x = 1 ,we are led to E | M εj ( t ) − M εj ( s ) | . | t − s | . Finally we have E | θ εj ( t ) − θ εj ( s ) | . (1 + C j ) | t − s | . Now, note that with (2 . , C j . √ µ j + µ j . Since we took γ > d/ , we can conclude that E k θ εt − θ εs k S − γ ( R d ) . | t − s | . It easily follows that, for υ < / , E k θ ε k W υ, (0 ,T,S − γ ( R d )) . so that by the Sobolevembedding W υ, (0 , T, S − γ ( R d )) ⊂ C ,τ (0 , T, S − γ ( R d )) which holds true whenever τ < υ − / , we obtain that E w ( θ ε , δ ) . δ τ for a certain positive τ .Thus, we deduce, with (7 . , E w ( ρ ε , δ ) ≤ E sup t ∈ [0 ,T ] k ρ εt − θ εt k S − γ ( R d ) + E w ( θ ε , δ ) . C λ ε + ε α + ε α + λ + δ τ . To conclude, we then have lim δ → lim sup ε → P ( w ( ρ ε , δ ) > σ ) ≤ lim δ → lim sup ε → σ − E w ( ρ ε , δ ) . σ − λ, and since λ > was arbitrary, we just proved (7 . . This concludes the proof. (cid:3) Convergence
In this section, we conclude the proof of Theorem 2.4. The idea is now, by thetightness result proved above and Prokhorov’s Theorem, to take a subsequence of ( ρ ε ) ε> that converges in law to some probability measure. Then we show thatthis limit probability is actually uniquely determined thanks to the convergencesto the limit generator L proved above.Let us fix η > . By Proposition and Prokhorov’s Theorem, there exist asubsequence of ( ρ ε ) ε> , still denoted ( ρ ε ) ε> , and a probability measure P on C ([0 , T ] , S − η ( R d )) such that P ε → P weakly on C ([0 , T ] , S − η ( R d )) , where P ε stands for the law of ρ ε . We will now identify the probability measure P . Since C ([0 , T ] , S − η ( R d )) is separable, we can apply Skohorod representationTheorem [2], so that there exist a new probability space ( e Ω , e F , e P ) and randomvariables e ρ ε , e ρ : e Ω → C ([0 , T ] , S − η ( R d )) , with respective law P ε and P such that e ρ ε → e ρ in C ([0 , T ] , S − η ( R d )) , e P − a.s. Inthe sequel, for the sake of clarity, we do not write any more the tildes.Let us pass to the limit ε → in the left-hand side of (6.2), namely in the quantity E (cid:20)(cid:18) Ψ( ρ εt F ) − Ψ( ρ εs F ) − Z ts L Ψ( ρ εσ ) d σ (cid:19) G ( ρ εs , ..., ρ εs n ) (cid:21) =: E [ A ( ρ ε )] . Without loss of any generality, we may assume that the function G ∈ C b (( L ( R d )) n ) is also continuous on the space H − η ; this is always possible with an approximationargument: it suffices to consider G r := G (( I + rJ ) − η · , ..., ( I + rJ ) − η · ) as r → .Then, with the P − a.s. convergence of ρ ε to ρ in the space C ([0 , T ] , S − η ( R d )) , wehave that A ( ρ ε ) → A ( ρ ) , a.s.Furthermore, thanks to the uniform L F − bound (5.3) and the boundedness of G , ( A ( ρ ε )) ε> is uniformly integrable since it is bounded in L (Ω) , hence E A ( ρ ε ) → E A ( ρ ) . As a consequence, we can now pass to the limit ε → in (6.2) to discover (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20)(cid:18) Ψ( ρ t F ) − Ψ( ρ s F ) − Z ts L Ψ( ρ σ ) d σ (cid:19) G ( ρ s , ..., ρ s n ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) . C ϕ λ. Since this holds true for arbitrary λ > , it yields E (cid:20)(cid:18) Ψ( ρ t F ) − Ψ( ρ s F ) − Z ts L Ψ( ρ σ ) d σ (cid:19) G ( ρ s , ..., ρ s n ) (cid:21) = 0 . (8.1)31imilarly, we can pass to the limit ε → in (6.3); it gives E (cid:20)(cid:18) | Ψ | ( ρ t F ) − | Ψ | ( ρ s F ) − Z ts L | Ψ | ( ρ σ ) d σ (cid:19) G ( ρ s , ..., ρ s n ) (cid:21) = 0 . (8.2)Since (8 . and (8 . are valid for all n ∈ N , s ≤ ... ≤ s n ≤ s ≤ t ∈ [0 , T ] and all G ∈ C b (( L ( R d )) n ) , we deduce that N ( t ) := Ψ( ρ t F ) − Ψ( ρ F ) − Z t L Ψ( ρ σ ) d σ, t ∈ [0 , T ] , and S ( t ) := | Ψ | ( ρ t F ) − | Ψ | ( ρ F ) − Z t L | Ψ | ( ρ σ ) d σ, t ∈ [0 , T ] , are martingales with respect to the filtration generated by ( ρ s ) s ∈ [0 ,T ] . It impliesthat, see [7, Appendix 6.9], the quadratic variation of N is given by h N i t = Z t (cid:2) L | Ψ | ( ρ σ ) − ρ σ ) L Ψ( ρ σ ) (cid:3) d σ, t ∈ [0 , T ] . Furthermore, we have L | Ψ | ( ρ σ ) − ρ σ ) L Ψ( ρ σ ) = − Z E ( ρ σ n, ϕ ) x ( ρ σ M − I ( n ) , ϕ ) x d ν ( n )= 2 E [ Z ∞ ( ρ σ m , ϕ ) x ( ρ σ m t , ϕ ) x d t ]= E [ Z R ( ρ σ m , ϕ ) x ( ρ σ m t , ϕ ) x d t ]= Z R d Z R d ρ σ ( x ) ϕ ( x ) ρ σ ( y ) ϕ ( y ) k ( x, y ) d x d y = k ρ σ Q ϕ k L x . Here, we recall that Ψ( ρF ) = ( ρF, ϕF ) = ( ρ, ϕ ) x and that the results above arevalid for all ϕ ∈ S ( R d ) . As a consequence, the martingale N gives us that M ( t ) := ρ t − ρ − Z t [ − κ ( − ∆) α ρ σ − ρ σ H ] d σ, t ∈ [0 , T ] , is a continuous martingale in L ( R d ) with respect to the filtration generated by ( ρ s ) s ∈ [0 ,T ] with quadratic variation h M i t = Z t ( ρ σ Q )( ρ σ Q ) ∗ d σ, t ∈ [0 , T ] . Thanks to martingale representation Theorem, see [4, Theorem 8.2], up to a changeof probability space, there exists a cylindrical Wiener process W in L ( R d ) suchthat ρ t − ρ − Z t [ − κ ( − ∆) α ρ σ − ρ σ H ] d σ = Z t ρ σ Q d W σ , t ∈ [0 , T ] . ρ has the law of a weak solution to the equation (2 . withpaths in C ([0 , T ] , S − η ( R d )) . Since this equation has a unique solution with pathsin the space C ([0 , T ] , S − η ( R d )) ∩ L ∞ ([0 , T ] , L ( R d )) , and since pathwise unique-ness implies uniqueness in law, we deduce that P is the law of this solution andis uniquely determined. Finally, by the uniqueness of the limit, the whole se-quence ( P ε ) ε> converges to P weakly in the space of probability measures on C ([0 , T ] , S − η ( R d )) . (cid:3) Appendix A
Proof of Proposition 4.1
For the first bound, we write, thanks to Cauchy-Schwarzinequality, k χ ε F k = Z R d Z V (cid:18)Z + ∞ e − t ϕ ( x + εvt ) d t (cid:19) F ( v ) d v d x ≤ Z R d Z V Z + ∞ e − t ϕ ( x + εvt ) F ( v ) d t d v d x = k ϕ k L x Z V Z + ∞ e − t F ( v ) d t d v = k ϕ k L x . To prove the second estimate, we fix λ > . Since F is integrable with respect to v , we take C λ > such that R {| v |≥ C λ } F ( v ) d v < λ . We have k ( χ ε − ϕ ) F k = Z R d Z V (cid:18)Z + ∞ e − t [ ϕ ( x + εvt ) − ϕ ( x )] d t (cid:19) F ( v ) d v d x. Then we split the v -integral into two terms τ and τ : τ := Z R d Z | v |≥ C λ Z + ∞ e − z [ ϕ ( x + εvz ) − ϕ ( x )] F ( v ) d z d v d x ≤ Z R d Z | v |≥ C λ Z + ∞ e − z (cid:0) | ϕ ( x + εvz ) | + | ϕ ( x ) | (cid:1) F ( v ) d z d v d x = 4 k ϕ k L x Z | v |≥ C λ Z + ∞ e − z F ( v ) d z d v < k ϕ k L x λ ; τ := Z R d Z | v |≤ C λ Z + ∞ e − z [ ϕ ( x + εvz ) − ϕ ( x )] F ( v ) d z d v d x = Z R d Z | v |≤ C λ Z + ∞ e − z (cid:18)Z εzv · ∇ x ϕ ( x + tεzv ) d t (cid:19) F ( v ) d z d v d x ≤ ε Z R d Z | v |≤ C λ Z + ∞ Z e − z z | v | |∇ x ϕ ( x + tεzv ) | F ( v ) d t d z d v d x ≤ ε C λ k∇ x ϕ k L x , and this is the result. (cid:3) roof of Lemma 4.2 We fix λ > . Then we choose C such that, for all | v | ≥ C , (cid:12)(cid:12)(cid:12)(cid:12) F ( v ) − κ | v | d + α (cid:12)(cid:12)(cid:12)(cid:12) ≤ λκ | v | d + α . (8.3)Now, we write, for x ∈ R d , ε − α Z V Z + ∞ e − t [ ϕ ( x + εvt ) − ϕ ( x )] F ( v ) d t d v = ε − α Z | v |≤ C Z + ∞ e − t [ ϕ ( x + εvt ) − ϕ ( x )] F ( v ) d t d v + ε − α Z | v |≥ C Z + ∞ e − t [ ϕ ( x + εvt ) − ϕ ( x )] κ | v | d + α d t d v + ε − α Z | v |≥ C Z + ∞ e − t [ ϕ ( x + εvt ) − ϕ ( x )] (cid:20) F ( v ) − κ | v | d + α (cid:21) d t d v =: I ( x ) + I ( x ) + I ( x ) . We begin by bounding k I k L x . Since F ( v ) = F ( − v ) , we rewrite I ( x ) as follows I ( x ) = ε − α Z | v |≤ C Z + ∞ e − t [ ϕ ( x + εvt ) − ϕ ( x ) − εvt · ∇ x ϕ ( x )] F ( v ) d t d v = ε − α Z | v |≤ C Z + ∞ Z e − t (cid:2) D ϕ ( x + εvts )( εvt, εvt ) (cid:3) F ( v ) d s d t d v. Then, with Cauchy-Schwarz inequality, we can write k I k L x = ε − α Z R d Z | v |≤ C Z + ∞ Z e − t (cid:2) D ϕ ( x + εvts )( εvt, εvt ) (cid:3) F ( v ) d s d t d v ! d x ≤ ε − α Z R d Z | v |≤ C Z + ∞ Z e − t ε | v | t | D ϕ ( x + εvts ) | F ( v ) d s d t d v d x = ε − α k D ϕ k L x Z | v |≤ C Z + ∞ Z e − t t | v | F ( v ) d s d t d v ≤ C ε − α k D ϕ k L x . We are now interested in I . We first rewrite I thanks to the change of variables w := εvtI ( x ) = ε − α Z + ∞ Z | w |≥ Cεt e − t [ ϕ ( x + w ) − ϕ ( x )] κ | εt | d + α | w | d + α dwε d t d d t = κ Z + ∞ Z | w |≥ Cεt e − t | t | α [ ϕ ( x + w ) − ϕ ( x )] dw | w | d + α d t. − ( − ∆) α ϕ ( x ) = c d,α PV Z V [ ϕ ( x + w ) − ϕ ( x )] dw | w | d + α = c d,α Z | w |≥ [ ϕ ( x + w ) − ϕ ( x )] dw | w | d + α + c d,α Z | w |≤ [ ϕ ( x + w ) − ϕ ( x ) − w · ∇ x ϕ ( x )] dw | w | d + α = L ( x ) + L ( x ) . It prompts us to use a similar decomposition of I ( x ) ; we thus write I ( x ) = κ Z / ( Cε )0 e − t | t | α Z | w |≥ [ ϕ ( x + w ) − ϕ ( x )] dw | w | d + α d t + κ Z / ( Cε )0 e − t | t | α Z Cεt ≤| w |≤ [ ϕ ( x + w ) − ϕ ( x ) − w · ∇ x ϕ ( x )] dw | w | d + α d t + κ Z + ∞ / ( Cε ) e − t | t | α Z | w |≥ Cεt [ ϕ ( x + w ) − ϕ ( x )] dw | w | d + α d t = J ( x ) + J ( x ) + J ( x ) . We recall the definition (2 . of κ : κ = κ c d,α Z + ∞ e − t | t | α d t. Then, with Cauchy-Schwarz inequality, k J − κL k L x = Z R d κ Z + ∞ / ( Cε ) e − t | t | α Z | w |≥ [ ϕ ( x + w ) − ϕ ( x )] dw | w | d + α d t ! d x ≤ κ Z | w |≥ dw | w | d + α ! Z R d Z + ∞ / ( Cε ) e − t | t | α Z | w |≥ [ ϕ ( x + w ) − ϕ ( x )] dw | w | d + α d t d x ≤ κ Z | w |≥ dw | w | d + α ! k ϕ k L x Z + ∞ / ( Cε ) e − t | t | α d t. To continue, we decompose J ( x ) − κL ( x ) into two terms τ ( x ) + τ ( x ) − κ Z / ( Cε )0 e − t | t | α Z ≤| w |≤ Cεt Z D ϕ ( x + ws )( w, w ) d s dw | w | d + α d t − κ Z + ∞ / ( Cε ) e − t | t | α Z | w |≤ Z D ϕ ( x + ws )( w, w ) d s dw | w | d + α d t. We work on k τ k L x , using Cauchy-Schwarz inequality, and the change of variables35 = w/ ( εt ) : k τ k L x = Z R d κ Z / ( Cε )0 e − t | t | α Z ≤| w |≤ Cεt Z D ϕ ( x + ws )( w, w ) d s dw | w | d + α d t ! d x ≤ Z R d κ Z / ( Cε )0 e − t | t | α Z ≤| w |≤ Cεt Z | D ϕ ( x + ws ) | d s dw | w | d + α − d t ! d x ≤ κ Z | w |≤ dw | w | d + α − Z R d Z / ( Cε )0 e − t | t | α Z ≤| w |≤ Cεt Z | D ϕ ( x + ws ) | d s dw | w | d + α − d t d x ≤ κ Z | w |≤ dw | w | d + α − k D ϕ k L x Z + ∞ e − t | t | α Z ≤| w |≤ Cεt dw | w | d + α − d t = κ Z | w |≤ dw | w | d + α − Z + ∞ e − t | t | α +2 d t Z | v |≤ C d v | v | d + α − ε − α k D ϕ k L x . With the same kind of computations, we are led to k τ k L x ≤ κ Z | w |≤ dw | w | d + α − ! k D ϕ k L x Z + ∞ / ( Cε ) e − t | t | α d t, and k J k L x ≤ κ Z | w |≥ dw | w | d + α ! k ϕ k L x Z + ∞ / ( Cε ) e − t | t | α d t. Finally, about the case of I , thanks to (8 . , we can do the same work as for I ;then we just have to put together all the bounds obtained to get the result. Thisconcludes the proof. (cid:3) Proof of Lemma 4.3
First, we write ε − α ( Lf + εAf, χ ε F ) = ε − α Z R d Z V ρF χ ε − f χ ε − εv · ∇ x f χ ε d v d x = ε − α Z R d Z V ρF χ ε − f ( χ ε − εv · ∇ x χ ε ) d v d x = ε − α Z R d Z V ρF χ ε − f ϕ d v d x = Z R d ρ Z V ε − α [ χ ε − ϕ ] F d v d x, where we used an integration by part and (4 . . Furthermore, we have ( − κ ( − ∆) α f, ϕF ) = ( f, − κ ( − ∆) α ϕF )= − κ Z R d Z V f ( − ∆) α ϕ d v d x = − κ Z R d ρ ( − ∆) α ϕ d v d x.
36s a consequence, with Cauchy-Schwarz inequality, we get ε − α ( εAf + Lf, χ ε F ) + ( κ ( − ∆) α f, ϕF ) = Z R d ρ (cid:20)Z V ε − α [ χ ε − ϕ ] F d v + κ ( − ∆) α ϕ (cid:21) d x ≤ k ρ k L x (cid:13)(cid:13)(cid:13)(cid:13)Z V ε − α [ χ ε − ϕ ] F d v + κ ( − ∆) α ϕ (cid:13)(cid:13)(cid:13)(cid:13) L x ≤ k f k (cid:13)(cid:13)(cid:13)(cid:13)Z V ε − α [ χ ε − ϕ ] F d v + κ ( − ∆) α ϕ (cid:13)(cid:13)(cid:13)(cid:13) L x , and an application of Lemma . then concludes the proof. (cid:3) References [1] G. Basile and A. Bovier.
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