Asymptotic behavior of a class of multiple time scales stochastic kinetic equations

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Asymptotic behavior of a class of multiple time scales stochastic kinetic equations

Charles-Edouard Bréhier, Shmuel Rakotonirina–Ricquebourg

To cite this version:

Charles-Edouard Bréhier, Shmuel Rakotonirina–Ricquebourg. Asymptotic behavior of a class of multiple time scales stochastic kinetic equations. 2021. �hal-03258628�

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Asymptotic behavior of a class of multiple time scales stochastic kinetic equations

Charles-Edouard Bréhier and Shmuel Rakotonirina-Ricquebourg

^∗

June 11, 2021

Abstract

We consider a class of stochastic kinetic equations, depending on two time scale separation parametersεandδ: the evolution equation contains singular terms with respect toε, and is driven by a fast ergodic process which evolves at the time scalet{δ². We prove that when pε, δq Ñ p0,0q the density converges to the solution of a linear diffusion PDE. This is a mixture of diffusion approximation in the PDE sense (with respect to the parameterε) and of averaging in the probabilistic sense (with respect to the parameterδ). The proof employs stopping times arguments and a suitable perturbed test functions approach which is adapted to consider the general regimeε‰δ.

1 Introduction

Multiscale and/or stochastic models are popular in all fields of science and engineering. In this paper, we consider a stochastic kinetic partial differential equation of the type

Btf^ε,δ`1

εapvq ¨∇xf^ε,δ`bpvq ¨∇xf^ε,δ`σpm^δpt, xqqf^ε,δ“ 1

ε²Lf^ε,δ, (1) with initial conditionf^ε,δp0q “f₀^ε,δ. The unknow f^ε,δ is a function of timetě0, positionxPT^d (the flat d-dimensional torus) and velocityvPV. Assumptions on the velocity fieldsaandband on the mappingσare given below (see Section2). Note thatf^ε,δpt, x, vqmay be interpreted as a density of particles with positionxand velocity v at time t; the system is not conservative (the integral of f^ε,δ is not constant) due to the source termσpm^δqf^ε,δ. In addition, the linear operatorLdescribes interactions between the particles: in this paper, we assume thatLis the Bhatnagar-Gross-Krook operator, given by

Lf “ρM´f,

where µis aσ-finite measure onV, the spatial density is defined byρ .

“hfi .

“ş

V f dµ, and where MPL¹pV, dµqis a density function, often called the Maxwellian (see Assumption1).

The evolution equation (1) depends on the so-called driving processm^δ, defined as follows:

one hasm^δpt, xq “mpt{δ², xq.

∗Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France,brehier@math.univ-lyon1.fr,rakoto@math.univ-lyon1.fr

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The stochastic evolution equation (1) depends on two parametersεandδ. In this paper, we are interested in the asymptotic behavior when pε, δq Ñ p0,0q. We prove the following result:

ρ^ε,δ“ f^ε,δ

converges to the solutionρof the following partial differential equation:

Btρ`J¨∇ρ`σ ρ“divpK∇ρq, (2)

where J, σ andK are defined below, with initial condition ρp0q “ρ₀ “limρ^ε,δp0q. We refer to the main results of this paper, Theorems3.2 and3.3 for rigorous statements, in particular concerning the mode of convergence. Let us describe how the form of the limit equation (2) arises in the asymptotic regime.

On the one hand, the parameterεdrives the behavior of the deterministic part of the evolution.

Under appropriate assumptions (including a centering condition for the velocity field a), the spatial densityρ^ε,δ“hfi^ε,δconverges whenεÑ0 to the solution of a diffusion partial differential equation, where the velocity variablevhas been eliminated. In the literature, such convergence results are referred to as diffusion approximation results, see for instance [DGP00]. The result is also partly an averaging result, since the termbpvq ¨∇x is replaced in the limit equation by J¨∇x, where J is the average of b(with respect to an appropriate measure).

On the other hand, the parameterδis a time-scale separation parameter which determines the random part of the evolution. When the processm such thatm^δpt, xq “mpt{δ², xqis assumed to be ergodic (see Section2, for instance one may consider an Ornstein-Uhlenbeck process), the randomness may be eliminated when δÑ 0: only the averageσ of σpmqwith respect to the invariant distribution remains in the limit evolution equation (it is a law of large numbers effect).

In the literature, such convergence results are referred to as averaging principle results.

In this paper, we thus prove the mixture of an diffusion approximation result in the PDE sense, and of an averaging principle result in the probability sense, when simultaneouslyεÑ0 andδÑ0. To the best of our knowledge, this regime has not been considered in the literature so far. Note that one of the major tasks in the analysis is to consider the general case whenεandδ go to 0 independently. Indeed, the analysis would be simpler ifε“δ. Note also that it would be simpler ifε“0 and δÑ0, or if εÑ0 and δ“0,i.e. if the limits are taken successively. The latter case is not included in the analysis but may be handled with similar techniques. The limit equation (2) is the same in all regimes.

For deterministic problems (σ“0), diffusion approximation results have been extensively studied. We refer for instance to [LK74,BLP79]. Kinetic models with small parameters appear in various situations, for example when studying semi-conductors [GP92] and discrete velocity models [LT97], or as limits for description of systems of particles, either with a single particle [GR09] or multiple particles [PV03]. The asymptotic behavior of stochastic kinetic multiscale problems have also been recently studied: we refer to the seminal article [DV12], and the more recent contributions [DV20,DRV20,RR20]. In those works, the authors have obtained diffusion approximation results both in the PDE and the probabilistic senses: the limit equation is a stochastic linear diffusion PDE driven by a Wiener process (with Stratonovich interpretation).

Indeed, in those worksσpm^δq(withδ“ε) is replaced bym^δ{δin (1), and the authors assume that the driving processm^δ satisfies an appropriate centering condition. In the present article, we consider a law of large numbers regime (hence the averaging principle result), instead of a central limit theorem regime. In spite of this fundamental difference, the setting is very close to [RR20]: in particular a major technical difficulty which is solved in this paper is to avoid boundedness assumptions on the driving processm, using only moment conditions, which allow us to encompass for instance Ornstein-Uhlenbeck processes.

The literature concerning the averaging principle for stochastic differential equations and stochastic partial differential equations is huge. The averaging principle in the SDE case has been introduced in the seminal reference [Kha68], see also the monograph [PS08] and references

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therein. In the SPDE case, authors have mainly studied the averaging principle for parabolic semilinear SPDE systems, see for instance [Cer09,CF09,Bré12,Bré20,RXY20] and references therein. Let us also mention the recent preprints [CX20,XY21] where diffusion approximation results are proved for such systems. The list of references above is not exhaustive.

The first main result of this manuscript is Theorem 3.2: the convergence ofρ^ε,δ “ f^ε,δ to ρ is understood as convergence in distribution (in the probabilistic sense), in the space C⁰pr0, Ts, H^´ςpT^dqq, for all arbitrarily small positive ς. Under an additional assumption (which allows us to employ an averaging lemma), the convergence holds in the spaceL²pr0, Ts, L²pT^dqq, see the second main result of this manuscript, Theorem 3.3. The functional spaces above are the standard spaces where convergence holds in the deterministic case. The convergence in distribution is the natural mode of the convergence for the probabilistic variable. However, since the limit equation (2) is deterministic, if the limit initial conditionρ₀ is also deterministic, then the convergence holds in probability.

Let us now describe the main tools for the proof of the main results of this manuscript. We follow a martingale problem approach combined with the perturbed test functions method, as in the classical article [PSV77] (see also [Kus84,EK86,FGPSl07,PS08, dBG12]). Perturbed test functions in the context of PDEs with diffusive limits applies in various situations, for instance in the context of viscosity solutions [Eva89], nonlinear Schrödinger equations [dBG12], a parabolic PDE [PP03] or, as in this article, kinetic SPDEs [DV12,DV20,DRV20,RR20].

The idea of the perturbed test function is to identify the limit generator L of the limit equation (2) as

Lϕ“ lim

pε,δqÑp0,0qL^ε,δϕ^ε,δ,

where ϕ “ϕpρq is an arbitrary test function (depending only on the spatial density variable ρ“hfiappearing in the limit equation), andϕ^ε,δ “ϕ^ε,δpf, mqis the perturbed test function given by

ϕ^ε,δ“ϕ`εϕ1,0`ε²ϕ2,0`δ²ϕ0,2`εδ²ϕ1,2.

We refer to Proposition4.3for a rigorous statement. Note that due to the assumption that in generalε‰δ, the construction of the perturbed test function requires to define the correctorϕ1,2

(corresponding to the term of the orderεδ²). This corrector does not appear in the analysis if ε“δ. The construction ofϕ_1,2 is one of the novelties of this manuscript.

In addition, like in the preprint [RR20], the driving process is not assumed to be bounded (like in [DV12] for instance), and only moment conditions are satisfied. This generalization allows us for instance to consider Ornstein-Uhlenbeck processes. The introduction of stopping times arguments is required, hence the need to control the asymptotic behavior of the stopping times τ^δ whenδÑ0. Note that we prove below thatτ^δ Ñ 8in probability: the arguments to prove convergence are thus simpler than in [RR20].

The main result of this manuscript is the convergence ofρ^ε,δ toρ. In future works, it may be interesting to study the fluctuations,i.e. to prove thatρ^ε,δ´ρ, properly rescaled, converges in distribution to a Gaussian process, solution of a linear stochastic evolution equation. Again the use of the perturbed test functions approach may be a suitable approach. It would also be interesting to investigate rates of convergence, both in the strong and weak senses, in the spirit of [Bré12,Bré20] concerning parabolic systems, using Kolmogorov equations techniques. Finally, the validity of diffusion approximation and averaging principle results is fundamental for the efficient numerical simulation of the systems. In the deterministic setting, there has been a lot of activity to develop asymptotic preserving and uniformly accurate numerical methods, see for instance [Jin99,Jin12]. An asymptotic preserving scheme has been proposed in [AF19] for a class of kinetic stochastic equations in the diffusion approximation regime. In a future work [BHRR21],

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we plan to investigate the generalization of the asymptotic preserving schemes introduced and analyzed in the recent preprint [BRR20] for stochastic differential equations.

The manuscript is organized as follows. The setting is described in Section2(in particular precise assumptions concerning the driving processmare provided). The main results, Theo- rems3.2and3.3, are stated and discussed in Section3. Section4 is devoted to the proofs of the main results, using martingale problem formulations, tightness arguments and identification of the limit. Auxiliary fundamental results are stated there: first, Proposition4.1concerning the asymptotic behavior of the stopping time; second, Proposition4.2providing an a priori estimate in an appropriate weightedL²norm, uniformly with respect toε, δ; third, Proposition4.3giving the details on the perturbed test functions. The proofs of those three auxiliary results are given in Section5. Note that these three results are essential, and their proofs are given in a separate section since they are the most original technical contributions of this manuscript (compared with the more standard strategy described in Section 4).

2 Setting

2.1 Notation

The solution f^ε,δ of the stochastic kinetic problem considered in this article is a processf^ε,δ : pt, x, vq P r0,8q ˆT^dˆV ÑR, whereV is a measurable space, equipped with aσ-finite measure µ, andT^d denotes the flatd-dimensional torus.

Let us introduce the standard Hilbert spaces L²_x “L²pT^d,Rqand L² “L²pT^dˆV,Rq of real-valued functions, with inner products defined as follows:

ph, kq_L2 x “

ż

T^d

hpxqkpxqdx, ph, kq_L2 “ ż

T^dˆV

hpx, vqkpx, vqdxdµpvq.

The associated norms are denoted by}¨}_L2

x and}¨}_L2 respectively.

The following notation is used in the sequel: for allf PL¹pT^dˆVq, letρPL¹pT^dqbe defined by

ρ .

“hfi .

“ ż

V

f dµ.

In addition, for allT P p0,8q, ςP p0,1s, iPN0 andpP r1,8s, introduce the Banach spaces C_xⁱ .

“CⁱpT^d,Rq,C_T⁰H_x^´ς .

“C⁰pr0, Ts, H^´ςpT^d,RqqandL^p_TL²_x .

“L^ppr0, Ts, L²pT^d,Rqq.

Finally, the state spaceE of the driving stochastic process is assumed to be a Banach space, with norm denoted by }¨}_E.

2.2 Assumptions on the coefficients

Let us now state the assumptions concerning the linear operatorLand the mappingsa,bandσ.

Assumption 1. 1. Let the mappingMPL¹pV, µq, be such thatMpvq ą0 for allvPV, and normalized such thatş

VMpvqdµpvq “1.

2. The linear operatorLis defined as follows: for allf PL¹pV, µq

Lf “ρM´f “hfi M´f. (3)

3. The mappingsa:V ÑR^dandb:V ÑR^dare bounded. In additionasatisfies the centering condition

ż

V

apvqMpvqdµpvq “0.

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4. The linear operatorsAandB are defined by Afpx, vq .

“apvq ¨∇xfpx, vq, Bfpx, vq .

“bpvq ¨∇xfpx, vq.

5. The mappingσ:EÑCx^td{2u`2 is Lipschitz continuous.

Let us also introduce the weighted Hilbert spaceL²pM^´1q, with the inner product ph, kq_L2pM^´1q“

ż

T^dˆV

hpx, vqkpx, vqdxdµpvq Mpvq.

The associated norm is denoted by }¨}_L2pM^´1q. Observe that applying the Cauchy-Schwarz inequality yields the following results: for allf PL²pM^´1q, one hasf PL¹pT^dˆVq,ρPL²_x, and Lf PL²pM^´1q, with

ĳ

T^dˆV

|fpx, vq|dµpvqdxď }f}L²pM^´1q

}ρM}_L2pM^´1q“ }ρ}_L2

x ď }f}L²pM^´1q. In addition, if f1PL²pM^´1q XL²and f2PL²pMq XL², then

|pf1, f2q_L2| ď }f1}_L2pM^´1q}f2}_L2pMq

as a consequence of the Cauchy-Schwarz inequality.

Example 2.1. The conditions in Assumption1above are satisfied in the following two examples:

1. Continuous velocities

• The spaceV “R^d is equipped with the Lebesgue measuredµpvq “dv.

• The functionMpvq “ p2πq^´d{2expp´}v}²{2qfor allvPR^dis the standard Maxwellian.

• The velocityais a bounded odd function. For instance, relativistic particles satisfy apvq “? ^v

1`}v}² in convenient units.

2. Discrete velocities

• The spaceV “ t˘1u^d is equipped with the counting measure.

• The functionMis constant: Mpvq “ _2d¹ for allvP t˘1u^d

• The velocityais an odd function. For instance, the isotropic discrete velocity is given byapvq “v for allvP t˘1u^d.

In both examples, the functionσis defined either as

• σp`qpxq “σ₁p`pxqqfor all`PE“Cx^td{2u`2andxPT^d, with a mappingσ₁:RÑR,

• or asσp`qpxq “σ2px, `qfor`PE “RandxPT^d, with a mappingσ2:T^dˆRÑR, where σ1, σ2 are of classC^td{2u`3 with bounded derivatives of all orders.

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2.3 Assumptions on the driving process

The driving process is a Markov process, which satisfies the conditions below.

Assumption 2. The family pm_`ptqq_`PE defines aE-valued Markov process, where one has the initial conditionm_`p0q “`. LetL_mdenote its infinitesimal generator, with domain denoted by DpL_mq.

We assume that this Markov process is ergodic, and that its unique invariant distribution, denoted byν, is integrable: ş

E}`}dνp`q ă 8.

The following notation is used throughout the article: for all Lipschitz continuous mappings θ:EÑR, set

θ“ ż

E

θp`qdνp`q.

Let `0 PE be a given initial condition, in the sequel the value of `0 is omitted to simplify notation: for alltě0, let

mptq “m`₀ptq.

Assumption2is sufficient to state the main convergence results below. However, the analysis of the asymptotic behavior off^ε,δ whenδÑ0 requires additional technical assumptions. Since they are not needed to state the convergence results below, they may be skipped by the reader, until they are used to prove auxiliary results.

Assumption 3. The Markov process introduced in Assumption 2 satisfies the appropriate moment bounds: there existsγP p2,8qsuch that

sup

iPN0

E

« sup

tPri,i`1s

}mptq}^γ_E ff

ă 8. (4)

The assumption thatγą2 is crucial in the analysis. Observe that Assumption3implies the following results:

ż

}`}^γdνp`q ă 8 sup

tě0 Er}mptq}^γ_Es ă 8.

A mixing assumption is employed below to have quantitative information on the large time behavior of the driving process.

Assumption 4. The Markov process introduced in Assumption2satisfies a mixing property:

there exists a nonnegative functionγmix PL¹pR^`qsuch that, for all initial conditions`1, `2PE, there exists a couplingpm^˚_`

1, m^˚_`

2qofpm`₁, m`₂q, satisfying the inequality E“›

›m^˚_`

1ptq ´m^˚_`

2ptq›

›E

‰ďγ_mixptq }`₁´`₂}_E for alltě0.

Let us recall that aEˆE-valued random processpm^˚_`

1, m^˚_`

2qis a coupling ofpm`₁, m`₂qif the marginals safistym^˚_`

1

“d m_`₁ andm^˚_`

2

“d m_`₂ (where equality is understood in distribution). As a consequence of Assumption4, it is straightforward to obtain the following result: ifθ:EÑRis Lipschitz continuous, then for all`PE and alltě0, one has

ˇ

ˇErθpm`ptqqs ´θˇ

ˇďmaxp1, ż

E

››`¹›

›Edνp`¹qqLippθqγmixptqp1` }`}_Eq,

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where Lippθqdenotes the Lipschitz constant ofθ. Due to this consequence of the mixing property (Assumption4), the resolvent operatorR0 introduced below is well-defined.

Definition 2.1 (Resolvent operator). Let

´ Cx^td{2u`2

¯˚

be the set of continuous linear forms on Cx^td{2u`2 and let E^˚pσq .

“

"

u˝σ|uP

´ Cx^td{2u`2

¯˚*

ĂLippE,Rq. The resolvent operator R0 is defined as follows: for allθPE^˚pσq

R0pθ´θqp`q “ ż8

0

E“

θpm`ptqq ´θ‰ dt.

The function ψ_θ .

“R₀pθ´θqis the unique solution of the Poisson equation

´Lmψθ“θ´θ, (5)

satisfying the conditionψ_θ“0.

Note that the functions ψθsatisfy the following bound: there existsCP p0,8q, such that for allθPE^˚pσqand for all`PE, one has

|ψθp`q| ďCLippθqp1` }`}_Eq, (6) The remaining assumption deals with the infinitesimal generator Lmof the driving process.

Assumption 5. For all θ₁, θ₂ P E^˚pσq, assume that ψ_θ₁ψ_θ₂ is in the domain DpLmq of the generatorLmof the driving process, and thatLmpψ_θ₁ψ_θ₂qhas at most polynomial growth:

sup

`PE

|Lmpψθ₁ψθ₂q p`q|

1` }`}²_E ă 8.

The assumptions above are satisfied if the driving process is a E-valued Ornstein-Uhlenbeck process, as explained below.

Example 2.2. Letpm`ptqq_`PE be defined by

dm_`,t“ ´pm_`,t´mqdt`dW_t, m_`,0“`, (7) where W is an E-valued Wiener process. It satisfies Assumption 2since it is ergodic and its unique invariant distribution is a normal distribution, hence integrable. Moreover, we have

m`ptq “`e^´t`m`

1´e^´t˘

` żt

0

e^s´tdWs. (8)

Assumption3is satisfied for anyγP p2,8q. The couplingpm^˚_`

1, m^˚_`

2qof Assumption4is obtained by driving both processes by the same Wiener processW. Indeed, (8) becomes

m^˚_`

1ptq ´m^˚_`

2ptq “ p`₁´`₂qe^´t,

and Assumption4is satisfied withγ_mixptq “e^´t. Finally, with the notation of Assumption5, we have

ψθp`q “ ż8

0

pθp`q ´θpmqqe^´tdt“θpmq ´θp`q.

Since the infinitesimal generator is given byL_mϕp`q “Dϕp`q¨pm´`q`¹₂Tr`

D²ϕp`q˘

, Assumption 5is also satisfied.

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3 Main result

3.1 Description of the model and of the limit problem

The multiscale stochastic problem considered in this article depends on two parametersεand δ. Since the objective of this work is to prove a convergence result whenpε, δq Ñ p0,0q, without loss of generality it is assumed thatεP p0, ε₀sandδP p0, δ₀s, where ε₀, δ₀are fixed – a precise condition is stated below. To simplify notation, we use the following convention: `

X^ε,δ˘

ε,δstands for the family of random variables`

X^ε,δ˘

εPp0,ε0s,δPp0,δ0s.

First, for all δP p0, δ₀s, the fast driving processm^δ is defined as follows: for alltě0, set

m^δptq “mpt{δ²q, (9)

where mis the driving process given by Assumption2, with the initial conditionm^δp0q “mp0q “

`0, which is assumed to be independent ofδ.

We study the asymptotic behavior whenpε, δq Ñ p0,0qof the solutionf^ε,δ of the following stochastic kinetic problem

Btf^ε,δ`

´1

εapvq `bpvq

¯

¨∇xf^ε,δ`σpm^δqf^ε,δ“ 1

ε²Lf^ε,δ. (10) with initial conditionf^ε,δp0q “f₀^ε,δ.

For any fixedεP p0, ε0s,δP p0, δ0s, the problem (10) is globally well-posed in the following sense.

Proposition 3.1. Introduce the linear operator A^ε “ ε^´1A`B, with domain DpA^εq .

“ f PL²pM^´1q | px, vq ÞÑ`

ε^´1apvq `bpvq˘

¨∇_xfpx, vq PL²pM^´1q(

, for all εP p0, ε₀s.

LetT P p0,8q, εP p0, ε₀sand δP p0, δ₀s. For any f₀^ε,δPL²pM^´1q, there exists, almost surely, a unique mild solutionf^ε,δ of (10) inC⁰pr0, Ts;L²pM^´1qq, in the sense that, almost surely, for all tP r0, Ts, one has

f^ε,δptq “e^´tA^εf₀^ε,δ` żt

0

e^´pt´sqA^ε ˆ1

ε²Lf^ε,δpsq ´σpm^δpsqqf^ε,δpsq

˙ ds.

The proof of Proposition3.1is based on a standard fixed point argument, combined with the following observation:

sup

tPr0,Ts

››m^δptq›

›Eď sup

iďT δ^´2`1

sup

tPri,i`1s

}mptq}_Eă 8

owing to Assumption3on the moments of the driving processm. The proof of Proposition3.1is thus omitted.

Note that the statement of Proposition 3.1is given for fixedεą0 andδą0, and does not provide uniform estimates of the solutionf^ε,δ with respect to these parameters. Proving such estimates needs extra arguments, which are not needed to state the main results of the article.

We refer to Proposition... below for the statement of an appropriate a priori estimate in the L²pM^´1qnorm, which requires the introduction of a stopping timeτ^δ defined below.

Let us now introduce the so-called averaged equation. First, set K“

ż

V

apvq bapvqMpvqdµpvq PSym^`pdq J “

ż

V

bpvqMpvqdµpvq PR^d

σ“ ż

E

σp`qdνp`q PC_x^td{2u`2.

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Note thatKandJ are well-defined sinceaandbare assumed to be bounded (see Assumption1).

In addition, σ is well-defined since σ is globally Lipschitz continuous from E to Cx^td{2u`2 (see Assumption1) and since the probability distributionν is integrable (see Assumption2).

The unknownρof the averaged equation is a mapping defined onr0,8q ˆT^d. We are finally in position to write the averaged equation:

Btρ`J¨∇ρ`σ ρ“divpK∇ρq, (12)

with initial condition ρp0q “ ρ₀. In (12) above, ∇ and div are the gradient and divergence operators with respect to the variablexrespectively. We consider solutions of (12) in the weak sense, see Definition3.1 below. Note that the solution may be a random process, even if the evolution is deterministic: it may happen that the initial conditionρ₀ is random.

Definition 3.1. LetT P p0,8qandρ₀ be aL²_x-valued random variable. A stochastic processρ is a weak solution of (12) inL²_x ifρPL⁸_TL²_x almost surely and if, for allξPH_x² andtP r0, Ts, almost surely,

pρptq, ξq_L2

x“ pρ₀, ξq_L2 x`

żt 0

pρpsq,divpK∇ξq `J¨∇ξ´σξq_L2

xds. (13)

For anyL²_x-valued random variable, the averaged equation (12) admits a unique weak solution in the sense of Definition3.1.

3.2 Convergence results

Let us now state the main results of this article, concerning the asymptotic behavior ofρ^ε,δ and f^ε,δ whenpε, δq Ñ p0,0q. In the sequel, it is convenient to impose the following non restrictive conditions on the parametersε0 andδ0(such thatεP p0, ε0sandδP p0, δ0s):

ε0ămin ˆ

1,

´

4p}a}_L8` }b}_L8qp1`T}σ}_C1 xq

¯´1˙

, δ0ămin

´

1,}`0}^´1_E

¯

. (14) Note that the condition forδ0 depends on the initial condition`0 of the driving process. This is one of the reasons why it is convenient to assume that it is deterministic and that it does not depend onδ. Extensions to more general initial conditions for the driving process would require extra technical assumptions and computations, which are omitted to simplify the setting and focus on the main aspects of the analysis of the asymptotic behavior of the stochastic multiscale problem (10) whenpε, δq Ñ p0,0q.

The initial condition f₀^ε,δof (10) is assumed to satisfy the following conditions.

Assumption 6. The family of initial conditions

´ f₀^ε,δ

¯

ε,δ satisfies the following moment bound:

sup

εPp0,ε0s,δPp0,δ0s

E

„›

›

›f₀^ε,δ

›

12 L²pM^´1q

 ă 8.

In addition, the initial densityρ^ε,δ₀ PL²_xconverges in distribution whenpε, δq Ñ p0,0qinL²_xto ρ₀PL²_x: recall that this means that for any bounded continuous mapping Φ :L²_xÑR, one has

E

” Φpρ^ε,δ₀ q

ı ÝÝÝÝÑ

ε,δÑ0 ErΦpρ₀qs.

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Observe that in generalρ₀ is aL²_x-valued random variable.

We are now in position to state the two main convergence results of this article. First, see Theorem 3.2, ρ^ε,δ “

f^ε,δ

converges in distribution to the unique solution ρof the averaged equation (12), in the spaceC_T⁰H_x^´ςfor allς ą0. Second, under an additional technical assumption which allows to apply an averaging lemma, one obtains a stronger result, see Theorem 3.3: ρ^ε,δ converges in distribution toρin the spaceL²_TL²_x, andf^ε,δconverges in distribution toρMin the spaceL²_TL²pM^´1q. Moreover, the convergence results hold in probability if the initial condition ρ₀(given by Assumption 6) of the averaged equation (12) is deterministic.

Let us state the first main result of this article.

Theorem 3.2. Let Assumptions1 to6 be satisfied. Letf^ε,δ be the solution of (10). Then, when pε, δq Ñ p0,0q, the random variableρ^ε,δ “

f^ε,δ

converges in distribution to the unique weak solution ρof (12), in C⁰_TH_x^´ς.

In addition, if the initial condition ρ₀ “ lim

pε,δqÑp0,0q

ρ^ε,δ₀ P L²_x is deterministic, then the convergence ofρ^ε,δ toρholds in probability.

An additional assumption is required to state Theorem3.3, as in [RR20] (Assumption7below).

It allows us to apply a so-called averaging lemma (see [BD99, Theorem 2.3]), developed for the study of kinetic PDEs, and to obtain convergence in the spaceL²_x(in Theorem3.3) instead of H^´ς (in Theorem3.2).

Assumption 7. • The space for the velocity variable is given by pV, dµq “ pRⁿ,^dµ_dvpvqdvq, with a Radon-Nikodym derivative (with respect to Lebesgue measure) satisfying ^dµ_dv P H¹pRⁿq.

• The mappingais locally Lipschitz continuous

• There exist Ca,µ P p0,8qand ςa,µ P p0,1s such that, for alluP S^d´1, all λP R and all ηP p0,8q, one has

ż

λăapvq¨uăλ`η

˜ˇ ˇ ˇ ˇ

dµ dvpvq

ˇ ˇ ˇ ˇ

2

` ˇ ˇ ˇ ˇ

∇dµ dvpvq

ˇ ˇ ˇ ˇ

2¸

dvďCa,µη^ς^a,µ.

We are now in position to state the second main result of this article.

Theorem 3.3. Let Assumptions1to6and Assumption7be satisfied.

Then, when pε, δq Ñ p0,0q, the random variableρ^ε,δ“ f^ε,δ

converges in distribution to the unique weak solution ρof (12), in the spaceL²_TL²_x. Moreover, whenpε, δq Ñ p0,0q, the random variablef^ε,δ converges in distribution toρM, in the spaceL²_TL²pM^´1q.

In addition, if the initial condition ρ₀ “ lim

pε,δqÑp0,0q

ρ^ε,δ₀ P L²_x is deterministic, then the convergence ofρ^ε,δ toρand of f^ε,δ toρM hold in probability.

3.3 Discussion

The main results, Theorems3.2and 3.3, state that diffusion approximation (in the PDE sense) and averaging (in the probabilistic sense) results hold, whenpε, δq Ñ p0,0q. These results are natural generalizations of previously obtained results, either in the deterministic case (εÑ0, σ“0), or in the probabilistic case (δÑ0,εą0 fixed). In fact, using the same arguments as in Section4below, one may obtain the following results, where the limitsεÑ0 andδÑ0 are taken successively.

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On the one hand, if δą0 is fixed, thenρ^ε,δ“hfi^ε,δ converges whenεÑ0, to the solution ρ^0,δ of the evolution equation

Btρ^0,δ`σpm^δqρ^0,δ“divpK∇ρ^0,δq ´J¨∇ρ^0,δ.

That result is a standard diffusion approximation result in the PDE sense. Then, whenδÑ0,ρ^0,δ converges to the solutionρof the limit equation (12), owing to the standard averaging principle for stochastic problems.

On the other hand, if ε ą0 is fixed, then owing to the standard averaging principle,f^ε,δ converges whenδÑ0, to the solution f^ε,0of the evolution equation

Btf^ε,0`1

εapvq ¨∇xf^ε,0`bpvq ¨∇xf^ε,0`σf^ε,0“ 1 ε²Lf^ε,0.

Then, when εÑ0,ρ^ε,0“hfi^ε,0converges to the solutionρof the limit equation (12), owing to the standard diffusion approximation result in the PDE sense.

To the best of our knowledge, the results above have not been rigorously proved in the literature, however they are variants of well-studied results. The proofs of Theorems3.2and3.3 do not encompass those regimes when eitherε“0 orδ“0: essentially this would require to adapt the construction of the perturbed test function. Indeed, below we directly focus on the behavior ofρ^ε,δ, thus the convergencef^ε,δ Ñf^ε,0 whenδÑ0 cannot be covered directly, for instance. In addition, one of the arguments of the proofs is the convergenceτ^δ Ñ 8when δÑ0 (whereτ^δ are stopping times defined below), thus the convergenceρ^ε,δÑρ^0,δ whenεÑ0 cannot

be covered without substantial modifications.

Still, one obtains the following result:

δÑ0lim lim

εÑ0ρ^ε,δ“lim

εÑ0 lim

δÑ0ρ^ε,δ“ lim

pε,δqÑp0,0qρ^ε,δ“ρ,

where the convergence is understood in the appropriate sense. The analysis presented in this article thus departs from the setting of [DV12, DV20,DRV20,RR20]. where in all cases there is only one small parameter ε“ δ. Our result is expected but important: it shows that the diffusion approximation and the averaging principle can be decoupled. As already mentioned in the introduction, considering the general caseε‰δrequires new arguments, in particular the construction of the perturbed test functions needs an additional corrector.

In the setting, it is assumed that the initial condition m^δp0q of the fast driving process is deterministic and independent ofδ: m^δp0q “m₀. It would be possible to extend the results to more general initial conditions, under appropriate modified moment conditions. This would for instance allow us to include the case wherem^δp0qis random and distributed following the ergodic invariant distributionν. Note also that if either the mappingσor the processm would have a bounded support, the analysis would be simplified.

As explained in the introduction, we have left open several questions for future works: first, the analysis of fluctuations, second, the identification of (strong and weak) rates of convergence.

4 Description of the proof

In the sequel, the following convention is employed: given variables uand parameters λ, the notationX1puq ÀX2puqmeans that for all parametersλ, there existsCpλq P p0,8qsuch that one hasX1puq ďCpλqX2puqfor allu. From the context the identification of variables (typically ε, δ, f, `) and parameters (typicallyT, ϕ) will be clear.

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First, we introduce two of the most important tools of the proofs of the main results: the stopping timeτ^δ (see Section4.1), and the perturbed test functionϕ^ε,δ“ϕ`εϕ1,0`ε²ϕ2,0` δ²ϕ0,2`εδ²ϕ1,2 (see Section 4.2), constructed for a class of admissible functions ϕsuch that ϕpf, `q “ϕphfiq “ϕpρq. Sections4.1and4.2 contain important auxiliary results, which require technical arguments in their proofs: the proofs are therefore postponed to Section5.

The use of the stopping timeτ^δ is instrumental to obtain appropriate moment bounds of the solutions, uniformly with respect toε, δ. We prove thatτ^δ Ñ 8in probability whenδÑ0. In the arguments, as a consequence of Slutsky’s lemma (see more details below), it is then sufficient to consider the stopped processes defined by f^ε,δ,τ^δ “f^ε,δp¨ ^τ^δq, ρ^ε,δ,τ^δ “ρ^ε,δp¨ ^τ^δq and m^δ,τ^δ “m^δp¨ ^τ^δq.

The use of the perturbed test function method is standard in the analysis of multiscale stochastic problems, see for instance [DV12, DV20, DRV20, RR20] in a similar context of stochastic kinetic equations. Compared to those references, note that it is necessary to consider two parametersε, δ which may be independent. Whenδ“ε, the construction of the corrector ϕ1,2 is not needed.

We then proceed to the proof of the main results of this article. The arguments are standard.

We first check a tightness property for` ρ^ε,δ˘

ε,δ,τ^δ in the appropriate function space. We then check that the Markov process pf^ε,δ,τ^δ, m^δ,τ^δq is the solution of a martingale problem for all εą0, δą0, and lettingpε, δq Ñ p0,0q, using the perturbed test function, we prove that any limit point of the family

´ ρ^ε,δ,τ^δ

¯

ε,δis a weak solution of the averaged equation, in the sense of (13).

Since τ^δ Ñ 8 in probability when δ Ñ 0, a uniqueness argument for the averaged equation then concludes the proof of Theorem3.2. Theorem3.3 is then obtained by the application of the averaging lemma. The fact that convergence holds in probability whenρ₀ is deterministic is a straightforward well-known consequence of Portmanteau Theorem (and is not specific to the PDE framework of this paper): in that case the solutionρof the averaged equation is also deterministic.

4.1 Stopping times and a priori estimates

In this section, first we give the definition of the stopping timeτ^δ, second we state thatτ^δ Ñ 8 in probability whenδÑ0. Finally, we state an a priori estimate forf^ε,δ,τ^δ inL²pM^´1q, which is uniform with respect toε, δ. The proofs are postponed to Section 5.

For allδP p0, δ₀sand alltě0, set ζ^δptq “ 1

δ żt

0

`σpm^δpsqq ´σ˘

dsPC_x^td{2u`2. (15)

Definition 4.1. LetαP p²_γ,1q, then for allδP p0, δ0s, define τ^δ“τ_m^δ ^τ_ζ^δ, where

τ_m^δ “inf tP r0, Ts |›

›m^δptq›

›Eěδ^´α( , τ_ζ^δ“inf

!

tP r0, Ts |›

›ζ^δptq›

›C_x¹ ěδ^´1 )

.

Note thatτ^δ is a stopping time for the filtration` F_t^δ˘

tPR^` generated by the driving process m^δ, with F_t^δ “ σ`

m^δpsq˘

0ďsďt. In the definition above, γ is the parameter introduced in Assumption3.

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The initial conditions of the stopped processesm^δ,τ^δ andζ^δ,τ^δ satisfy

›

›m^δ,τ^δp0q

›

›E

“ }`₀}_Eď δ^´1₀ ď δ^´α (since δ ďδ₀ ď1 and αă1), and ζ^δ,τ^δp0q “0. As a consequence, almost surely τ^δ ą0, and the following estimates form^δ andζ^δ hold: for alltě0 and allδP p0, δ₀s, one has

›

›m^δ,τ^δptq

›

›_E “›

›m^δpt^τ^δq›

›Eďδ^´α, (16)

and ›

›

›ζ^δ,τ^δptq

›

›C_x¹ “›

›ζ^δpt^τ^δq›

›C¹_xďδ^´1. (17)

Let us now study the behavior of the stopping timeτ^δ whenδÑ0.

Proposition 4.1. WhenδÑ0,τ^δ Ñ 8 in probability: for allT ą0, one has P`

τ^δ ăT˘

ÝÝÝÝÝÑ

pε,δqÑ0

0.

The proof of Proposition 4.1is postponed to Section5.

Finally, let us state the following a priori estimate in the L²pM^´1qnorm forf^ε,δ,τ^δptq, in an almost sure sense.

Proposition 4.2. For T P p0,8q, there exists CpTq P p0,8q, such that for all t P r0, Ts, εP p0, ε0sandδP p0, δ0s, almost surely one has

›

›f^ε,δ,τ^δptq

›

2 L²pM^´1q

` 1 2ε²

żt^τ^δ 0

›

›Lf^ε,δ,τ^δpsq

›

2 L²pM^´1q

dsďCpTq

›

›f₀^ε,δ

›

2 L²pM^´1q

. (18) Let us emphasize that the constant CpTqappearing on the right-hand side of (18) is deterministic and does not depend onεandδ.

The proof of Proposition 4.2 is posponed to Section 5. Note that the a priori estimate

›

›f^ε,δ,τ^δptq

›

›L²pM^´1q À

›

›f₀^ε,δ

›

›L²pM^´1q

is instrumental in all the arguments of the proof of Theo- rem3.2, whereas the upper bound for the integral term on the left-hand side of (18) is used only in the proof of Theorem3.3.

4.2 Perturbed test functions

In this section, we describe the construction of a perturbed test function ϕ^ε,δ, such that the following properties are satisfied:

ϕ^ε,δÝÝÝÝÑ

ε,δÑ0 ϕ L^ε,δϕ^ε,δÝÝÝÝÑ

ε,δÑ0 Lϕ,

where ϕis any sufficiently smooth function such thatϕpf, `q “ϕphfiq, andL^ε,δ andL are the infinitesimal generators associated with the Markov processespf^ε,δ, m^δqandρ, solving (10) and (12) respectively.

To state more rigorously and more precisely the properties mentioned above, we first introduce two appropriate classes of test functions, such thatϕPΘlim andϕ^ε,δPΘ for allε, δ, and such that the errors|ϕ^ε,δ´ϕ|and |L^ε,δϕ^ε,δ´Lϕ|are quantified in terms ofε, δ.

Let us first describe the class of functions Θlim.

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Definition 4.2. Let Θ_limbe the class of real-valued test functionsϕsuch that for allf PL²pM^´1q and`PE, one has

ϕpf, `q “ϕpρq “χppρ, ξq_L2

xq, (19)

where we recall thatρ“hfi“ş

V f dµ, with arbitraryχPC³_bpR,RqandξPC_x³.

Recall thatK,J andσare defined by (11). The infinitesimal generatorLassociated with the limit problem (12) is defined by

Lϕpρq “Dϕpρq ¨ pdivxpK∇xρq ´J¨∇xρ´σρq, (20) for allρPL²_xand allϕPDpLq, with the domainDpLqgiven by

DpLq “ ϕPC⁰pL²_xq |LϕPC⁰pL²_xq( .

Note that ΘlimĂDpLq. In addition, for allϕPΘlim, of the form (19), one has for allρPL²_x Lϕpρq “χ¹ppρ, ξq_L2

xq pρ,divxpK∇xξq `J¨∇xξ´σξq_L2

x. (21)

Let us now describe the class of functions Θ.

Definition 4.3. Let Θ be the class of test functions ϕ : L²pM^´1q ˆE Ñ R satisfying the following conditions.

• For all`PE,ϕp¨, `q PC¹pL²pM^´1qq.

• For allf PL²pM^´1q,ϕpf,¨q PC⁰pEq.

• For alliP t1,2uand allf PL²pM^´1q,ϕpf,¨qⁱPDpLmqandLmpϕⁱq PC⁰pL²pM^´1q ˆEq.

• For allf PL²pM^´1qand`PE, denote by∇fϕpf, `q PL²pM^´1qthe gradient ofϕatpf, `q, which is defined defined such that

@hPL²pM^´1q,p∇fϕpf, `q, hq_L2pM^´1q“Dfϕpf, `q ¨h.

Then, for allf PL²pM^´1qand`PE, one has ż ż

}∇x∇fϕpf, `qpx, vq}²dxdµpvq Mpvqă 8.

• There exists C_ϕp0,8qsuch that, for allf, hPL²pM^´1qand`₁, `₂PE,

|ϕpf, `1q| ` |Dfϕpf, `1qpAhq| ` |Dfϕpf, `1qpBhq| ` |Dfϕpf, `1qp`2hq| ` |Dfϕpf, `1qpLhq|

ďC_ϕ

´

1` }f}³_L2pM^´1q` }h}³_L2pM^´1q

¯ ´

1` }`₁}²_E` }`₂}²_E

¯ .

The infinitesimal generator L^ε,δ associated with the stochastic problem (10) with driving processm^δ, satisfies the following multiscale expansion in terms ofε, δ:

L^ε,δ“L0`ε^´1L1`ε^´2L2`δ^´2Lm (22) where the infinitesimal generator of the driving processL_m is introduced in Assumption2above, andL₀,L₁ andL₂ are defined as follows: for allf PL²pM^´1qand all`PE, set

L0ϕpf, `q “ ´Dfϕpf, `q ¨ pσp`qf`Bfq, L1ϕpf, `q “ ´D_fϕpf, `q ¨ pAfq,

L2ϕpf, `q “Dfϕpf, `q ¨Lf,