Conditional propagation of chaos for mean field systems of interacting neurons

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of interacting neurons

Xavier Erny, Eva Löcherbach, Dasha Loukianova

To cite this version:

Xavier Erny, Eva Löcherbach, Dasha Loukianova. Conditional propagation of chaos for mean field

systems of interacting neurons. 2020. �hal-02280882v3�

Conditional propagation of chaos for mean eld systems of interacting neurons

Xavier Erny^∗, Eva Löcherbach^† and Dasha Loukianova^∗

∗Université Paris-Saclay, CNRS, Univ Evry, Laboratoire de Mathématiques et Modélisation d'Evry, 91037, Evry, France

†Statistique, Analyse et Modélisation Multidisciplinaire, Université Paris 1 Panthéon-Sorbonne, EA 4543 et FR FP2M 2036 CNRS

Abstract: We study the stochastic system of interacting neurons introduced inDe Masi et al.

(2015) and inFournier and Löcherbach(2016) in a diusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to0and all other neurons receive an additional amount of potential which is a centred random variable of order 1{?

N . In between successive spikes, each neuron's potential follows a deterministic ow. We prove the convergence of the system, as N Ñ 8, to a limit nonlinear jumping stochastic dierential equation driven by Poisson random measure and an additional Brownian motionW which is created by the central limit theorem. This Brownian motion is underlying each particle's motion and induces a common noise factor for all neurons in the limit system. Conditionally on W, the dierent neurons are independent in the limit system. This is the conditional propagation of chaos property. We prove the well-posedness of the limit equation by adapting the ideas of Graham (1992) to our frame. To prove the convergence in distribution of the nite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system.

MSC 2010 subject classications: 60J75, 60K35, 60G55, 60G09.

Keywords and phrases: Multivariate nonlinear Hawkes processes with variable length mem- ory, Mean eld interaction, Piecewise deterministic Markov processes, Interacting particle systems, Propagation of chaos, Exchangeability, Hewitt Savage theorem.

Introduction

This paper is devoted to the study of the Markov processX_t^N pX_t^N,1, . . . , X_t^N,Nq taking values in R^N and having generatorA^N which is dened for any smooth test functionϕ:R^N ÑRby

A^Nϕpxq α

¸N i1

Bxⁱϕpxqxⁱ

¸N i1

fpxⁱq

»

R

νpduq

ϕpxxⁱei

¸

ji

?u

Nejq ϕpxq

, where x px¹, . . . , x^Nq and where ej denotes the jth unit vector inR^N. In the above formula, α¡0is a xed parameter andν is a centred probability measure onRhaving a second moment.

Informally, the process pX^N,jq1¤j¤N solves X_t^N,iX₀^N,iα

»t 0

X_s^N,ids

»t 0

X_s^N,idZ_s^N,i 1

?N

¸

ji

»t 0

U^jpsqdZ_s^N,j, (1) where U^jpsq are i.i.d. centred random variables distributed according to ν, and where for each 1¤j¤N, Z^N,j is a simple point process onR having stochastic intensitysÞÑf

X_s^N,j .

1

The particle system (1) is a version of the model of interacting neurons considered in De Masi et al. (2015), inspired byGalves and Löcherbach(2013), and then further studied inFournier and Löcherbach (2016) and Cormier, Tanré and Veltz (2019). The system consists of N interacting neurons. In (1),Z_t^N,j represents the number of spikes emitted by the neuronj in the interval r0, ts and X_t^N,j the membrane potential of the neuronj at timet. Spiking occurs randomly following a point process of ratefpxq for any neuron of which the membrane potential equalsx.Each time a neuron emits a spike, the potentials of all other neurons receive an additional amount of potential.

In De Masi et al. (2015), Fournier and Löcherbach (2016) and Cormier, Tanré and Veltz (2019) this amount is of order N¹, leading to classical mean eld limits asN Ñ 8. On the contrary to this, in the present article we study a diusive scaling where each neuron j receives the amount U{?

N at spike times t of neuron i, i j, where U ν is a random variable. The variable U is centred modeling the fact that the synaptic weights are balanced. Moreover, right after its spike, the potential of the spiking neuroni is reset to 0, interpreted as resting potential. Finally, in between successive spikes, each neuron has a loss of potential of rate α.

Equations similar to (1) appear also in the frame of multivariate Hawkes processes with mean eld interactions. Indeed, if Z^N,i

1¤i¤N is a multivariate Hawkes process where the stochastic intensity of eachZ^N,i is given byf X_t^N

t with X_t^N e^αtX₀^N 1

?N

¸N j1

»t 0

e^αptsqU^jpsqdZ_s^N,j, (2) thenX^N satises

X_t^N X₀^Nα

»t 0

X_s^Nds 1

?N

¸N j1

»t 0

U^jpsqdZ_s^N,j,

which corresponds to equation (1) without the big jumps, i.e. without the reset to 0 after each spike.

The above system of interacting Hawkes processes with intensity given by (2) has been studied in our previous paper Erny, Löcherbach and Loukianova(2019). There we have shown rstly that X^N converges in distribution inDpR ,Rq to a limit processX¯ solving

dX¯_t αX¯_tdt σ b

f X¯_t

dW_t, (3)

and secondly that the sequence of multivariate counting processes Z^N,i

iconverges in distribution inDpR ,Rq^N to a limit sequence of counting processes Z¯ⁱ

i.EveryZ¯ⁱis driven by its own Poisson random measure and has the same intensity fpX¯tq

t,whereX¯ is the strong solution of (3) with respect to some Brownian motion W. Consequently, the processes Z¯ⁱ pi ¥ 1q are conditionally independent given the Brownian motionW.

In the present paper we add the reset term in (1) that forces the potential X^N,i of neuron i to go back to 0 at each jump time of Z^N,i. This models the well-known biological fact that right after its spike, the membrane potential of the spiking neuron is reset to a resting potential. From a mathematical point of view, this reset to 0 induces a de-synchronization of the processes X^N,i (1¤i¤N). In terms of Hawkes processes, it means that in (2), the processX_t^N has been replaced by

X_t^N,i 1

?N

¸N j1

»t Lⁱ_t

e^αptsqU^jpsqdZ_s^N,j e^αtX₀^N,i1Lⁱ_t0,

whereLⁱ_tsupt0¤s¤t: ∆Z_s^N,i1u is the last spiking time of neuronibefore timet,with the convention supH:0. Thus the integral over the past, starting from 0 in (2), is replaced by an integral starting at the last jump time before the present time. Such processes are termed being of variable length memory, in reminiscence ofRissanen(1983). They are the continuous-time analogues of the model considered inGalves and Löcherbach(2013), and we are thus considering multivariate Hawkes processes with mean eld interactions and variable length memory. As a consequence, on the contrary to the situation inErny, Löcherbach and Loukianova(2019), the point processesZ^N,i (1¤i¤N) do not share the same stochastic intensity. The reset term in (1) is a jump term that survives in the limitN Ñ 8.

Before introducing the exact limit equation for the system (1), let us explain informally how the limit particle system associated to X^N,i

1¤i¤N should a priori look like. Suppose for the moment that we already know that there exists a process pX¯¹,X¯²,X¯³, . . .q PDpR ,Rq^N such that for all K¡0,weak convergenceLpX^N,1,, . . . , X^N,Kq ÑLpX¯¹, . . . ,X¯^Kq inDpR ,Rq^K,asN Ñ 8,holds.

In equation (1) the only term that depends onN is the martingale term which is approximately given by

M_t^N 1

?N

¸N j1

»t 0

U^jpsqdZ_s^N,j.

Then in the innite neuron model, each process X¯ⁱ should solve the equation (1), where the term M_t^N is replaced byM_t: lim

NÑ8M_t^N. Because of the scaling in N^1{2,the limit martingale M_t will be a stochastic integral with respect to some Brownian motion, and its variance the limit of

E

pM_t^Nq² σ²

»t 0

E

1 N

¸N j1

fpX_s^N,jq

ds,

whereσ² is the variance ofU^jpsq.Therefore, the limit martingale (if it exists) must be of the form

Mtσ

»t 0

gf felim

NÑ8

1 N

¸N j1

f

Xs^N,j dWsσ

»t 0

b lim

NÑ8µ^N_s pfqdWs, whereµ^N_s is the empirical measure of the system X_s^N,j

1¤j¤N andW is a one-dimensional standard Brownian motion.

Since the law of the Nparticle system pX^N,1, . . . , X^N,Nq is symmetric, the law of the limit systemX¯ pX¯¹,X¯²,X¯³, . . .q must be exchangeable, that is, for all nite permutationsσ,we have that LpX¯^σp1q,X¯^σp2q, . . .q LpX¯q. In particular, the theorem of Hewitt-Savage, see Hewitt and Savage(1955), implies that the random limit

µ_s: lim

NÑ8

1 N

¸N i1

δX¯ⁱ_s (4)

exists. Supposing thatµ^N_s converges, it necessarily converges towardsµs.Therefore,X¯ should solve the limit system

X¯_tⁱX¯₀ⁱα

»t 0

X¯_sⁱds

»t 0

X¯_sⁱdZ¯_sⁱ σ

»t 0

aµspfqdWs, iPN, (5)

where eachZ¯ⁱ has intensitytÞÑfpX¯_tⁱq,and whereµs is given by (4).

The above arguments are made rigorous in Sections2.1and2.2below.

Let us briey discuss the form of the limit equation (5). Analogously to Erny, Löcherbach and Loukianova(2019), the scaling inN^1{2in (1) creates a Brownian motionW in the limit system (5).

We will show that the presence of this Brownian motion entails a conditional propagation of chaos, that is the conditional independence of the particles givenW. In particular, the limit measureµs

will be random. This diers from the classical framework, where the scaling is in N¹ (see e.g.

Delattre, Fournier and Homann (2016), Ditlevsen and Löcherbach (2017) in the framework of Hawkes processes, andDe Masi et al.(2015),Fournier and Löcherbach(2016) andCormier, Tanré and Veltz (2019) in the framework of systems of interacting neurons), leading to a deterministic limit measureµ_sand the true propagation of chaos property implying that the particles of the limit system are independent.

This is not the rst time that conditional propagation of chaos is studied in the literature; it has already been considered e.g. inCarmona, Delarue and Lacker(2016),Coghi and Flandoli(2016) and Dermoune(2003). But in these papers the common noise, represented by a common (maybe innite dimensional) Brownian motion, is already present at the level of the nite particle system, the mean eld interactions act on the drift of each particle, and the scaling is the classical one inN¹.On the contrary to this, in our model, this common Brownian motion, leading to conditional propagation of chaos, is only present in the limit, and it is created by the central limit theorem as a consequence of the joint action of the small jumps of the nite size particle system. Moreover, in our model, the interactions survive as a variance term in the limit system due to the diusive scaling inN^1{2.

Now let us discuss the form of µs, which is the limit of the empirical measures of the limit system X¯_sⁱ

i¥1. The theorem of Hewitt-Savage,Hewitt and Savage(1955), implies that the law of X¯_sⁱ

i¥1is a mixture directed by the law ofµs. As it has been remarked byCarmona, Delarue and Lacker(2016) and Coghi and Flandoli(2016), this conditioning reects the dependencies between the particles.

We will show that the variablesX¯ⁱare conditionally independent given the Brownian motionW.

As a consequence, µs will be shown to be the conditional law of the solution given the Brownian motion, that is,Palmost surely,

µspq PpX¯_sⁱ P |pWtq0¤t¤sq PpX¯_sⁱP |Wq, (6) for anyiPN. Equation (5) together with (6) gives a precise denition of the limit system.

The nonlinear SDE (5) is not clearly well-posed, and our rst main result, Theorem1.2, gives appropriate conditions on the system that guarantee pathwise uniqueness and the existence of a strong solution to (5).

We then prove, in Sections 2.1 and 2.2, our main Theorem 1.7stating the convergence in dis- tribution of the sequence of empirical measuresµ^N N¹°N

i1δ_pXN,i

t qt¥0,inPpDpR ,Rqq,to the random limit µ PppX¯_tqt¥0 P |Wq. To do so, we rst prove that under suitable conditions on the parameters of the system, the sequenceµ^N is tight (see Proposition2.1below). We then follow a classical road and identify every possible limit as solution of a martingale problem. Since the random limit measureµwill only be the directing measure of the limit system (that is, the condi- tional law of each coordinate, but not its law), this martingale problem is not a classical one. It is in particular designed to reect the correlation between the particles and to describe all possible limits of couples of neurons.

Classical representation theorems imply that any coordinate of the limit process must satisfy an equation of the type (5). The fact that our martingale problem describes correlations within couples

of neurons allows to show that each coordinate is driven by its own Poisson random measure and that all coordinates are driven by the same underlying Brownian motionW.But it is not yet clear that µs is of the form (6). In other words, it has to be proven that the only common randomness is the one present in the driving Brownian motion W. To prove this last point, we introduce an auxiliary particle system which is a mean eld particle version of the limit system, constructed with the same underlying Brownian motion, and we provide an explicit control on the distance (with respect to a particularL¹norm) between the two systems.

Let us nally mention that the random limit measureµsatises the following nonlinear stochastic PDE in weak form: for any test function ϕPC_b²pRq,the set ofC²-functions onRsuch that ϕ, ϕ¹ andϕ² are bounded, for anyt¥0,

»

R

ϕpxqµ_tpdxq

»

R

ϕpxqµ₀pdxq

»t 0

»

R

ϕ¹pxqµ_spdxq a

µ_spfqdW_s

»t 0

»

R

rϕp0q ϕpxqsfpxq αϕ¹pxqx 1

2ϕ²pxqµspfq µspdxqds.

Organisation of the paper. In Section 1, we state the assumptions and formulate the main results. Section2 is devoted to the proof of the convergence of µ^N :°N

j1δ_XN,j (Theorem 1.7).

In particular, we introduce our new martingale problem in Section2.2and prove the uniqueness of the limit law in Theorem2.6. Finally, in Appendix, we prove some auxiliary results.

1. Notation, Model and main results 1.1. Notation

We use the following notation throughout the paper.

IfE is a metric space, we note:

• PpEq the space of probability measures onE endowed with the topology of the weak conver- gence,

• C_bⁿpEq the set of the functionsg which arentimes continuously dierentiable such thatg^pkq is bounded for each0¤k¤n,

• C_cⁿpEq the set of functionsgPC_bⁿpEq that have a compact support.

In addition, in what follows DpR ,Rq denotes the space of càdlàg functions from R to R, endowed with Skorohod metric, and C and K denote arbitrary positive constants whose values can change from line to line in an equation. We writeCθ andKθ if the constants depend on some parameter θ.

1.2. The nite system

We consider, for each N ¥1, a family of i.i.d. Poisson measures pπⁱpds, dz, duqqi1,...,N on R R Rhaving intensity measuredsdzνpduq where ν is a probability measure onR, as well as an i.i.d. family pX₀^N,iqi1,...,N ofR-valued random variables independent of the Poisson measures. The object of this paper is to study the convergence of the Markov process X_t^N pX_t^N,1, . . . , X_t^N,Nq

taking values inR^N and solving, for i1, . . . , N, fort¥0,

$' '' ''

&

'' '' '%

X_t^N,i X₀^N,iα

»t 0

X_s^N,ids

»

r0,tsR R

X_s^N,i1_tz¤fpX^N,i

squπⁱpds, dz, duq

?1 N

¸

ji

»

r0,tsR R

u1_tz¤fpX_s^N,jquπ^jpds, dz, duq, X₀^N,i ν0.

(7)

The coecients of this system are the exponential loss factor α ¡0, the jump rate function f : RÞÑR and the probability measuresν andν0.

In order to guarantee existence and uniqueness of a strong solution of (7), we introduce the following hypothesis.

Assumption 1. The function f is Lipschitz continuous.

In addition, we also need the following condition to obtain a priori bounds on some moments of the process X^N,i

1¤i¤N.

Assumption 2. We assume that³

Rxdνpxq 0,³

Rx²dνpxq   8, and³

Rx²dν0pxq   8. Under Assumptions 1 and 2, existence and uniqueness of strong solutions of (7) follow from Theorem IV.9.1 of Ikeda and Watanabe (1989), exactly in the same way as in Proposition 6.6 of Erny, Löcherbach and Loukianova(2019).

We now dene precisely the limit system and discuss its properties before proving the convergence of the nite to the limit system.

1.3. The limit system The limit system X¯ⁱ

i¥1is given by

$' '' '&

'' ''

%

X¯_tⁱ X¯₀ⁱα

»t 0

X¯_sⁱds

»

r0,tsR R

X¯_sⁱ1_tz¤fpX¯_sⁱ quπⁱpds, dz, duq σ

»t 0

b E

f X¯_sⁱW dW_s, X¯₀ⁱ ν0.

(8)

In the above equation, pWtqt¥0is a standard one-dimensional Brownian motion which is independent of the Poisson random measures, andWσtW_t, t¥0u. Moreover, the initial positionsX¯₀ⁱ, i¥1, are i.i.d., independent ofW and of the Poisson random measures, distributed according toν₀which is the same probability measure as in (7). The common jumps of the particles in the nite system, due to their scaling in 1{?

N and the fact that they are centred, by the Central Limit Theorem, create this single Brownian motionW_twhich is underlying each particle's motion and which induces the common noise factor for all particles in the limit.

The limit equation (8) is not clearly well-posed and requires more conditions on the rate func- tionf. Let us briey comment on the type of diculties that one encounters when proving trajec- torial uniqueness of (8). Roughly speaking, the jump terms demand to work in anL¹framework, while the diusive terms demand to work in anL²framework.Graham (1992) proposes a unied approach to deal both with jump and with diusion terms in a non-linear framework, and we shall

rely on his ideas in the sequel. The presence of the random volatility term which involves conditional expectation causes however additional technical diculties. Finally, another diculty comes from the fact that the jumps induce non-Lipschitz terms of the formX¯_sⁱfpX¯_sⁱq.For this reason a classical Wasserstein-1coupling is not appropriate for the jump terms. Therefore we propose a dierent distance which is inspired by the one already used inFournier and Löcherbach(2016). To do so, we need to work under the following additional assumption.

Assumption 3. 1. We suppose that inff ¡0.

2. There exists a function aPC²pR,R q, strictly increasing and bounded, such that, for a suitable constant C,for allx, yPR,

|a²pxq a²pyq| |a¹pxq a¹pyq| |xa¹pxq ya¹pyq| |fpxq fpyq| ¤C|apxq apyq|. Note that Assumption3implies Assumption1as well as the boundedness of the rate functionf.

Proposition 1.1. Suppose thatfpxq c darctanpxq, wherec¡d^π₂, d¡0. Then Assumption3 holds with af.

Proof. We quickly check that |xa¹pxqya¹pyq| ¤C|apxqapyq|.We have thata¹pxq ₁^d_x2,whence xa¹pxq ya¹pyq dp₁^x_x2 ₁^y_y2q. We use thatdx^d

x

1 x²

_p1¹x^x22q² ¤ 1¹x². Suppose w.l.o.g.

that x¤y.As a consequence,

|xa¹pxq ya¹pyq| d »y

x

1t² p1 t²q²dt

¤d

»y x

1

1 t²dt |arctanpyq arctanpxq| d|apxq apyq|. The other points of Assumption 3follow immediately.

Under these additional assumptions we obtain the well-posedness of each coordinate of the limit system (8), that is, of the pFtqt adapted process pX¯tqthaving càdlàg trajectories which is solution of the SDE

$&

%

dX¯t αX¯tdtX¯t

»

R R1_tz¤fpX¯_tquπpdt, dz, duq σa

µtpfqdWt, X¯0 ν0, µtpfq E

f X¯tW E

f X¯tWt

.

(9)

Here,Ftσtπpr0, ss Aq, s¤t, APBpR Rqu _Wt,WtσtWs, s¤tu andWσtWs, s¥0u. Theorem 1.2. Grant Assumption 3.

1. Pathwise uniqueness holds for the nonlinear SDE (9).

2. If additionally, ³

Rx²dν0pxq   8, then there exists a unique strong solution pX¯tqt¥0 of the nonlinear SDE (9), which is pFtqt adapted with càdlàg trajectories, satisfying for everyt¡0,

E

sup

0¤s¤t

X¯_s²

  8. (10)

Remark 1.3. Notice that the stochastic integral ³t 0

aµ_spfqdW_s is well-dened sincesÞÑa µ_spfq is an pW_tqtprogressively measurable process.

In what follows we just give the proof of Item 1. of the above theorem since its arguments are important for the sequel. We postpone the rather classical proof of Item 2. to Appendix.

Proof of Item 1. of Theorem1.2. Consider two solutions p pXtqt¥0and p qXtqt¥0,pFtqtadapted, de- ned on the same probability space and driven by the same Poisson random measure π and the same Brownian motion W, and with pX0 qX0. We consider Zt : ap pXtq ap qXtq. Denote p

µspfq Erfp pXsq|Wss and qµspfq Erfp qXsq|Wss. Using Ito's formula, we can write

Zt α

»t 0

Xpsa¹p pXsq qXsa¹p qXsq ds 1 2

»t 0

pa²p pXsqpµspfq a²p qXsqqµspfqqσ²ds

»t 0

pa¹p pXsqa p

µspfq a¹p qXsqa q

µspfqqσdWs

»

r0,tsR Rrap pXsq ap qXsqs1_{tz¤fpxXsq^fp|Xsqu}πpds, dz, duq

»

r0,tsR Rrap0q ap pX_sqs1_{tfp|Xsq z¤fpxXsqu}πpds, dz, duq

»

r0,tsR Rrap qXsq ap0qs1_{tfpxXsq z¤fp|Xsqu}πpds, dz, duq :At Mt ∆t, whereAtdenotes the bounded variation part of the evolution,Mtthe martingale part and∆tthe sum of the three jump terms. Notice that

M_t

»t 0

pa¹p pX_sqa p

µ_spfq a¹p qX_sqa q

µ_spfqqσdW_s is a square integrable martingale sincef anda¹ are bounded.

We wish to obtain a control on |Z_t|:sup_s¤t|Zs|.We rst take care of the jumps of |Zt|.Notice rst that, sincef andaare bounded,

∆px, yq: pfpxq ^fpyqq|apxq apyq| |fpxq fpyq||ap0q apyq| |ap0q apxq|

¤C|apxq apyq|, implying that

Esup

s¤t|∆s| ¤CE

»t 0

|ap pXsq ap qXsq|ds¤CtE|Z_t|. Moreover, for a constant Cdepending onσ²,}f}₈,}a}₈,}a¹}₈,}a²}₈ andα,

Esup

s¤t|As| ¤C

»t 0

E|a¹p pXsq pXsa¹p qXsq qXs|ds C

»t 0

|a²p pXsq a²p qXsq|ds

»t 0

|pµspfq qµspfq|ds

. We know that |a¹p pXsq pXsa¹p qXsq qXs| |a²p pXsq a²p qXsq| ¤C|ap pXsq ap qXsq| C|Zs|.Therefore,

Esup

s¤t|As| ¤CE

»t 0

|Zs|ds

»t 0

|pµspfq qµspfq|ds

.

Moreover,

|pµspfq qµspfq| E

fp pXsq fp qXsq|W ¤E

|fp pXsq fp qXsq||W ¤Ep|Zs||Wq,

and thus,

E

»t 0

|pµspfq qµspfq|ds¤E

»t 0

|Zs|ds¤tE|Z_t|.

Putting all these upper bounds together we conclude that for a constantC not depending ont, Esup

s¤t|As| ¤CtE|Z_t|.

Finally, we treat the martingale part using the Burkholder-Davis-Gundy inequality, and we obtain

Esup

s¤t|Ms| ¤CE

»t

0

pa¹p pXsqa p

µspfq a¹p qXsqa q

µspfqq²ds ¹{2

. But

pa¹p pXsqa p

µspfq a¹p qXsqa q

µspfqq²¤C

ppa¹p pXsq a¹p qXsqq² pa p

µspfq a q µspfqq²

¤C|Z_t|² Cpa p

µspfq a q

µspfqq², (11) where we have used that |a¹pxq a¹pyq| ¤C|apxq apyq| and thatf anda¹ are bounded.

Finally, sinceinff ¡0,

|a p

µ_spfq a q

µ_spfq|²¤C|pµ_spfq qµ_spfq|²¤CpEp|Z_s||Wsqq².

We use that |Z_s| ¤ |Z_t|, implying thatEp|Z_s||Wq ¤Ep|Z_t||Wq. Therefore we obtain the upper bound

|a p

µspfq a q

µspfq|²¤CpEp|Z_t||Wqq² for alls¤t,which implies the control of

Esup

s¤t|Ms| ¤C? tE|Z_t|.

The above upper bounds imply that, for a constantCnot depending ontnor on the initial condition, E|Z_t| ¤Cpt ?

tqE|Z_t|,

and therefore, fort1suciently small,E|Z_t1| 0.We can repeat this argument on intervals rt1,2t1s, with initial condition Xˆt₁, and iterate it up to any nite T because t1 does only depend on the coecients of the system but not on the initial condition. This implies the assertion.

Remark 1.4. Theorem 1.2states the well-posedness of the SDE (9). Under the same hypotheses, with almost the same reasoning, one can prove the well-posedness of the system (8).

In the sequel, we shall also use an important property of the limit system (8), which is the conditional independence of the processesX¯ⁱ (i¥1) given the Brownian motionW.

Proposition 1.5. If Assumption3holds and³

Rx²dν0pxq   8,then (i) for all N P N there exists a strong solution X¯ⁱ

1¤i¤N of (8), and pathwise uniqueness holds,

(ii) X¯¹, . . . ,X¯^N are independent conditionally to W, (iii) for allt¥0, almost surely, the weak limit of _N¹ °N

i1δX¯_|r0,tsⁱ is given bylimNÑ8 1 N

°N

i1δX¯ⁱ_|r0,ts PpX¯_|rⁱ_0,tsP |Wtq PpX¯_|rⁱ_0,tsP |Wq.

Let us nally mention that the random limit measureµsatises a nonlinear stochastic PDE in weak form. More precisely,

Corollary 1.6. Grant Assumption 3and suppose that ³

Rx²dν0pxq   8. Then the measure µ PppX¯tqt¥0P |Wq satises the following nonlinear stochastic PDE in weak form: for anyϕPC_b²pRq, for any t¥0,

»

R

ϕpxqµtpdxq

»

R

ϕpxqν0pdxq

»t 0

»

R

ϕ¹pxqµspdxq a

µspfqσdWs

»t 0

»

R

rϕp0q ϕpxqsfpxq αϕ¹pxqx 1

2σ²ϕ²pxqµspfq µspdxqds.

The proofs of Proposition1.5and of Corollary1.6are postponed to Appendix.

1.4. Convergence to the limit system We are now able to state our main result.

Theorem 1.7. Grant Assumptions1,2and3. Then the empirical measureµ^N _N¹ °N

i1δ_XN,i of the Nparticle system pX^N,iq1¤i¤N converges in distribution in PpDpR ,Rqq to µ :LpX¯¹|Wq, where pX¯ⁱqi¥1 is solution of (8).

Corollary 1.8. Under the assumptions of Theorem 1.7, pX^N,jq1¤j¤N converges in distribution to pX¯^jqj¥1 in DpR ,Rq^N.

Proof. Together with the statement of Theorem 1.7, the proof is an immediate consequence of Proposition 7.20 ofAldous(1983).

We will prove Theorem1.7in a two step procedure. Firstly we prove the tightness of the sequence of empirical measures, and then in a second step we identify all possible limits as solutions of a martingale problem.

2. Proof of Theorem 1.7

This section is dedicated to prove that the sequence pµ^NqN of the empirical measures µ^N :

°N j1δ_pXj

tqt¥0 converges in distribution toµ:LpX¯¹|Wq, where pX¯^jqj¥1 is solution of (8).

In a rst time, we prove that the sequence pµ^NqN is tight on PpDpR ,Rqq. The main step to prove the convergence of pµ^NqN is then to show that each converging subsequence converges to the same limit in distribution. For this purpose, we introduce a new martingale problem, and we show that every possible limit of µ^N is a solution of this martingale problem. Finally, we will show how the uniqueness of the limit law follows from the exchangeability of the system.

2.1. Tightness of pµ^NqN

Proposition 2.1. Grant Assumptions 1 and 2. For each N ¥ 1, consider the unique solution pX_t^Nqt¥0 to (7) starting from some i.i.d.ν0-distributed initial conditions X₀^N,i.

(i) The sequence of processes pX_t^N,1qt¥0 is tight in DpR ,Rq.

(ii) The sequence of empirical measuresµ^N N¹°N

i1δ_pXN,i

t qt¥0 is tight in PpDpR ,Rqq.

Proof. First, it is well-known that point (ii) follows from point (i) and the exchangeability of the system, see (Sznitman 1989, Proposition 2.2-(ii)). We thus only prove (i). To show that the family ppX_t^N,1qt¥0qN¥1 is tight inDpR ,Rq, we use the criterion of Aldous, see Theorem 4.5 ofJacod and Shiryaev(2003). It is sucient to prove that

(a) for all T ¡ 0, all ε¡ 0, lim_δÓ0lim sup_NÑ8sup_pS,S1qPA_δ,T Pp|X_S^N,1₁ X_S^N,1| ¡ εq 0, where Aδ,T is the set of all pairs of stopping times pS, S¹q such that0¤S¤S¹¤S δ¤T a.s., (b) for allT ¡0,limKÒ8sup_NPpsup_tPr0,Ts|X_t^N,1| ¥Kq 0.

To check (a), consider pS, S¹q PA_δ,T and write

X_S^N,1₁ X_S^N,1

»S¹ S

»

R

»₈

0

X_s^N,1 1_tz¤fpX^N,1

squπ¹pds, du, dzq α

»S¹ S

X_s^N,1ds

?1 N

¸N j2

»S¹ S

»

R

»₈

0

u1_tz¤fpX_s^N,jquπ^jpds, du, dzq, implying that

|X_S^N,11 X_S^N,1| ¤ |

»S¹ S

»

R

»₈

0

X_s^N,1 1_tz¤fpX_s^N,1quπ¹pds, du, dzq|

δα sup

0¤s¤T

X_s^N,1 | 1

?N

¸N j2

»S¹ S

»

R

»₈

0

u1_tz¤fpX_s^N,jquπ^jpds, du, dzq|

:|IS,S¹| δα sup

0¤s¤T

X_s^N,1 |JS,S¹|. We rst note that |IS,S¹| ¡0 implies thatI˜S,S¹ :³S¹

S

³

R

³₈

0 1_tz¤fpX_s^N,1quπⁱpds, du, dzq ¥1, whence Pp|IS,S1| ¡0q ¤PpI˜S,S1 ¥1q ¤ErI˜S,S1s ¤E

»^{S δ}

S

fpX_s^N,1qds

¤ ||f||₈δ,

sincef is bounded. We proceed similarly to check that Pp|JS,S1| ¥εq ¤ 1

ε²ErpJS,S1q²s ¤ σ² N ε²

¸N j2

E »^{S δ}

S

fpX_s^N,jqds ¤σ²

ε²}f}₈δ.

The term sup_0¤s¤T|X_s^N,1| can be handled using Lemma3.1.(ii).

Finally (b) is a straightforward consequence of Lemma3.1.(ii) and Markov's inequality.

2.2. Martingale problem

We now introduce a new martingale problem, whose solutions are the limits of any converging subsequence of µ^N _N¹ °N

j1δ_XN,j. In this martingale problem, we are interested in couples of trajectories to be able to put hands on the correlations between the particles. In particular, this will allow us to show that, in the limit system (8), the processesX¯ⁱ(i¥1) share the same Brownian motion, but are driven by Poisson measuresπⁱ (i¥1) which are independent. The reason why we only need to study the correlation between two particles is the exchangeability of the innite system.

Let Q be a distribution on PpDpR ,Rqq. Dene a probability measure P on PpDpR ,Rqq DpR ,Rq²by

PpABq:

»

PpDpR ,Rqq1^ApmqmbmpBqQpdmq. (12) We write any atomic eventωPΩ :PpDpR ,RqqDpR ,Rq²asω pµ, Yq,withY pY¹, Y²q. Thus, the law of the canonical variableµisQ, and that of pYtqt¥0 is

PY

»

PpDpR ,Rqq

Qpdmqmbmpq. Moreover we haveP almost surely

µLpY¹|µq LpY²|µq andLpY|µq µbµ.

Writingµ_t:³

DpR ,Rqµpdγqδ_γt for the projection onto thetth time coordinate, we introduce the ltration

G_tσpY_s, s¤tq _σpµ_spfq, s¤tq.

Denition 2.2. We say thatQPPpPpDpR ,Rqqq is a solution to the martingale problem pMq if the following holds.

(i) Qalmost surely,µ0ν0.

(ii) For allgPC_b²pR²q, M_t^g:gpYtq gpY0q ³t

0Lgpµs, Ysqdsis a pP,pGtqtqmartingale, where Lgpµ, xq αx¹Bx¹gpxq αx²Bx²gpxq σ²

2 µpfq

¸2 i,j1

Bx²ⁱx^jgpxq fpx¹qpgp0, x²q gpxqq fpx²qpgpx¹,0q gpxqq.

Remark 2.3. It is not clear if the martingale problem is well-posed, but we are not interested in proving uniqueness for it. However, we will have uniqueness within the class of all possible limits in distribution of µ^N.More precisely, we shall prove that, if µis a limit in distribution of µ^N such that Lpµq is solution to pMq, then µLpX¯|Wq, with X¯ the strong solution of (9). Equivalently, dening the problem pMq for all nite-dimensional distributions, and not only for two coordinates, whereYⁱ (i¥1) are dened as a mixture directed byµ, would lead to uniqueness.

Let pX¯ⁱqi¥1 be the solution of the limit system (8) and µLpX¯¹|Wq. Then we already know that Lpµq is a solution of pMq.Let us now characterise any possible solution of pMq.

Lemma 2.4. Grant Assumption 3. Let Q P PpPpDpR ,Rqqq be a solution of pMq. Let pµ, Yq be the canonical variable dened above, and write Y pY¹, Y²q. Then there exists a standard pGtqtBrownian motion W and on an extension pΩ,˜ pG˜tqt,P˜q of pΩ,pGtqt, Pq there exist pG˜tqt Poisson random measuresπ¹, π²onR R having Lebesgue intensity such thatW, π¹andπ² are independent and

dY_t¹ αY_t¹dt σa

µtpfqdWtY_t¹

»

R

1t^z¤fpYt¹quπ¹pdt, dzq, dY_t² αY_t²dt σa

µ_tpfqdW_tY_t²

»

R

1t^z¤fpY_t²quπ²pdt, dzq.

Proof. Item (ii) of of pMq together with Theorem II.2.42 of Jacod and Shiryaev(2003) imply that Y is a semimartingale with characteristics pB, C, νq given by

Bⁱ_t α

»t 0

Y_sⁱds

»t 0

Y_sⁱfpY_sⁱqds, 1¤i¤2, C_t^i,j

»t 0

µ_spfqds, 1¤i, j¤2, νpdt, dyq dtpfpY_t¹qδ_pY1

t,0qpdyq fpY_t²qδ_p0,Y2 tqpdyqq.

Then we can use the canonical representation of Y (see Theorem II.2.34 ofJacod and Shiryaev (2003)) with the truncation functionhpyq y for every y: Y_tY₀B_tM_t^c M_t^d,whereM^c is a continuous local martingale andM^da purely discontinuous local martingale. By denition of the characteristics, xM^c,i, M^c,jyt C_t^i,j. In particular, xM^c,iyt ³t

0µspfqds (i 1,2). Consequently, applying Theorem II.7.1 of Ikeda and Watanabe (1989) to both coordinates, we know that there exist Brownian motions W¹, W² such that

M_t^c,i

»t 0

aµ_spfqdW_sⁱ, i1,2.

We now prove thatW¹W².Letρbe the correlation betweenW¹andW². Classical computations give xW¹, W²yt ρt, implying that xM^c,1, M^c,2yt ρ³t

0µspfqds. In addition xM^c,1, M^c,2yt C_t^1,2 ³t

0µ_spfqds, and this implies that ρ1 andW¹ W², since³t

0µ_spfqds ¡0 because f is lower-bounded.

We now prove the existence of the independent Poisson measures π¹, π². We know that M^d h pµ^Y νq, whereµ^Y °

s1t∆Ys0uδ_ps,Ysq is the jump measure ofY andν is its compensator.

We rely on Theorem II.7.4 ofIkeda and Watanabe(1989). Using the notation therein, we introduce Z R , mLebesgue measure onZ and

θpt, zq: pY_t¹,0q1t^z¤fpYt¹ qu p0,Y_t²q1t^{||f||8 z¤||f||8 fpY}t²qu.

According to Theorem II.7.4 of Ikeda and Watanabe(1989), there exists a Poisson measureπ on R R having intensitydtdzsuch that, for allEPBpR²q,

µ^Ypr0, ts Eq

»t 0

»₈

0

1_tθps,zqPEuπpds, dzq. (13)

Now let us consider two independent Poisson measures π˜¹,π˜² (independent of everything else) on r||f||₈,8r having Lebesgue intensity. We then dene π¹ in the following way: for any A P BpR r0,||f||₈sq, π¹pAq πpAq, and for A P BpR s||f||₈,8rq, π¹pAq ˜π¹pAq. We dene π² in a similar way: For A P BpR r0,||f||₈sq, π²pAq πptpt,||f||₈ zq : pt, zq P Auq, and for A P BpR s||f||₈,8rq, π²pAq π˜²pAq. By denition of Poisson measures, π¹ and π² are independent Poisson measures on R² having Lebesgue intensity, and together with (13), we have

M_t^d,i

»

r0,tsR

Y_sⁱ1t^z¤fpY_sⁱ quπⁱpds, dzq

»t 0

Y_sⁱfpY_sⁱqds, 1¤i¤2.

Moreover we have the following

Theorem 2.5. Assume that Assumptions 1,2and3hold. Then the distribution of any limitµ of the sequence µ^N : _N¹ °N

j1δ_XN,j is solution of item (ii) of pMq.

Proof. Step 1. We rst check that for any t ¥ 0, a.s., µptγ : ∆γptq 0uq 0. We assume by contradiction that there exists t ¡0 such thatµptγ : ∆γptq 0uq ¡0 with positive probability.

Hence there are a, b ¡ 0 such that the event E : tµptγ : |∆γptq| ¡ auq ¡ bu has a positive probability. For everyε¡0, we haveE€ tµpB_a^εq ¡bu, whereB^ε_a: tγ: sup_{sPptε,t εq}|∆γpsq| ¡au, which is an open subset ofDpR ,Rq. ThusP_a,b^ε : tµPPpDpR ,Rqq:µpB_a^εq ¡bu is an open subset ofPpDpR ,Rqq. The Portmanteau theorem implies then that for anyε¡0,

lim inf

NÑ8 Ppµ^N PP_a,b^ε q ¥PpµPP_a,b^ε q ¥PpEq ¡0. (14) Firstly, we can write

J^N,ε,i: sup

tε s t ε

∆X_s^N,i G^ε,i_N _S_N^ε, whereG^ε,i_N :max_sPDε,i

N |X_s^N,i| is the maximal height of the big jumps ofX^N,i,withD_N^ε,i: ttε¤ s¤t ε:πⁱptsu r0, fpX_s^N,iqs R q 0u.Moreover,S_N^ε :maxt|U^jpsq|{?

N:sP”

1¤j¤ND_N^ε,ju is the maximal height of the small jumps ofX^N,i, whereU^jpsq is dened forsPD_N^ε,j, almost surely, as the only real number that satisesπ^jptsu r0, fpX_s^N,j qs tU^jpsquq 1.

We have that

µ^NpB_a^εq ¡b(

# 1 N

¸N j1

1tJ^N,ε,j¡au¡b +

. Consequently, by exchangeability and Markov's inequality,

P µ^NpB^ε_aq ¡b

¤ 1 bE

1_tJ^N,ε,1¡au

1

bP J^N,ε,1¡a

¤1 b

P

G^ε,1_N ¡a PpS_N^ε ¡aq . (15) The number of big jumps of X^N,1 in stε, t εr is smaller than a random variable ξ having Poisson distribution with parameter2ε||f||8.Hence

P G^ε_N,1¡a

¤Ppξ¥1q 1e^2ε||f||8¤2ε||f||8. (16)