Annealed limit for a diffusive disordered mean-field model with random jumps

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arXiv:2210.13128v4 [math.PR] 14 Sep 2023

Annealed limit for a diﬀusive disordered

mean-ﬁeld model with random jumps

Xavier Erny

Ecole polytechnique, Centre de math´ematiques appliqu´ees (CMAP), 91128 Palaiseau

Abstract: We study a sequence of N−particle mean-ﬁeld systems, each driven by Nsimple

point processes ZN,i in a random environment. Each ZN,i has the same intensity (f(XN

t−))t

and at every jump time of ZN,i,the process XNdoes a jump of height Ui/√Nwhere the Ui

are disordered centered random variables attached to each particle. We prove the convergence

in distribution of XNto some limit process ¯

Xthat is solution to an SDE with a random

environment given by a Gaussian variable, with a convergence speed for the ﬁnite-dimensional

distributions. This Gaussian variable is created by a CLT as the limit of the patial sums of

the Ui.To prove this result, we use a coupling for the classical CLT relying on the result

of Koml´os, Major and Tusn´ady, that allows to compare the conditional distributions of XN

and ¯

Xgiven the environment variables, with the same Markovian technics as the ones used

in Erny, L¨ocherbach and Loukianova (2022).

MSC2020 subject classiﬁcations:60K37, 60J35, 60J25, 60J60; secondary 60F05, 60G50,

60G55.

Keywords and phrases: Annealed limit in random environment, Central limit theorem

coupling, Piecewise deterministic Markov processes, Mean-ﬁeld model.

1. Introduction

The term disordered comes from physics litterature to designate some ”asymmetric systems”. Par-

ticle systems in random environment can often be seen as disordered models, since, in this kind

of models, it is possible to attach to each particle (or each pair of particles) a random variable,

interpreted as a disordered variable, creating asymmetric interactions between the particles. This is

for example the case of models with spin glass dynamics, where for each pair of particle (i, j),there

exists a disordered variable of the form SiSjVij , where Siand Sjare typically {−1,+1}−valued

variables modelling the spins of the particles iand j, and Vij is a random variable modelling an in-

teraction strength between the two particles. This kind of model has been introduced in the seminal

paper of Sherrington and Kirkpatrick.

In the model we study in this paper, unlikely with spin glass dynamics model, we attach disor-

dered variables to each particle (and not to each pair of particles) that are i.i.d. and independent of

the time, like in the very similar model of Pfaﬀelhuber, Rotter and Stiefel. More precisely, we study

the solutions of stochastic diﬀerential equations with a drift term and a jump term that depends on

the disordered variables. There is a natural application of this kind of model in neurosciences (as

explained in Section 4 of Pfaﬀelhuber, Rotter and Stiefel), where the equations model the dynam-

ics of the membrane potentials of neurons, the jump times model the times at which the neurons

receive spikes by another neuron of the network, and the jump heights (given by the disordered

variables) model the synaptic weights in the network. The drift term models the (deterministic)

dynamics of the membrane potential between its jump times.

The property we want to prove is the convergence in distribution of the disordered system as

the number of particles of the system goes to inﬁnity. As we work in a random environment,

X. Erny/Annealed limit in random environment 2

the convergence of the processes can be understood in two ways: a quenched convergence (i.e.

convergence conditionally on the environment) or an annealed convergence (i.e. convergence when

the environment variables are averaged). We refer to Ben Arous and Guionnet and Guionnet (1997)

for the deﬁnitions of the terms ”quenched” and ”annealed” that we use in this paper.

Formally, the aim of the paper is to prove the convergence in distribution of a process (XN

t)tde-

ﬁned in a random environment. Let us deﬁne rigorously this process. If the environment (u[N]

j)1≤j≤N∈

RNis ﬁxed, we deﬁne (XN

t(u[N]))tas the solution of the following SDE:

dXN

t(u[N]) = b(XN

t(u[N]))dt +1

√N

j=1

u[N]

jZR+

1{z≤f(XN

t−(u[N]))}dπj(t, z),(1)

where band fare deterministic functions and πj(1 ≤j≤N) are independent Poisson measures

on [0,+∞)2of intensity dt ·dz, that are independent of XN

0.We note νN

0the distribution of XN

By deﬁnition, the process (XN

t)tis deﬁned as

t:= XN

t(U[N]),(2)

where the variables U[N]

j(1 ≤j≤N) are i.i.d. centered random variables with ﬁnite variance σ2,

that are independent of the Poisson measures πj(1 ≤j≤N) and of the initial condition XN

0. We

note µthe law of these variables. This law is independent of Nand is a parameter of the model. For

the sake of notation, instead of deﬁning for each N−particle system a family of Nvariables U[N]

(1 ≤j≤N), we introduce a countable sequence of random variables Uj(j≥1). By deﬁnition, we

set U[N]:= (U1, ..., UN) (e.g. the ﬁrst Nvariables of U[N+1] are exactly U[N]). In particular the [N]

in superscript will be dropped.

The limit of (XN

t)tis shown to be a process ( ¯

Xt)tthat is also deﬁned in a random environment.

For a ﬁxed w∈R,let ( ¯

Xt(w))tbe deﬁned as the solution of

d¯

Xt(w) = b(¯

Xt(w))dt +wf(¯

Xt(w))dt +σqf(¯

Xt(w))dBt,(3)

where Bis a standard one-dimensional Brownian motion that is independent of ¯

X0.We note ¯ν0

the distribution of ¯

X0.The limit process ( ¯

Xt)tis deﬁned as

Xt:= ¯

Xt(W),(4)

where Wis a Gaussian variable with parameter (0, σ2) that is independent of B, ¯

X0.

Heuristically, a simple way to obtain the limit equation (3)−(4) from (1)−(2) consists in writing

the SDE of (XN

t)tas

dXN

t=b(XN

t)dt +1

√N

j=1

UjZR+

1{z≤f(XN

t−)}d˜πj(t, z) + 



√N

j=1

Uj

f(XN

t)dt, (5)

where ˜πj(dt, dz) := πj(dt, dz)−dtdz is the compensated Poisson measure of πj(1 ≤j≤N).

Under this form, the second term of the SDE above is a local martingale (conditionally on the

environment) whose jump heights vanish as Ngoes to inﬁnity. So, in the limit equation, it creates

the Brownian term in (3). And the third term in the SDE above is a drift term that corresponds

X. Erny/Annealed limit in random environment 3

to the second term of (3) since, according to the CLT, the variable N−1/2PN

j=1 Ujconverges in

distribution to W∼ N(0, σ2).

The main result of the paper is the convergence in distribution of the process (XN

t)tto the

process ( ¯

Xt)t.Our approach only allows to prove the annealed convergence of the processes, because

there is a tricky problem concerning the quenched result: for a convergence to be true in our model,

we need to guarantee N−1/2PN

j=1 Ujto converge. This convergence holds true in distribution but

cannot be true almost surely. We still manage to prove some properties related to a quenched

convergence.

The dynamics (1)−(2) and (3)−(4) are similar to respectively equations (3.9) and (3.10) of

Pfaﬀelhuber, Rotter and Stiefel. Indeed, if we consider b(x) := −αx for the drift function in our

model, we obtain exactly the same dynamics as Pfaﬀelhuber, Rotter and Stiefel for the convolution

kernel ϕ(t) := e−αt.Consequently, our main result Theorem 1.2 can be compared to Theorem 2

of Pfaﬀelhuber, Rotter and Stiefel. However, the proof of Pfaﬀelhuber, Rotter and Stiefel cannot

be used to prove Theorem 1.2 since it relies on the assumption that the environment variables Uj

(j≥1) follow Rademacher distribution, whereas we only assume the environment variables to

be centered with some exponential moments. Note that it would also not be possible to prove

Theorem 2 of Pfaﬀelhuber, Rotter and Stiefel with the proof of our Theorem 1.2 since it requires

to have some Markovian structure (conditionally to the environment variables), which is not the

case for Hawkes processes with general convolution kernel as in Pfaﬀelhuber, Rotter and Stiefel.

Another interesting point of Theorem 1.2 is that we have an explicit convergence speed.

The model of this paper is also close to the one of Erny, L¨ocherbach and Loukianova (2022),

except that in Erny, L¨ocherbach and Loukianova (2022) the environment variables Uj(1 ≤j≤N)

in (1) were replaced by centered marks of the point processes (ZN,j

t)tdeﬁned as

ZN,j

t:= Z[0,t]×R+

1{z≤f(XN

s−)}dπj(s, z).

In term of neurosciences, the model of Erny, L¨ocherbach and Loukianova (2022), contrary to the

model (1)−(2) and Pfaﬀelhuber, Rotter and Stiefel, is not consistent with the following biological

property: the role of a synapse cannot change (i.e. it is either always excitatory or always inhibitory).

Mathematically, if the rate function fis monotone, it means that every time some particle iinteracts

with another particle jthrough a synaptic strength Ui, the sign of Uiis always the same. This

property is not guaranteed in Erny, L¨ocherbach and Loukianova (2022), where the Uiare marks of

some point processes (and hence they change at each spiking time of the same neuron), but it holds

true in the model of this paper where they are ﬁxed environment variables.

The model of this paper is a priori harder to study because, in Erny, L¨ocherbach and Loukianova

(2022), the processes (XN

t)t(and their limit) are Markov processes and semimartingales, which is

not the case here because of the random environment. In order to apply results from the theories

of Markov processes and semimartingales, we need to work conditionally on the environment and,

in a second time, to integrate over the environment.

To work conditionally on the environment in a proper manner, we use a coupling between the

variables Uj(j≥1) of (2) and the Gaussian variable Wof (4), corresponding to a coupling result for

the classical CLT. To be more precise and formal, we construct a sequence of identically distributed

and non-independent Gaussian variables (W[N])Nsuch that, for every N≥2,



√N

j=1

Uj−W[N]≤Kln N

√N,(6)

X. Erny/Annealed limit in random environment 4

where K > 0 is a random variable independent of N. This coupling and the control that we use on

the random variable Krely on Theorem 1 of Koml´os, Major and Tusn´ady. Indeed, if we assume

that the distribution µadmits some exponential moments: there exists α > 0 such that

eα|x|dµ(x)<∞,

then, it is possible (following the reasonning of Section 7.5 of Ethier and Kurtz (2005) that relies on

Theorem 1 of Koml´os, Major and Tusn´ady) to construct on the same probability space (possibly

enlarged) as (Uj)j≥1a standard one-dimensional Brownian motion (βt)tsuch that, the random

variable Kdeﬁned as

K:= sup

N≥2

j=1

Uj−σβN

/ln N

is ﬁnite almost surely and admits exponential moments (this is stated and proved in Lemma A.1

for self-containedness). Then, deﬁning W[N]in the following way

W[N]:= σβN/√N

gives exactly (6). This construction is recalled at Appendix A.

The coupling (6) allows us to compare the conditional distributions of the processes (XN

t)t

and ( ¯

t)tgiven the environment variables (where ¯

XNis deﬁned as ¯

X(W[N]) in (4)). This results

can be used to prove that the diﬀerence of the ﬁnite-dimensional distributions vanishes almost surely

w.r.t. the randomness of the environment. However, we cannot obtain a quenched convergence in the

usual sense, since the version of the limit to which we compare XNis ¯

XNwhich still depends on N.

More precisely, the problem is the fact that, the sequence of Gaussian variables W[N]has (almost

surely) many subsequential limits: indeed, recalling that W[N]:= σβN/√Nfor some standard

Brownian motion β, it is of common knowledge that the limit superior (resp. inferior) of W[N]is

inﬁnity (resp. minus inﬁnity) almost surely.

To prove this result, we need to introduce formally Ethe sigma-ﬁeld related to the random

environment:

E:= σ(Uj:j≥1) ∨σW[N]:N∈N∗.

And, in order to compare the conditional distributions of (XN

t)tand ( ¯

t),we introduce their

(conditional) semigroups (PN

E,t)tand ( ¯

E,t)t, and their inﬁnitesimal generators AN

Eand ¯

Ew.r.t. E

(see Section 1.1 for the deﬁnitions). Using our coupling, we obtain a bound for the diﬀerence of the

generators for suﬃciently smooth test-functions, and deduce a bound for the semigroups using the

following formula (which can be found in Lemma 1.6.2 of Ethier and Kurtz (2005) and in equa-

tion (3.1) of Erny, L¨ocherbach and Loukianova (2022) with diﬀerent terminologies and hypotheses):

for gsmooth enough, t≥0 and x∈R,

E,tg(x)−¯

E,tg(x) = Zt

E,t−sAN

E−¯

E¯

E,sg(x)ds. (7)

Note that, to use the formula above, one needs to guarantee some regularity properties on

the limit semigroup ( ¯

E,t)t. This is proved using the regularity of the stochastic ﬂow of the pro-

cess ( ¯

t)t(with a proof similar as the one of Proposition 3.4 of Erny, L¨ocherbach and Loukianova

(2022)).

X. Erny/Annealed limit in random environment 5

Let us ﬁnally mention that the same kind of model (i.e. particle systems directed by SDEs where

the jump term depends on random environment variables) has already been studied in normal-

ization N−1in Chevallier et al. and more recently in Agathe-Nerine. In both of these references,

the random environment relies on the spatial structure of the particle system. To the best of our

knowledge, Pfaﬀelhuber, Rotter and Stiefel is the ﬁrst paper about the convergence of this type of

model in normalization N−1/2.However this kind of convergence concerning models where a drift

term is driven by a random environment in normalization N−1/2have already been proved: e.g.

Ben Arous and Guionnet,Guionnet (1997) and Dembo, Lubetzky and Zeitouni. And, in a similar

framework, Lu¸con has proved a quenched convergence of the ﬂuctuations of a particle system (with

a drift term driven by a random environment) in normalization N−1,which is a similar regime.

Note that the tricky problem concerning the quenched convergence mentionned in the previous

paragraph (also related to Remark A.2) is also encountered in Lu¸con.

Organization. In Section 1.1, we introduce the notation that we use throughout the paper.

The assumptions, the main result about the annealed converge (i.e. Theorem 1.2) and a quenched

control (i.e. Proposition 1.3) are stated in Section 1.2. Section 2is dedicated to prove our main result

Theorem 1.2, while Proposition 1.3 is proved in Section 3. Finally, the CLT coupling is formally

recalled and its main property is proved in Appendix A, and Appendix Bgathers the proofs of

some technical lemmas.

1.1. Notation

In the paper, we use the following notation:

•For T > 0,we note D([0, T ],R) (resp. D(R+,R)) the set of c`adl`ag functions deﬁned on [0, T ]

(resp. R+) endowed with Skorohod topology (see for example Section 12 (resp. Section 16) of

Billingsley (1999)).

•For n∈N∗, Cn

b(R) denotes the set of real-valued functions deﬁned on Rthat are Cnsuch

that all their derivatives (up to order n) are bounded.

•For n∈N∗,and g∈Cn

b(R),we note

||g||n,∞:=

k=0 g(k)∞.

•For ν1, ν2distributions on Rwith ﬁnite ﬁrst order moments, we use the notation dKR(ν1, ν2)

for the Kantorovich-Rubinstein metric between ν1and ν2.This quantity is deﬁned as:

dKR(ν1, ν2) := sup

gZR

g(x)dν1(x)−ZR

g(x)dν2(x),

where the supremum is taken over the Lipschitz continuous functions g:R→Rwhose

Lipschitz constants are non-greater than one.

•If (Xt)t≥0is a real-valued Markov process, (X(x)

t)t≥0,x∈Rdenotes its stochastic ﬂow. In other

words, for x∈R,(X(x)

t)tis the process starting at position x∈Rthat is deﬁned by the

dynamics of (Xt)t≥0.In addition, if the stochastic ﬂow is ntimes diﬀerentiable w.r.t. the

space variable xat time t, we note ∂n

xX(x)

tthe related derivative.

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摘要：

arXiv:2210.13128v4[math.PR]14Sep2023Annealedlimitforadiﬀusivedisorderedmean-ﬁeldmodelwithrandomjumpsXavierErnyEcolepolytechnique,Centredemath´ematiquesappliqu´ees(CMAP),91128PalaiseauAbstract:WestudyasequenceofN−particlemean-ﬁeldsystems,eachdrivenbyNsimplepointprocessesZN,iinarandomenvironment.EachZ...

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Annealed limit for a diffusive disordered mean-field model with random jumps

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