# Some Fluctuation Results for Weakly Interacting Multi-type Particle Systems

###### Abstract

A collection of -diffusing interacting particles where each particle belongs to one of different populations is considered. Evolution equation for a particle from population depends on the empirical measures of particle states corresponding to the various populations and the form of this dependence may change from one population to another. In addition, the drift coefficients in the particle evolution equations may depend on a factor that is common to all particles and which is described through the solution of a stochastic differential equation coupled, through the empirical measures, with the -particle dynamics. We are interested in the asymptotic behavior as . Although the full system is not exchangeable, particles in the same population have an exchangeable distribution. Using this structure, one can prove using standard techniques a law of large numbers result and a propagation of chaos property. In the current work we study fluctuations about the law of large number limit. For the case where the common factor is absent the limit is given in terms of a Gaussian field whereas in the presence of a common factor it is characterized through a mixture of Gaussian distributions. We also obtain, as a corollary, new fluctuation results for disjoint sub-families of single type particle systems, i.e. when . Finally, we establish limit theorems for multi-type statistics of such weakly interacting particles, given in terms of multiple Wiener integrals.

AMS 2000 subject classifications: 60F05; 60K35; 60H30; 60J70.

Keywords: Mean field interaction, common factor, weakly interacting diffusions, propagation of chaos, central limit theorems, fluctuation limits, exchangeability, symmetric statistics, multiple Wiener integrals.

CLT for Multi-type Particle Systems

August 3, 2019

## 1 Introduction

For , let be -valued stochastic processes, representing trajectories of particles, each of which belongs to one of types (populations) with the membership map denoted by , namely -th particle is type if . The dynamics is given in terms of a collection of stochastic differential equations (SDE) driven by mutually independent Brownian motions (BM) with each particle’s initial condition governed independently by a probability law that depends only on its type. The stochastic processes interact with each other through the coefficients of the SDE which, for the -th process, with , depend on not only the -th state process and the -th type, but also the empirical measures , and a stochastic process that is common to all particle equations (common factor). Here is the total number of particles that belong to the -th type. The common factor is an -dimensional stochastic process described once more through an SDE driven by a BM which is independent of the other noise processes. Such stochastic systems are commonly referred to as weakly interacting diffusion processes and have a long history. Classical works that study law of large number (LLN) results and central limit theorems (CLT) include McKean [14, 15], Braun and Hepp [2], Dawson [7], Tanaka [21], Oelschaläger [17], Sznitman [19, 20], Graham and Méléard [10], Shiga and Tanaka [18], Méléard [16]. All the above papers consider exchangeable populations, i.e. , and a setting where the common factor is absent. Motivated by approximation schemes for Stochastic Partial Differential Equations (SPDE) the papers [12, 13] considered a setting where the common factor is modeled as a Brownian sheet that drives the dynamics of each particle. The paper [12] studied LLN and [13] considered fluctuations about the LLN limit. In a setting where particle dynamics are given through jump-diffusions and the common factor is described by another jump-diffusion that is coupled with the particle dynamics, a CLT was recently obtained in [3]. The fluctuation limit theorems in [13, 3], although allowing for a common factor, are limited to exchangeable populations. The goal of the current work is to study fluctuations for multi-type particle systems. Since these systems are not exchangeable (there is also no natural way to regard the system as a -vector of -dimensional exchangeable particles), classical techniques for proving CLT, developed in the above papers [19, 18, 16, 13], are not directly applicable .

Multi-type systems have been proposed as models in social sciences [5], statistical mechanics [4], neurosciences [1], etc. In particular the last paper [1], considers interacting diffusions of the form studied in the current work and establishes a LLN result and a propagation of chaos property. Our results in particular will provide asymptotic results on fluctuations from the LLN behavior obtained in [1]. Systems with a common factor also arise in many different areas. In Mathematical Finance, they have been used to model correlations between default probabilities of multiple firms [6]; in neuroscience modeling these arise as systematic noise in the external current input to a neuronal ensemble [9]; and for particle approximation schemes for SPDE, the common factor corresponds to the underlying driving noise in the SPDE [12, 13]. The goal of this work is to study a family of multi-type weakly interacting diffusions with a common factor. Our main objective is to establish a suitable CLT where the summands are quite general functionals of the trajectories of the particles with suitable integrability properties. Specifically, in the case where there is no common factor, letting denote the set of indices such that and

for functions on the path space of the particles, we will establish (see Theorem 3.1) the weak convergence of the family , in the sense of finite dimensional distributions, to a mean Gaussian field . Here is a family of functions on the path space that are suitably centered and have appropriate integrability properties (see Section 3.3 for definitions.) In the presence of a common factor the centering term is in general random (a function of the common factor) and denoting by these suitably randomly centered and normalized sums of (see (4.9)), we prove that under suitable conditions, , where is once again a collection of functions on the path space with suitable integrability, converges in the sense of finite dimensional distributions to a random field whose distribution is given in terms of a Gaussian mixture.

CLT established in this work also leads to new fluctuation results for the classical single type setting. Consider for example the case with no common factor and suppose one is interested in the joint asymptotic distribution of , where

and . Existing results on central limit theorems for (eg. [18, 19, 16]) do not give information on the joint limiting behavior of the above random variables. Indeed, a naive guess that the propagation of chaos property should imply the asymptotic independence of and is in general false. In Section 3.4 we will illustrate through a simple example how one can characterize the joint asymptotic distribution of the above pair.

We are also interested in asymptotic behavior of path functionals of particles of multiple type. Specifically, in the no common factor case, we will study in Section 3.6 the limiting distribution of multi-type statistics of the form

(1.1) |

where is a suitably centered function on the path space of dimensional stochastic processes with appropriate integrability. In the classical case (cf. [8]) where the particles have independent and identical dynamics, limit distributions of analogous statistics (with ) are given through certain multiple Wiener integrals (see Section 3.5). In the setting considered here are neither independent nor identical and we need to suitably extend the classical result for U-statistics to a multi-type setting and apply techniques as in the proof of Theorem 3.1 to establish weak convergence of (1.1) and characterize the limit distributions.

The central idea in our proofs is a change of measure technique based on Girsanov’s theorem that, in the case of single type populations with no common factor, goes back to the works [18, 19]. For this case, the technique reduces the problem to a setting with i.i.d. particles and the main challenge is to suitably analyze the asymptotic behavior of the Radon-Nikodym derivative. In the multi-type setting (with no common factor), although one can similarly reduce to a problem for independent particles (however not identically distributed), the Radon-Nikodym derivative is given in terms of quantities that involve particles of different types and the classical results on asymptotics for symmetric statistics [8] that are used in [18, 19] are not directly applicable and one needs to suitably lift the problem to a higher dimensional space. For this, we first treat the easier setting where for all , so that under the new measure one can view the whole population as a collection of i.i.d. -vector particles with -th coordinate belonging to the -th type. We then extend the result to the setting where the number of particles of different types may not be the same. For proofs in this general case, the property that, under the new measure, the particles are independent of each other together with classical results for multiple Wiener integrals, play an important role. Note that when , one can view under the original measure as a -vector of exchangeable particles. For this case one can apply results of [18, 19] to deduce a CLT. However it seems hard to use this result directly to treat the setting with different numbers of particles in different populations. One of the key ingredients in the approach taken in [18, 19] for the single type setting is to identify the limit of Radon-Nikodym derivatives in terms of a suitable integral operator. In the multi-type case (with no common factor) there is an analogous operator, however describing it and the Hilbert space on which it acts requires more work. Roughly speaking the Hilbert space corresponds to the space of -square integrable functions on the path space of dimensional continuous stochastic processes, where and for is the limit law of processes associated with particles of type .

The setting with a common factor (see Sections 4, 6 and 7) presents several additional challenges. The factor process is fully coupled with the -particle dynamics in the sense that the coefficients depend not only on its own state but also on the particle empirical measures . Due to the presence of the common factor, the limit of will in general be a random measure. As a result, the centering in the fluctuation theorem will typically be random as well and one expects the limit law for such fluctuations to be not Gaussian but rather a “Gaussian mixture”. Our second main result (Theorem 4.2) establishes such a CLT under appropriate conditions. Proof proceeds by first considering a closely related collection of stochastic processes which, conditionally on a common factor, are independent with distribution that only depends on its type. Unlike the setting with no common factor, there is no convenient change of measure under which this collection has the same law as that of the original collection of processes. We instead consider a change of measure, under which the distribution of the random centering remains invariant and one can view the distribution of a suitably perturbed form of the original vector of scaled and centered sums in terms of conditionally i.i.d. collections associated with the types of particles. Asymptotics of this latter collection can be analyzed in a manner analogous to the no-common-factor case, however in order to deduce the asymptotic properties of the original collection, one needs to carefully estimate the error introduced by the perturbation (cf. Section 7.2). Once again, to identify the limits of the Radon-Nikodym derivative, integral operators on suitable -path spaces are employed. A new aspect here is that we need to first consider limit laws conditional on the common factor (almost surely), which are now characterized in terms of certain random integral operators. Finally, the description of the limit of the (un-conditional) Radon-Nikodym derivatives and synthesis of the limiting random field requires a measurable construction of multiple Wiener integrals with an additional random parameter (see Section 7.5).

Central limit theorems for systems of weakly interacting particles with a common factor have previously been studied in [13, 3]. Both of these papers consider the case . The paper [13] establishes a fluctuation result for centered and scaled empirical measures in the space of suitably modified Schwartz distributions. Such a result does not immediately yield limit theorems for statistics that depend on particle states at multiple time instants (see [3] for a discussion of this point). In contrast the approach taken in [3], and also in the current paper, allows to establish limit theorems for quite general square integrable path functionals. Furthermore, the paper [3] sketches an argument for recovering the weak convergence of empirical measures in the Schwartz space from the CLT for path functionals. Although not pursued here, with additional work and in an analogous manner, for the current setting as well one can establish convergence of suitably centered and normalized empirical measures in an appropriate product Schwartz space.

The paper is organized as follows. In Section 2 we begin by introducing our model of multi-type weakly interacting diffusions where the common factor is absent. A basic condition (Condition 2.1) is stated, under which both SDE for the pre-limit -particle system and for the corresponding limiting nonlinear diffusion process have unique solutions, and a law of large numbers and a propagation of chaos property holds. These results are taken from [1]. For simplicity we consider here the case where the dependence of the drift coefficients on the empirical measures is linear. A more general nonlinear dependence is treated in Section 4. In Section 3 we present a CLT (Theorem 3.1) for the no-common-factor case. As noted previously, this CLT gives new asymptotic results for a single type population as well. This point is illustrated through an example in Section 3.4. We also give a limit theorem (Theorem 3.3) for multi-type statistics of the form as in (1.1) in Section 3.6. Proofs of Theorems 3.1 and 3.3 are provided in Section 5. But before, in Section 4, we state our main results for the setting where a common factor is present. Specifically, Section 4.1 states a basic condition (Condition 4.1), which will ensure pathwise existence and uniqueness of solutions to both SDE for the -particle system and a related family of SDE describing the limiting nonlinear Markov process. Main result for the common factor setting is Theorem 4.2 which appears in Section 4.2. For the sake of the exposition, some of the conditions for this theorem (Conditions 6.1 and 6.2) and related notation appear later in Section 6. Finally Section 7 contains proofs of Theorem 4.2 and related results.

The following notation will be used. Fix . All stochastic processes will be considered over the time horizon . We will use the notations and interchangeably for stochastic processes. Denote by the space of probability measures on a Polish space , equipped with the topology of weak convergence. A convenient metric for this topology is the bounded-Lipschitz metric defined as

where is the collection of all Lipschitz functions that are bounded by and such that the corresponding Lipschitz constant is also bounded by 1, and for a signed measure on and -integrable function . Let be the Borel -field on . Space of functions that are continuous from to will be denoted as and equipped with the uniform topology. For , let denote and let for , . Space of functions that are continuous and bounded from to will be denoted as . For a bounded function from to , let . Probability law of an -valued random variable will be denoted as . Expected value under some probability law will be denoted as . Convergence of a sequence of -valued random variables in distribution to will be written as . For a -finite measure on a Polish space , denote by as the Hilbert space of -square integrable functions from to . When and are not ambiguous, we will merely write . The norm in this Hilbert space will be denoted as . Given a sequence of random variables on probability spaces , , we say converges to in if as . We will usually denote by , the constants that appear in various estimates within a proof. The value of these constants may change from one proof to another. Cardinality of a finite set will be denoted as . For , let .

## 2 Model

Consider an infinite collection of particles where each particle belongs to one of different populations. Letting , define a function by if the -th particle belongs to -th population. For , let . For , let and denote by as the number of particles belonging to the -th population, namely . Assume that as . Let .

For fixed , consider the following system of equations for the -valued continuous stochastic processes , , given on a filtered probability space . For , ,

(2.1) |

where and are suitable functions, and . Here are mutually independent -dimensional -Brownian motions. We assume that are -measurable and mutually independent, with for .

Conditions on the various coefficients will be introduced shortly. Along with the -particle equation we will also consider a related infinite system of equations for -valued continuous stochastic processes , , given (without loss of generality) on . For and ,

(2.2) |

where . We assume that are -measurable and mutually independent, with for and .

The existence and uniqueness of pathwise solutions of and can be shown under following conditions on the coefficients (cf. [1]).

###### Condition 2.1.

For all , the functions are locally Lipschitz in the second variable, uniformly in : For every there exists such that for all , and :

There exists such that for all , and :

For all , are bounded Lipschitz functions: There exists such that for all and in , we have:

The paper [1] also proves the following propagation of chaos property: For any -tuple , , with , ,

(2.3) |

in , where for and , and is as in (2.2). Using the above result and a straightforward argument similar to [20] one can show the following law of large numbers. Proof is omitted.

###### Theorem 2.1.

Note that Theorem 2.1 implies in particular that for all and , as ,

(2.5) |

In this work we are concerned with the fluctuations of expressions as in the LHS of (2.4) and (2.5) about their law of large number limits given by the RHS of (2.4) and (2.5), respectively. Limit theorems that characterize theses fluctuations are given in Theorems 3.1 and 3.3. We will also establish central limit theorems in a setting where there is a common factor that appears in the dynamics of all the particles. This setting, which in addition to a common factor will allow for a nonlinear dependence of the drift coefficient on the empirical measure, is described in Section 4 and our main result for this case is given in Theorem 4.2.

## 3 Fluctuations for multi-type particle system

Throughout this section Condition 2.1 will be assumed and will not be noted explicitly in statement of results.

### 3.1 Canonical processes

We now introduce the following canonical spaces and stochastic processes. Let . For , denote by the law of where and is given by (2.2). Let . Define for the probability measure on as

For , let . Abusing notation,

(3.1) |

Also define the canonical processes on as

(3.2) |

### 3.2 Some integral operators

We will need the following functions for stating our first main theorem. Define for and , the function from to as

(3.3) |

Define for , the function from to ( a.s.) as

(3.4) |

We now define lifted functions from to ( a.s.) for as follows: For and in , let

(3.5) |

Now consider the Hilbert space . For , define integral operators and on as follows: For and ,

(3.6) |

It’s clear from (3.5)-(3.6) that . Denote by the identity operator on . The following lemma will be proved in Section 5.1.

###### Lemma 3.1.

. Trace for all . is invertible.

### 3.3 Central limit theorem

We can now present the first main result of this work. For , let be the collection of such that . Denote by the collection of all measurable maps such that . For , let . Given and , define lifted function as follows:

(3.7) |

###### Theorem 3.1.

converges as to a mean Gaussian field in the sense of convergence of finite dimensional distributions, where for and ,

with , defined as in (3.7), and , given similarly.

Proof of the theorem is given in Section 5.2.

### 3.4 An application to single-type particle system

Consider the special case of (2.1) where . Here, Theorem 3.1 can be used to describe joint asymptotic distributions of suitably scaled sums formed from disjoint sub-populations of the particle system. As an illustration, consider the single-type system given through the following collection of equations:

(3.8) |

Suppose that we are interested in the joint asymptotic distribution of for suitable path functionals and , where

The propagation of chaos property in (2.3) says that any finite collection of summands on the right sides of the above display are asymptotically mutually independent. However, in general, this does not guarantee the asymptotic independence of and . In fact, one can use Theorem 3.1 with to show that converges in distribution to a bivariate Gaussian random variable and provide expressions for the asymptotic covariance matrix. We illustrate this below through a toy example.

###### Example 3.1.

Suppose , , , and , where satisfies Condition 2.1 with replaced by . Further suppose that is an odd function, namely for all . Due to the special form of , one can explicitly characterize the measure , . Indeed, noting that for a one dimensional BM, ,

we see that for . Consider for ,

Note that since is odd, . For this example one can explicitly describe the asymptotic distribution of by regarding (3.8) as a -type population with and . Following (3.5) and (3.7), define for ,

and for ,

The operator is then defined by (3.6). The special form of allows us to determine , . Indeed, for , let

Then

This shows that for , . From Theorem 3.1 we then have that converges in distribution to a bivariate Gaussian random variable with mean and covariance matrix , where

and is a one dimensional BM.

### 3.5 Asymptotics of symmetric statistics

The proofs of Theorems 3.1 and 3.3 in Section 3.6 crucially rely on certain classical results from [8] on limit laws of degenerate symmetric statistics. In this section we briefly review these results.

Let be a Polish space and let be a sequence of i.i.d. -valued random variables having common probability law . For , let be the space of all real valued square integrable functions on . Denote by the subspace of centered functions, namely such that for all ,

Denote by the subspace of symmetric functions, namely such that for every permutation on ,

Also, denote by the subspace of centered symmetric functions in , namely . Given define the symmetric statistic as

In order to describe the asymptotic distributions of such statistics consider a Gaussian field such that

For , define as

and set .

The MWI of , denoted as , is defined through the following formula. For ,

(3.9) |

The following representation gives an equivalent way to characterize the MWI of :