Limits of One-dimensional Interacting Particle Systems with Two-scale Interaction

Tong ZHAO

Abstract This paper characterizes the limits of a large system of interacting particles distributed on the real line.The interaction occurring among neighbors involves two kinds of independent actions with different rates.This system is a generalization of the voter process, of which each particle is of type A or a.Under suitable scaling, the local proportion functions of A particles converge to continuous functions which solve a class of stochastic partial differential equations driven by Fisher-Wright white noise.To obtain the convergence, the tightness of these functions is derived from the moment estimate method.

Keywords Interacting particle systems, Stochastic partial differential equations,Two-scale interaction, Tightness

1 Introduction

This paper studies the dynamics of a large system of interacting particles distributed on the real line.In our model, the particles are placed at grids ρ?1Z, where the parameter ρ denotes the numbers of grids in a unit interval.Each grid is occupied by one and only one particle, and each particle is of type A or a.Each particle interacts with its neighbors in a certain way that will be specified later; here we say two particles x and y are neighbors (denoted by x ～y)if their distance is less than a given bound D.Our model can be regarded as a generalization of the voter process studied in [11].

The central problem in this paper is how to characterize the limit behavior of the system when the grids are more and more dense, say, ρ →∞.To be more specified, let ρnbe a sequence tending to infinity,and Dnbe the corresponding bounds for neighborhood.Define the local proportion of type A around a grid x ∈Z at time t as,where Nn≈2Dnρnis the number of neighbors of the particle at x, and(x)indicates the type of the particle at (t,x)(1 for A and 0 for a).We extend un(t,x)to the entire space R+×R by linear interpolation.Our goal is to investigate the convergence of unas n tends to infinity, and to characterize the limit if it exists.

The dynamics of undepends on the interacting manner of the particle system.In our model,each particle interacts with its neighbors independently according to a Poisson process.We further assume that the interaction(i.e.,two-scale interaction)involves two types of independent actions: The regular one-to-one interaction with higher-rate Hnand the rare interaction with lower-rate Ln.In the first type, the particle chooses one neighbor randomly and duplicates its type.In contrast, the other type of action is much more flexible: We assume that the particle at grid x and time t updates its type to i ∈{A,a} with probability pi(un(t,x))and with rate Fi(un(t,x)), where pi: [0,1] →[0,1] and Fi: [0,1] →[0,+∞)are given bounded and measurable functions; in this case, Ln(t,x)=FA(un(t,x))+Fa(un(t,x)).In other words,the rare interaction can be state-dependent (see [6, Theorem I.3.9]), which endows the model with the ability to capture various features in specific applications.For example, the voter process studied in [11] is associated with the setting that pA(u)= u, Fa≡0, and FAis a positive constant.When applying to population genetics, it can model various effects in gene frequency diffusion, such as mutation (e.g., piand Fiare all constants), selection (e.g.,pA(u)= u, pa(u)= 1 ?u, and FA> Faif A is advantaged), Allee’s effect (e.g., pA(u)= u,pa(u)= (1 ?u), and FA(u)= u,Fa(u)= (1 ?u)), and so on (see [4, 14, 17]); all these effects can be overlaid.

The main result of this paper is to obtain the convergence of unin a proper way.Define the λ-normand the following topological vector space of continuous functions

with norm ‖·‖λ, where λ<0.

Theorem 1.1Let ρn= n,Dn=,Hn= 2n, and un(0,·)converges to f0in C as n →∞.Then unconverges in distribution to a C-valued continuous process u which satisfies the following equation with initial condition u(0,x)=f0:

Remark 1.1The higher-rate interaction contributes to the second-order termby random walk with generator (see [5])

Remark 1.2This conclusion is able to be carried over to a case on ring if measures,functions and noise are periodic.However, the corresponding result for high-dimension case does not come true.Because super Brownian motion in higher dimensions exists as singular measure-valued process rather than a density value process in one dimension case which can be expressed as in Theorem 1.1 (see [12, Theorem III.4.2]).

Remark 1.3The coefficients are a little different from those in[11], which results from the choice of parameters.

(1.1)can be given rigorous meaning in terms of an integral equation as explained in [16,Chapter 3].Our theorem gives existence of solutions to the associate martingale problem (see[13]), while uniqueness in law also holds in a general condition, due to the following result from[13].

Lemma 1.1If piand Fiwith i ∈{A,a} are Lipschitz continuous, the solution to the martingale problem associated with (1.1)is unique.

The proof strategy of Theorem 1.1 is adopted from[11], and the key idea is to show that the model satisfies a martingale problem that approximates the martingale problem for the limiting processes.Tightness is proved through estimating moments of small increments for the local proportion and arguing as in the Kolmogorov tightness criterion; to this end, we establish an approximate Green’s function representation (2.12)for the local proportion un(t,x), which is analogous to the one for the solution to (1.1)but with certain error terms.The introduction of probabilities pA(u)and pa(u)not only generalizes the model, but also helps us simplify the proof of Green’s function representation, comparing to that for the voter process in [11].

It is worth noting that various properties of solutions to SPDEs like(1.1)have been studied in the literature;for instance,the compact support property for solutions was discovered by[9];as a stochastic version of reaction-diffusion equations, random traveling waves were introduced and investigated in[10,15], and further analysis of the traveling speed was carried out in[7–8],etc.

The rest of the paper is all devoted to the proof of Theorem 1.1.After figuring out the dynamics of(x), we decompose each term in the expansion of a functional of(x)into the sum of a fluctuation term and an average term in Subsection 2.1.Green’s function representation is derived in Subsection 2.2, and tightness of unis proved in Subsection 2.3.The limit is characterized in Subsection 2.4, which concludes the proof.

2 Proof of Theorem 1.1

Let us introduce three independent Poisson processes associated with x,y ∈Z, i.e.,Pt(x,y)with rate Hn/Nn,(x)with rate FA(ut?(x)), and(x)with rate Fa(ut?(x)),characterizing the events of x interviewing y, x updating its type to A, and x updating its type to a, respectively.

In what follows, we simply write

Moreover,we denote eλ(x):=eλ|x|.

According to the setting of our model, the process(x)satisfies

Take a test function φ : [0,∞)×Z →R with t →φt(x)being continuously differentiable and satisfying

Define the measure valued process

and then using the integration by parts, for t ≤T, we have

By symmetry, one can see that

so the previous formula can be written as

2.1 The Doob-Meyer decomposition

We need to decompose each of the four terms(2.1),denoted by J1,J2,J3and J4,respectively,into the sum of a fluctuation term and an average term.In the following argument, we will omit superscript n without ambiguity and denote un(t,x)as ut(x).Define

For J1, one computes that

Alternatively, we bound it by

Hereafter, C denotes a generic positive constant that may change from line to line.

For J2, one has that

For J3, one obtains that

For J4, one obtains that

where Zt(φ)is a martingale with brackets process given by

Combining (2.2), (2.5), (2.7)and (2.9), one gains that

2.2 Green’s function representation

Remark 2.1A continuous-time random walk with generator ?nis of variance

Set φs=for s ≤t and substitute it into(2.11),then the second term on the right-hand side vanishes, andSo

Lemma 2.1For T ≥0, p ≥2,λ>0, we have

ProofFrom [11, Lemma 3(a)], the greatest jumps of the martingales(φ)are bounded bya.s., where C is a constant.

For i=1, using Burkholder’s inequality, (2.3)and (2.4), we have

By [11, Lemma 3(a,c)], one has

and

Finally,

Similarly, for i=2,3, by (2.6), (2.8)and [11, Lemma 3(c)], we get

According to (2.13)and (2.14), one has

where 1 ≤i ≤3.The proof is complete.

2.3 Tightness

Our objective is to prove the tightness of un.Define

Lemma 2.2For 0 ≤s ≤t ≤T, y,z ∈Z, |t ?s|,|y ?z|≤1, λ>0, p ≥2, we have

ProofSet

From (2.12), we have

By Lemma 2.1 and using H?lder’s inequality, we estimate that

For the term T1, using [11, Lemma 3(c, e)] one has that

Similarly, for the term T2, one obtains

For the term T3, use [11, Lemma 3(c, e)] and (2.10), we find

Combining the estimates for T1,T2and T3and noticing the definition of δ, we have

Also using Lemma 2.1, (2.10)and H?lder’s inequality, we can similarly estimate that

For terms T4and T6, we use [11, Lemma 3(a)] and get

For terms T5and T7, we use [11, Lemma 3(c, f)] and obtain

For term T8, using [11, Lemma 3(c)], we have

For term T9, we use [11, Lemma 3(c, f)] and obtain

Combining the estimates for Ti(4 ≤i ≤9)and noticing the definition ofwe have

Put (2.15)–(2.16)together and get

The proof is complete.

To get the tightness of un(·,·),define(t,x):=(t,x)on the grid z ∈Z, t ∈N/(nρn),then linearly interpolate it first in x and then in t to obtain a continuous C valued process.Using Lemma 2.2, it is easy to find that

because that means to find an m such that.If ρn=n, m should be 24.

The following result is taken from [11, Lemma 7].

Lemma 2.3For any λ>0,T <∞,

On any given compact subset K ?R+×R,(t,x)is uniformly bounded and equicontinuous by using Kolmogorov’s continuity criterion(see[16,Corollary 1.2(ii)])almost surely.Therefore,we get the tightness of(t,x)as continuous C-valued process.Then the tightness of un(t,x)follows from the above lemma.Also, the continuity of all limit points follows.

2.4 Characterizing limit points

Taking a continuous function φ:R →R with compact support, we define

Because of simultaneous convergence of subsequence of the pairs(un(t),),by Skorokhod’s theorem(see[3,Theorem 2.1.8]),we can find random variables with the same distribution as,which converges almost surely.We can still label it as(un(t),)since our interest is to identify the distribution of the limit.Since the limits are continuous,the almost sure convergence holds not only in Skorokhod sense but also in uniform sense on compact sets.Thus, with probability one, for any T <∞, λ>0 and function φ with compact support, we have

where vt(dx)=ut(x)dx for all t ≥0, from (2.17).

When n goes to infinity, we know that(φ)(1 ≤i ≤3)tend to zero almost surely for all t by(2.3), (2.6)and (2.8), and also know ?n(φ)(x)tends touniformly by Taylor’s expansion.Therefore, Zt(φ)tends to a continuous local martingale zt(φ), where

From (2.10), we know the following process

is a martingale, where

Fortunately, we have a bound

So

is a continuous local martingale.Since(R)is dense in(R), (2.18)–(2.19)hold for any φ ∈C2c(R).Hence, the solution u(t,x)to the martingale problem associated with the following SPDE

comes.

Remark 2.2Instead of taking φ ∈(R)directly, we use φ ∈(R).The reason is when we conclude ?n(φ)(x)tends to, Taylor’s expansion is required.

AcknowledgementsThe author thanks his Ph.D.supervisor Shanjian Tang, researcher Kai Du and vice professor Jing Zhang for their advice in the past 5 years.Meanwhile, he also cherishes the anonymous reviewers’ valuable and helpful comments which improve the quality of this article .