Particle systems with quasi-homogeneous initial states and their occupation time fluctuations

Occupation time fluctuation limits of particle systems in R^d with independent motions (symmetric stable Levy process, with or without critical branching) have been studied assuming initial distributions given by Poisson random measures (homogeneous and some inhomogeneous cases). In this paper, with d=1 for simplicity, we extend previous results to a wide class of initial measures obeying a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of Z), by means of a new unified approach. In previous papers, in the homogeneous Poisson case, for the branching system in"low"dimensions, the limit was characterized by a long-range dependent Gaussian process called sub-fractional Brownian motion (sub-fBm), and this effect was attributed to the branching because it had appeared only in that case. An unexpected finding in this paper is that sub-fBm is more prevalent than previously thought. Namely, it is a natural ingredient of the limit process in the non-branching case (for"low"dimension), as well. On the other hand, fractional Brownian motion is not only related to systems in equilibrium (e.g., non-branching system with initial homogeneous Poisson measure), but it also appears here for a wider class of initial measures of quasi-homogeneous type.


Introduction
In a series of papers [3,4,5,6,7,8,9,10] we studied particle systems in R d starting from a configuration determined by a random point measure ν, and independently moving according to a standard α-stable Lévy process (0 < α ≤ 2). In some models the particles additionally undergo critical branching. The evolution of the system is described by the empirical process N = (N t ) t≥0 , where N t (A) is the number of particles in the set A ⊂ R d at time t. The main object of interest is the limit of the time-rescaled and normalized occupation time fluctuation process X T defined by as T → ∞ (i.e., as time is accelerated), where F T is a suitable deterministic norming. The process X T is signed measure-valued, but we regard it as a process with values in the space of tempered distributions S ′ (R d ) for technical convenience, and also because in some cases the limit is genuinely S ′ (R d )-valued. In all the cases considered in the abovementioned papers the initial measure ν was a Poisson field, homogeneous or not. This assumption permitted to investigate convergences conveniently with the help of the Laplace transform (due to infinite divisibility). The results always exhibited the same type of phase transition: for "low" dimensions d the limit process was the Lebesgue measure multiplied by a real long-range dependent process, whereas for "high" dimensions the limit was an S ′ (R d )-valued process with independent increments. A natural question is what happens for non-Poisson initial measures ν. Mi loś [27,28,29] considered (critical) branching systems where ν was an equilibrium measure (see [19]). In that model the limits have a similar dimension phase transition; moreover, for high dimensions they are the same as in the homogeneous Poisson case, while for low dimensions they are different. The conclusion was that in low dimensions the occupation time fluctuation process "remembers" the initial state of the system. Since the equilibrium states of the branching system are somewhat similar to homogeneous Poisson measures (they are infinitely divisible random point measures with uniform intensity; distributions of this kind are called "equilibrium distributions of Poisson type" in [26]), the Laplace transform method was also useful in [27,28,29].
The aim of the present paper is to investigate what happens with initial measures of other types, for example, some measures that are deterministic or almost deterministic. For simplicity we consider d = 1 and assume that the motions are either without branching or with the simplest critical binary branching. In [4,5] we proved for such motions, with general d, that if ν is a homogeneous Poisson measure, then the following results hold (where λ denotes Lebesgue measure and K is a different constant in each case): in the non-branching system: if d < α, then X T converges in distribution (in C([0, τ ], S ′ (R d )) for any τ > 0) to a process Kλξ, where ξ is a fractional Brownian motion; if d = α, then the limit process is Kλβ, where β is a standard Brownian motion; if d > α, then the limit is a time-homogeneous Wiener process in S ′ (R d ); in the branching system: if α < d < 2α, then the limit is Kλζ, where ζ is a sub-fractional Brownian motion (the case d ≤ α requires a slightly different treatment based on high-density models, see [9]); if d = 2α, then the limit is Kλβ; if d > 2α, then the limit is a time-homogeneous Wiener process in S ′ (R d ), different from the one in the non-branching case.
In this paper we define a class M of initial measures ν which contains in particular homogeneous Poisson measures (which are "completely random" [24]), and quasi-homogeneous deterministic measures (e.g., the measure defined by one atom at each j ∈ Z, which is "completely deterministic"), and we develop a unified approach that permits to obtain limits of X T for all ν ∈ M. By a quasi-homogeneous deterministic point measure on R we mean any measure defined by the following procedure: Given a positive integer k, in each interval [j, j + 1), j ∈ Z, we fix k points. For a general ν ∈ M, each interval [j, j + 1) contains θ j points chosen at random, and θ j , j ∈ Z, are i.i.d. random variables (see Section 2 for a rigorous definition). The main feature of those measures is this form of quasi-homogeneity and independence on the family of intervals [j, j + 1).
For each ν ∈ M we obtain the limit of the corresponding X T and in this way we recover the results of [4,5] for the homogeneous Poisson case (for d = 1, but there is no doubt that the results for higher dimensions can be obtained analogously), and we also derive limits for many other initial measures. It seems interesting that the idea of the proofs in this general framework is simpler than that in our previous papers, and is based on the central limit theorem. This is a significant change of methodology. However, some technical points in those papers are employed again here. An argument using the non-linear equation associated with the occupation time of the branching system, which can be obtained by means of the Feynman-Kac theorem, again plays an important role, but now in a different way: it is a key step in moment estimates in order to apply the Lyapunov theorem in the branching case. The equilibrium measures for the branching system do not belong to M because the branching introduces spatial dependence.
Some of the results we obtain are unexpected. It turns out that the only case where new limits appear is the non-branching case with (d = )1 < α. They have the form Kλξ, where ξ is the sum of two independent processes, one of them is a sub-fractional Brownian motion (see (2.3)), and the second one is a new (centered continuous with long range dependence) Gaussian process (see (2.4)). The process ξ depends on the initial measure ν only through Eθ 0 and Var θ 0 . In particular, for a deterministic initial measure this process reduces to a sub-fractional Brownian motion, and in the homogeneous Poisson case (as well as for any ν with Eθ 0 = Var θ 0 ) it yields a fractional Brownian motion (see Theorem 2.2). This result seems surprising since in all earlier papers sub-fractional Brownian motion was related only to branching systems, and was consequently attributed to the branching, but now, in the present context, this process turns out to be more "natural" than fractional Brownian motion. On the other hand, fractional Brownian motion, which is typically related to systems in equilibrium (in particular the non-branching system with initial homogeneous Poisson measure), now appears also for a wider class of quasi-homogeneous initial measures, as noted above.
In all the remaining cases the limits are (up to constants) the same, and with the same normings F T , as those recalled above for homogeneous Poisson models.
The results show that within the class M the fluctuations caused by the branching are so large that X T "forgets" the randomness of the initial state of the system (it "remembers" Eθ 0 only). On the other hand, for low dimensions it does distinguish between ν ∈ M and the equilibrium initial state (which is not in M). Another conclusion is that for high dimensions (which for d = 1 amounts to small α), the fluctuation process "forgets" the initial measure, as long as it is in some sense homogeneous (i.e., ν ∈ M), and this property holds for branching and non-branching systems; it is also preserved for branching systems in equilibrium.
In this paper we are interested mainly in identifying the limit processes, therefore we have not attempted to prove convergences in their strongest, functional form; in most cases we prove only convergence of finite-dimensional distributions. Presumably, convergence in distribution also holds in C([0, τ ], S ′ (R)) for any τ > 0. As an example, we give one result of this type (Proposition 2.6).
We have not found results in the literature concerning occupation times for particle systems starting from a deterministic or quasi-deterministic point measure. Some kinds of quasi-homogeneity of initial configurations for systems of independent particles, different from those in this paper, appear in other contexts in [33] and [20] (see Remark 2.5 (f)). It may be that systems of independent particles with α-stable motion and the initial conditions of [33] lead to the same results as with initial homogeneous Poisson distribution.
The following notation is used in the paper.
S(R): space of C ∞ rapidly decreasing function on R.
Generic constants are written C, C i , with possible dependencies in parenthesis.
In Section 2 we describe the particle system, formulate the results and discuss them. Section 3 contains the proofs.

Results
We start with detailed description of the particle system.
Let θ be a non-negative integer-valued random variable with distribution P (θ = k) = p k , k = 0, 1, 2, . . . , (2.1) such that Eθ 3 < ∞. This moment condition is a technical assumption satisfied by all cases of interest in this paper, but we suppose that finiteness of the second moment could be sufficient.
Let θ j , j ∈ Z, be independent copies of θ, and for each j ∈ Z and k = 1, 2, . . ., let ρ j k = (ρ j k,1 , . . . , ρ j k,k ) be a random vector with values in [j, j + 1) k . We assume that (θ j , (ρ j k ) k=1,2 , . . .), j ∈ Z, are independent. These objects determine a random point measure ν on R in the following way: For each j, θ j is the number of points in the interval [j, j + 1), and for each k, if θ j = k, the positions of those points are determined by ρ j k . In other words, where κ j,n = ρ j θ j ,n and δ a is the Dirac measure at a ∈ R. We denote by M the class of all such measures ν.
Remark 2.1 (a) If θ ≡ k and, for each j, the ρ j k are not random, then ν is a quasi-homogeneous deterministic measure mentioned in Introduction. The simplest example is ν = j∈Z δ j .
(b) If θ is a standard Poisson random variable and, for each j, ρ j k,1 , . . . , ρ j k,k are independent, uniformly distributed on [j, j + 1), then ν given by (2.2) is the homogeneous Poisson point measure (with intensity measure λ).
Fix α ∈ (0, 2] and ν ∈ M. Assume that at the initial time t = 0 there is a collection of particles in R with positions determined by ν. As time evolves, these particles move independently according to the standard α-stable Lévy process. We consider systems either without branching, or with critical binary branching (i.e., 0 or 2 particles with probability 1/2 each case) at rate V . For the corresponding empirical process, we define an S ′ (R)-valued process X T by (1.1).
Before stating the first theorem we recall the definition of sub-fractional Brownian motion. A sub-fractional Brownian motion with parameter H (0 < H < 1) is a centered continuous Gaussian process ζ H with covariance See [3,35] for properties of this process. It appears in [16] in a different context, and it has also been investigated in [1,31,36,37,38]. We will need another centered Gaussian process ϑ H with covariance Existence of this process for H = 1/2 follows from the formula which implies positive-definiteness of Q H .
Theorem 2.2 For the system without branching, (a) if 1 < α and where ζ H , ϑ H are independent, H = 1 − 1/2α, and then where β is a standard Brownian motion in R, and (c) if 1 > α and where X is an S ′ (R)-valued homogeneous Wiener process with covariance be the process in Theorem 2.2(a). If Eθ = Varθ, in particular if ν is homogeneous Poisson (see Remark 2.1(b)), then ξ H is, up to a constant, a fractional Brownian motion with Hurst parameter H, i.e., it has covariance C(s 2H + t 2H − |s − t| 2H ). Thus we recover Theorem 2.1 of [4]. On the other hand, if θ is deterministic, then ξ H is a sub-fractional Brownian motion. Moreover, in general randomness of the ρ's in the definition of ν ∈ M (see (2.2)) does not play any role in the limit. (b) The long time dependent behavior of Gaussian processes is usually characterized by the covariance of increments of the process on intervals separated by distance τ , as τ → ∞. For the process ϑ H that behavior is asymptotic decay like τ 2H−2 (the same as for fractional Brownian motion), for sub-fBm it is τ 2H−3 (see [3]). So, the long time dependent behavior of the process ξ H in Theorem 2.2(a) is determined by ϑ H in all cases where θ is random.
where ζ H is a sub-fractional Brownian motion with parameter H = (3 − 1/α)/2, and where β is a standard Brownian motion, and where X is an S ′ (R)-valued homogeneous Wiener process with covariance  [4] and [5]). (b) The condition α < 1 in part (a) of the last theorem corresponds to α < d in [4] and [5]. In the homogeneous Poisson case for d ≤ α, we obtained limits of the same form as for α < d < 2α, by introducing high density, i.e., considering systems with initial intensity H T λ, H T → ∞ sufficiently fast [9]; the high density counteracts the tendency to local extinction caused by the critical branching. The same procedure can be applied in the present case, yielding the limits for 1 ≤ α ≤ 2, if the intervals [j, j + 1) are replaced by [j/H T , (j + 1)/H T ).
(c) As in [4] and [5], Theorems 2.2 and 2.4 can be extended to systems in R d , where the intervals (d) Comparing parts (a) of Theorems 2.2 and 2.4, we see that the branching weakens the influence of the initial configuration.
(e) The previous results show that sub-fractional Brownian motion is a "natural" process for our model. So far it had appeared only in the context of branching systems, but now we see that it is intrinsically related to the non-branching systems as well for a large class of initial conditions. Fractional Brownian motion occured before only in the case of systems in equilibrium, but now it also appears wherever Eθ = Varθ.
(f) Theorems 2.2 and 2.4 can also be extended to other models. For example, in [20] a model is studied in a different context with independent α-stable motions without branching and initial positions of particles (j + ρ) j∈Z , where ρ is a random variable uniformly distributed on [0, 1], independent of the motions. It is easy to see, by a standard conditioning argument (considering the characteristic function and conditioning on ρ), that for models with or without branching and with this initial configuration, the limits are the same as for deterministic ρ, i.e., they are given by Theorems 2.2 and 2.4.
We have formulated Theorems 2.2 and 2.4 with convergence of finite-dimensional distributions only, as we are mostly interested in the limit processes, but we have no doubt that functional convergence also holds. As an example, let us consider the cases of large α. For simplicity we assume that the initial configuration is such that θ ≡ 1 and ρ j 1,1 is uniformly distributed on [j, j + 1), j ∈ Z. Proposition 2.6 For the model described above, the processes X T in Theorems 2.2(a) and 2.4(a) converge in law in C([0, τ ], S ′ (R)) for any τ > 0.

Auxiliary facts related to the stable density
We will often use the self-similarity property of the transition density p t of the standard α-stable process in R: Since p t (·) is decreasing on R + and symmetric, then by (3.1) we have For ϕ ∈ S(R) we have |ϕ(x)| ≤ C(ϕ, m)φ m (x). This, and an obvious inequality, In the sequel we will use various versions of this estimate, e.g., We will also need the following estimate ( [22], Lemma 5.3) for the potential operator G (see (1.3)). If α < 1, q > 1, and ϕ is a measurable function on R such that (3.7)

Scheme of proofs
The proofs of Theorems 2.2 and 2.4 are based on the central limit theorem and follow the scheme described presently.
Let N x denote the empirical process of the system (with or without branching) started from a single particle at x, and N (j) , j ∈ Z, be the empirical process for the particles which at time t = 0 belong to [j, j + 1), i.e., according to the description at the beginning of Section 2 (see (2.2)). Note that N (j) , j ∈ Z, are independent.
The process X T defined in (1.1) can be written as The first step in our argument is to prove that for any ϕ, ψ ∈ S(R), and s, t ≥ 0, where X is the corresponding limit process. Without loss of generality we may assume that ϕ, ψ ≥ 0. Using (3.9) we have r ′ , ψ dr ′ dr. (3.11) Using (3.8), (2.1) and the fact that E N x t , ϕ = T t ϕ(x) in both non-branching and (critical) branching cases, and defining, for x ∈ R, n ≤ k, random variables k,n − x, (3.12) where [x] is the largest integer ≤ x, we rewrite (3.11) as where 14) (in the first equality for II we used independence of systems starting from different points), In each case we will show convergence of I, II and III, thus proving (3.10). (It will be shown that I, II, III are bounded, so the passage to the limit in each sum in (3.13) is justified). Next, we show that X(t), ϕ ⇒ X(t), ϕ , ϕ ∈ S(R), t ≥ 0.
To this end, by (3.9) and (3.10) it suffices to prove that the Lyapunov condition is satisfied, and this property will follow if we show that It is clear that convergence in law of linear combinations m k=1 a k X T (t k ), ϕ k can be obtained analogously from (3.10) and (3.18), thus establishing the claimed convergence X T ⇒ f X.

Proof of Theorem 2.2(a)
Following the scheme we show that lim T →∞  Let η denote the standard α-stable Lévy process in R. As we consider the model without branching, we have, for r > r ′ , Putting this into (3.14) and omitting the subscripts k, n, we obtain where In I 1 we substitute r = r/T, r ′ = r ′ /T , use (2.5) and (3.1), and then we substitute x = T −1/α (x−y), arriving at By (3.17) and (3.1), the expression under the integrals converges pointwise, as T → ∞, to can be treated analogously, hence by (3.23) we obtain (3.20).
Next we take II. In (3.15) we substitute r = r/T, r ′ = r ′ /T , and by (2.5) we have II(T ; k, n, m) We use (3.1) and substitute x = T −1/α (x − z), obtaining The integrand converges pointwise to p r (x)p r ′ (x)ϕ(y)ψ(z), and (3.17) implies that for T > 1 , it is bounded by r −1/α p 1 (0)g r ′ (x)ϕ(y)ψ(z) (see (3.3)), which is integrable. As R p r (x)p r ′ (x)dx = (r + r ′ ) −1/α p 1 (0), we obtain the limit (3.21) for II. Note that in this argument the only property of h k,n we have used is (3.17), therefore it is immediately seen that the limit of III can be obtained in the same way (see (3.16)). This completes the proof of (3.10).
As explained in the scheme, to finish the proof it suffices to show (3.19).
The expression under lim T →∞ sup n,k in (3.19), similarly as in (3.22), can be written as (3.26) where 27) and φ 2 is given by (3.4). The last inequality in (3.26) is of the same type as (3.6), and can be obtained by an analogous argument using (3.5) and (3.17). Hence, for (3.19) it is enough to show that lim T →∞ J(T ) = 0. (3.28) Note that in this argument we have not used the assumption on α.
After obvious substitutions, using (2.5) and the invariance of Lebesgue measure for T t , we have By the self-similarity property (3.1),
It is now easy to see that Passing to I 3 defined by (3.32), we first estimate it similarly as I 1 (see (3.36)), then we change the order of integration drdr ′ , and substituter ′ = r ′ /T andr = r − Tr ′ , obtaining

Proof of Theorem 2.4
We recall first the following formula for the second moments of critical binary branching systems with branching rate V: (3.47) which is obtained using, e.g. Lemma 3.1 in [25], and the Markov property.
We have proved (3.10) in all the cases. According to the scheme, to finish the proof it remains to show (3.19). To this and we define the function It is known that the Feynman-Kac formula implies that v θ satisfies the non-linear equation (see e.g. [18], or the space-time approach used in [4,5]). Hence, by a similar argument as in (3.45)-(3.47) of [5], we obtain Without loss of generality we may assume t = 1.
We turn to J 4 , which requires more work. By (3.62), the Schwarz inequality and (3.67) we have

Proof of Proposition 2.6
Since the convergence of finite-dimensional distributions has been proved already, by virtue of the Mitoma theorem [30] it remains to show tightness of X T , ϕ , T > 1, in C([0, τ ], R) for any fixed ϕ ∈ S(R), ϕ ≥ 0. To this end we prove E( X T (t), ϕ − X T (s), ϕ ) 2 ≤ C(t − s) a , s < t ≤ τ, (3.69) for some a > 1. By (3.8) and (3.9) we have The latter expression is equal to E X Remark Recall that Proposition 2.6 refers to a special simple choice of ν. For a general ν ∈ M, the proof of (3.69) is similar but slightly more involved. One has to estimate an extra term and use an inequality of the type (3.6).