Clustering behavior of a continuous-sites stepping-stone model with Brownian migration

: Clustering behavior is studied for a continuous-sites stepping-stone model with Brownian migration. It is shown that, if the model starts with the same mixture of diﬁerent types of individuals over each site, then it will evolve in a way such that the site space is divided into disjoint intervals where only one type of individuals appear in each interval. Those intervals (clusters) are growing as time t ! 1 . The average size of the clusters at a ﬂxed time t > 0 is of the order of p t . Clusters at diﬁerent times or sites are asymptotically independent as the diﬁerence of either the times or the sites goes to inﬂnity.


Introduction
Stepping-stone models were first proposed by Kimura [9] as stochastic models in population genetics. Discrete-sites stepping-stone models describe the simultaneous evolutions of populations at different colonies, where it undergoes mutation, selection and resampling within each colony and migration among those colonies. They have been studied since by different authors (see Handa [8] and Sawyer [13]). Similar models (interacting Fleming-Viot models) were considered by Dawson, Greven and Vaillancourt [3]. Very loosely put, these models can be thought as collections of Fisher-Wright models or Fleming-Viot models with geographical structures. There is one model at each colony. Different populations interact with each other via migrations among colonies. Results on long-term behaviors of such models were obtained in [3,8].
Continuous-sites stepping-stone models with two types of individuals were first introduced in Shiga [14]. Cluster formation of such models was considered by Evans and Fleischmann [6] for a particular class of sites, namely, the continuous hierarchical group. Another continuous-sites stepping-stone model with infinitely many types was defined and discussed by Evans [5]. Further properties of this model can be found in Donnelly et al. [4]. Duality plays an important role in these studies.
Clustering is a phenomenon observed among such models, namely, individuals over sites close to each other tend to have the same type. In models with hierarchically structured site space the cluster formation was discussed by Fleischmann and Greven [7], Evans and Fleischmann [6] and Klenke [10] through studying the time-site scaling of the original models. In this paper we will focus on the infinitely many types stepping-stone model over the real line. Using the scaling property for stable processes, Evans [5] (also see [4]) showed that if the migration process is a stable process with index 1 < α ≤ 2, then there is only one type of individual appearing over each site as soon as time t > 0. In this paper we point out that the above mentioned phenomenon can actually occur across an interval. i.e. the system clusters. We call such an interval a cluster with a certain type.
When the migration is Brownian motion and the initial state of the model consists of the same mixture of different types of individuals over each site, the evolution of the clusters can be intuitively described as follows: If we start with the same mixture of different types of individuals over each site, then clustering happens across the site space simultaneously as soon as t > 0. The site space is divided into intervals where there is only one type of individuals over each interval. As time goes on, the clusters are getting bigger and bigger in size. The average size of those clusters is of the order of √ t at any fixed time t. The types of two clusters are asymptotically independent if they are separated by either a long distance or a long time. Those results are obtained by the moment duality and analysis of the dual process, the coalescing Brownian motion. If the initial mixing measure is diffuse, sharp results can be obtained. We remark that in this model the clustering phenomenon occurs in a clean-cut fashion in contrast to those in [7,6,10] where the clustering is described indirectly via scaled processes.
The clustering behavior described in Theorem 3.7 resembles the one in multi-type nearest neighbor voter models over the one-dimensional lattice Z (see Liggett [11] for an account on the two-type case). This suggests that the continuous-sites stepping-stone model should arise as an appropriate time-space scaling limit of voter models. We refer the reader to Mueller and Tribe [12] and Cox, Durrett and Perkins [1] for work along this line.
The rest of the paper is organized as follows. We first briefly introduce the setup and the moment duality of a continuous-sites stepping-stone model in Section 2. Then in Section 3 we apply the moment duality along with a result on coalescing Brownian motion flow to study the dynamics of cluster formation in this model. In Section 4 we prove a duality formula involving joint moments over different times, which will be used later to investigate the relationship between the types of clusters at different locations and different times.

Definitions and preliminary results
We first sketch the setup of an infinitely-many-types continuous-sites stepping-stone model X with Brownian migration.
Let real line R be the site-space. m denotes the Lebesgue measure on R. Let K := [0, 1]. We identify K with the coin-tossing space {0, 1} N . K equipped with the product topology serves as the type-space of X. Evans later points out that the above-mentioned topology could also be replaced by the usual topology on [0, 1]. Write M (K) for the Banach space of finite signed measures on K equipped with the total variation norm · M (K) . Let L ∞ (m, M (K)) denote the Banach space of (equivalence classes of) maps µ : R → M (K) such that ess sup{ µ(e) M (K) : e ∈ R} < ∞. Write C(K) for the Banach space of continuous functions on K equipped with the usual supremum norm · C(K) . To simplify notations we always write m(de) for de. Let L 1 (m, C(K)) denote the Banach space of (equivalence classes of) maps µ : R → C(K) such that µ(e) C(K) de < ∞. Then L ∞ (m, M (K)) is isometric to a closed subspace of the dual of L 1 (m, C(K)) under the pairing (µ, x) → µ(e), x(e) de, µ ∈ L ∞ (m, M (K)), x ∈ L 1 (m, C(K)). Write M 1 (K) for the closed subset of M (K) consisting of probability measures, and let Ξ denote the closed subset of L ∞ (m, M (K)) consisting of (equivalence classes of) maps with values in M 1 (K). Ξ equipped with the relative weak * topology is a compact, metrizable space. It serves as the state space of X.
The intuitive interpretation is that µ ∈ Ξ describes the relative frequencies of different populations at the various sites: µ(e)(L) is the "proportion of the population at site e ∈ R that has a type belonging to the set L ⊂ K".
More elaborate discussions on the set up of such processes can be found in [5].
The nth moment of µ ∈ Ξ corresponding to φ ∈ L 1 (m ⊗n , C(K n )) is defined as follows.
Now we are going to define coalescing Brownian motion which is dual to the steppingstone model we are interested in. Coalescing Brownian motion is a system of indexed one-dimensional interacting Brownian motions with the following intuitive description. All the processes evolve as independent Brownian motions until two of them first meet. After this moment, which we call a coalescing time, the process with higher index assumes the value of the process with lower index. We say the process with higher index is attached to the one with lower index which is still free. They move together according to a single Brownian motion independent of the others until the next coalescing time. The system then evolves in the same fashion.
To keep track of the interactions within the coalescing system we have to introduce more notations. Given a positive integer n, let P n denote the set of partitions of N n := {1, . . . , n}. That is, an element π of P n is a collection π = {A 1 (π), . . . , A h (π)} of disjoint subsets of N n such that i A i (π) = N n . The sets A 1 (π), . . . A h (π) are the blocks of the partition π. The integer h is called the length of π and is denoted by l(π). For convenience we always suppose that the blocks are indexed such that min A i (π) < min A j (π) for i < j, i.e. they are indexed according to the order of their smallest elements. Equivalently, we can think of P n as the set of equivalence relations on N n and write i ∼ π j if i and j belong to the same block of π ∈ P n .
Given π ∈ P n , Let What we really mean by N n is that it is the collection of indices of all the processes in a coalescing system. A partition π describes the interaction in the system at a fixed time. Each block in π corresponds to a free process. The block consists of the index of that free process together with the indices of all the other processes attached to it. a π (i) is just the index of the free process to which the ith process is attached. a i (π), i = 1, . . . , l(π), are all the indices of the free processes left.
R n π is just the effective state space of the coalescing system when the interaction is represented by π. Note thatŘ n π andŘ n π are disjoint for π = π . More precisely, let W e = (W 1 , . . . , W n ) be a n-dimensional Brownian motion starting from e ∈ R n . The n-dimensional coalescing Brownian motionW e = (W 1 , . . . ,W n ) can be constructed from W e inductively as follows. Suppose that times 0 =: . . , n}} have already been defined andW e has been defined on [0, Theorem 2.2 was first obtained in [5]. It will be used repeatedly in the present paper.
Consequently, there is a Hunt process, (X, Q µ ), with state-space Ξ and transition semigroup {Q t } t≥0 .
Remark 2.3. The duality formula (2.4) doesn't have exactly the same expression as that in Theorem 4.1 of [5]. But one can easily check that they turn out to be the same.
Because coalescing Brownian motion is dual to the stepping-stone model, it plays a crucial role in analyzing the clustering behavior. We first introduce two results on a system of coalescing Brownian motions. Given a < b, letW a,b,n := (W a,b,n 1 , . . . ,W a,b,n n ) be a collection of coalescing Brownian motions such that the initial valuesW a,b,n (0) = (W a,b,n n (0), . . . ,W a,b,n n (0)) are independent and uniformly distributed over interval [a, b]. We can defineW a,b,n , n = 1, 2, . . . , on the same probability space in such a way that Write |W a,b (t)| for the cardinality of the collection of coordinates ofW a,b (t), i.e. the total number of "free" Brownian motions left in the coalescing system W a,b,n by time t. WriteW a forW −a,a . Lemma 2.4 was first obtained in [16]. It plays a key role in analyzing the cluster formation and the sizes of those clusters.
Since P{|W a (t)| ≥ 2} is equal to the probability that two independent Brownian motions, with initial values −a and a respectively, have not met until time t. The next result is a consequence of reflection principle of Brownian motion. dx.

Clustering of the continuous-sites Stepping-stone model
For θ ∈ M 1 (K), write θ R for the element in Ξ such that θ R (e) = θ for m a.e. e ∈ R. δ {k} denotes the point mass at k ∈ K. δ δ {k} R is the point mass at δ {k} R ∈ Ξ. Then δ δ {k} R θ(dk) ∈ M 1 (Ξ) means that with probability θ(dk) only individuals of type k appear over the site space R. Write P t for the transition semigroup of one-dimensional Brownian motion. Proof. The necessity of (3.2) follow readily from the moment duality formula (2.4). We only need to show that (3.2) is sufficient. Write e for (e 1 , ..., e n ). Write ψ ⊗ χ for the tensor product of ψ ∈ L 1 (m ⊗n ) ∩ C(R n ) and χ ∈ C(K n ). Observe that in an n-dimensional coalescing Brownian motion, there will be only one free Brownian motion (with index 1) left eventually. It follows from moment duality (2.4) that Hence, (3.1) follows from Lemma 3.1 in [5].
Remark 3.2. By Theorem 3.1, if the initial value of X is θ R , θ ∈ M 1 (K), i.e. X starts with the same mixture of individuals over each site, then θ R Q t → δ δ {k} R θ(dk). As a result, we certainly expect that individuals of the same type clump together.
Notice that given G ⊂ K, µ Theorem 3.3. Given x ∈ R and t > 0, Q θ R almost surely, there exists a constant A > 0 and 1 ≤ i ≤ d such that X t (y)(G i ) = 1 for m a.e. y ∈ (x − A, x + A). Given x ∈ R and a > 0, Q θ R almost surely, there exists a time T > 0 and 1 ≤ i ≤ d such that X T (y)(G i ) = 1 for m a.e. y ∈ (x − a, x + a).
Proof. Suppose that x = 0. For any 1 ≤ i ≤ d, a > 0 and positive integer n, apply moment duality (2.4), we have where X t[−a,a] denotes the block average of X t and θ i : By Lemma 2.4, for fixed t > 0, |W a (t)| → 1 in probability as a → 0+. In addition, for fixed a > 0, |W a (t)| → 1 in probability as t → ∞. Then Notice that d i=1 θ i = 1, the assertion in this theorem is verified. Remark 3.4. It seems the initial value θ R is necessary to obtain the desired result in Theorem 3.3. This is similar to the study on voter models where a typical initial distribution is a renewal measure. Also notice the similar requirements on the initial values of related models in [7,6,10]. The following lemma says that clustering occurs not only locally, as described in   Proof. Set g x,a (µ) : The next result shows that clustering happens simultaneously over R. It also gives an estimate on the average size of the clusters at a fixed time t > 0. Notice that Lemma 3.5 does not exclude the possibility that N M (X t ) could be infinite.
Lemma 3.6. For any t > 0, Q θ R t is strong mixing. Therefore, it is ergodic with respect to τ x .
such that X t has a cluster of type G i for some 1 ≤ i ≤ d on each (b j−1 , b j ) and the clusters over neighboring intervals are of different types. Moreover, there exists a constant c t such that lim M →∞ Proof. For a > 0 and µ ∈ Ξ, write f x,a (µ) : (3.10) It is easy to see that Let a → 0+, by Lemma (2.5) we have Since N M (X t ) < ∞ Q θ R a.s., this together with Lemma 3.5 imply that Q θ R almost surely, except on a m-null set, the interval [−M, M ] is divided into finite subintervals where X t has one type over each interval. M is arbitrary, the first assertion of this theorem is thus proved.
On the other hand, Let a → 0+, it follows form Lemma (2.5) again that Write N m (µ), m > 0, for the total number of clusters of µ ∈ Ξ over the interval [0, m]. It follows from (3.13 By definition one can also verify that N m+n ≤ N m + τ m N n . Then the subadditive ergodic theorem (see Theorem 10.1 in [15]) implies that It follows from Lemma 3.6 that, Q θ R almost surely, lim M →∞ is a constant. Using the subadditive ergodic theorem again, we have Assume that the initial mixture θ ∈ M 1 (K) is a diffuse measure, i.e. θ({k}) = 0, k ∈ K, then Theorem 3.7 can be improved.
Theorem 3.8. Suppose that θ ∈ M 1 (K) is a diffuse measure, then given t > 0, Q θ R almost surely, there exists a sequence . .
Proof. For each positive integer n, let Π n := {[ i 2 n , i+1 2 n ) : 1 ≤ i ≤ 2 n − 1} be a partition of K. Since Π n is getting finer and finer as n increase, a cluster with respect to Π n can break into new clusters with respect to Π n+1 . Apply Theorem 3.7 to Π n and let (b  : −∞ < i < ∞} has no subsequence converging to a finite limit and we choose it as the collection of those b i s in the present theorem. Since any k 1 , k 2 ∈ K, k 1 = k 2 are separated by Π n for n big enough, then on each interval (b j , b j+1 ) X t can only have a cluster of a single type k ∈ K. The last assertion in this theorem follows from the subadditive ergodic theorem, (3.14), (3.15) and the fact that θ is diffuse.
For µ ∈ Ξ, define L(µ) and U (µ), U (µ) > 0 as the essential lower and upper bounds of the cluster of µ at 0. Given that θ is diffuse, we can obtain the joint distribution of L(X t ) and U (X t ).
Theorem 3.9. Suppose that θ ∈ M 1 (K) is diffuse, then for any a > 0, b > 0 and t > 0, Proof. Using the same partition Π n defined in the proof of Theorem 3.7, by (3.4) we have Remark 3.10. As an consequence of Theorem 3.9, the probability that two sites a and b belong to the same cluster of the stepping-stone model at time t is also the probability that two independent Brownian motions with initial values a and b respectively meet each other before time t.

Correlation between types over different sites
In the rest of the paper, we focus on understanding the relationship between the types of clusters over different sites and at different times. To accomplish that we need to generalize the moment duality formula to one involving joint moments over different times.
Remark 4.2. Lemma 4.1 can be generalized to a duality involving a n-fold joint (over different times) moment of X. We leave the details to the readers. Also notice that there is a similar duality in voter model. See [2] for related accounts.
Given z ∈ R and t > 0, write X [z] (t) := lim a→0+ X t[z−a,z+a] for the block average of X t at site z. Notice that X [z] (t) always exists by Theorem 3.3. Since X t (z) is defined for m-almost all z ∈ R, X [z] (t) seems to be more appropriate to describe the distribution of types at time t and at site z. The joint(over different times) moment duality in Lemma 4.1 can be used to study the correlation of X   Moreover, for any 0 < t 2 < t 1 , z 1 , z 2 ∈ R and sets A ⊂ K, B ⊂ K,  To prove (4.5) we apply moment duality, Remark 4.4. It follows from Theorem 4.3 that the types of the two clusters at site z 1 , z 2 and at time t 1 , t 2 respectively are asymptotically independent as |z 2 − z 1 | + |t 2 − t 1 | → ∞. This together with Theorem 3.1 shows that X(t) converges to θ R in distribution but not in probability.