The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction

In this paper we prove that the center of mass of a supercritical branching-Brownian motion, or that of a supercritical super-Brownian motion tends to a limiting position almost surely, which, in a sense complements a result of Tribe on the final behavior of a critical super-Brownian motion. This is shown to be true also for a model where branching Brownian motion is modified by attraction/repulsion between particles. We then put this observation together with the description of the interacting system as viewed from its center of mass, and get the following asymptotic behavior: the system asymptotically becomes a branching Ornstein Uhlenbeck process (inward for attraction and outward for repulsion), but the origin is shifted to a random point which has normal distribution, and the Ornstein Uhlenbeck particles are not independent but constitute a system with a degree of freedom which is less by their number by precisely one.


Introduction
We start with some basic notation.
Notation 1.In this paper M f (R d ) and M 1 (R d ) denote the space of finite measures and the space of probability measures, respectively, on R d .For µ ∈ M f (R d ), we define µ := µ(R d ).
1.1.A model with self-interaction.Consider a dyadic (i.e.precisely two offspring replaces the parent) branching Brownian motion (BBM) in R d with unit time branching and with the following interaction between particles: if Z denotes the process and Z i t is the ith particle, then Z i t feels the drift 1 n t 1≤j≤nt γ Z j t − • , where γ = 0 , that is the particle's infinitesimal generator is (Here and in the sequel, n t is a shorthand for 2 ⌊t⌋ , where ⌊t⌋ is the integer part of t.)If γ > 0, then this means attraction, if γ < 0, then it means repulsion.
To be a bit more precise, we can define the process by induction as follows.Z 0 is a single particle at the origin.In the time interval [m, m + 1) we define a system of 2 m interacting diffusions, by the following system of SDE's: where {W i τ , τ ∈ [0, 1)} are independent Brownian motions for i = 1, 2, . . ., 2 m and they start at the position of their parents at the end of the previous step (at time m − 0).Existence and uniqueness follows from the fact that these 2 m interacting diffusions can be considered as a single 2 Remark 2. It may seem natural to replace the interaction we defined by the gravitational force between particles 1 , however this would lead to a randomized version of the (notoriously difficult) 'n-body problem.' ⋄ 1.2.Results on the self-interacting model.We are interested in the long time behavior of Z, and also,whether we can say something about the number of particles in a given compact set for n large.In the sequel we will use the standard notation Z t , g := nt i=1 g(Z i t ).In this paper we will show (Theorem 14) that Z asymptotically becomes a branching Ornstein Uhlenbeck process (inward for attraction and outward for repulsion), but (1) the origin is shifted to a random point which has d-dimensional normal distribution N d (0, 2), and 1 I.e. the forces are given by Newton's law of universal gravitation: they vary as the inverse square of the distance between the particles.
(2) the Ornstein Uhlenbeck particles are not independent but constitute a system with a degree of freedom which is less by their number by precisely one.For the local behavior we formulate and motivate a conjecture (Conjecture 15).1.3.An extension of Tribe's result on critical super-Brownian motion.In the proof of Theorem 14 we will first show that Z t := 1 nt nt i=1 Z i t , the center of mass for Z satisfies lim t→∞ Z t = N , where N ∼ N d (0, 2).In fact, the proof will reveal that Z moves like a Brownian motion, which is nevertheless slowed down tending to a final limiting location (see Lemma 5 and its proof).
Since this is also true for γ = 0 (BBM with unit time branching and no selfinteraction), our first natural question is whether we can prove a similar result for the supercritical super-Brownian motion.
Another motivation for the same goal is as follows.Tribe [7] proved that a critical super-Brownian motion near its extinction time ξ behaves like a single Brownian path stopped at ξ.More precisely, X t → δ F as t → ∞ a.s. in the weak topology, where F is a d-dimensional random variable and its distribution is the same as that of a Brownian motion at time ξ.
We would like to extend Tribe's result to the supercritical super-Brownian motion X.We are interested in whether we can obtain a similar result on the survival set.Of course, then X t does not shrink to a point in any sense, however, we may hope to get an analogous result regarding the center of mass, defined as (Since f (x) = x is not a bounded function we may not hope to use Tribe's techniques though.) In the sequel we will prove that X will be a time changed Brownian motion on a finite (but random) time interval.
At the beginning of this subsection we referred to a result on Z.In fact we now see that the results on X and Z are completely analogous: in both cases the center of mass is a Brownian motion slowed down in such a way that the time interval [0, ∞) is compressed into a finite one.A slight difference is that, in case of Z, the terminal time is deterministic, because so is the offspring distribution.(The terminal time is t = 2.) Let X be the ( 1 2 ∆, β, α; R d )-superdiffusion with α, β > 0 (supercritical super-Brownian motion).Let P denote the corresponding probability.Let us restrict Ω to the survival set Our main result is that the center of mass for X stabilizes as t → ∞, and furthermore the path of the center of mass is a finite piece of a Brownian path (with a different time parametrization).Theorem 3. Let α, β > 0 and let X denote the center of mass process for the ( 1 2 , β, α; R d )-superdiffusion X.Then, on S, (i) X t converges P δx -almost surely as t → ∞.
(ii) In fact, the finite path {X t } t≥0 is the same as the path of a time changed Brownian motion on [0, T ∞ ), where T ∞ is a positive and finite random variable.More precisely, there exists a d-dimensional Brownian motion B (on an enlarged space) such that X = B • T where T : R + → R + is a random time change satisfying lim t→∞ T (t) < ∞ P δx − a.s.

Remark 4.
(a) A heuristic argument for (i) is as follows.Obviously, the center of mass is invariant under H-transforms whenever H is spatially (but not temporarily) constant.Let H(t) := e −βt .Then X H is a ( 1 2 ∆, 0, e −βt α; R d )superdiffusion, that is, a critical super-Brownian motion with a clock that is slowing down.Therefore, heuristically it seems plausible that X H , the center of mass for the transformed process stabilizes, because this is obviously true in case of extinction, and otherwise the center of mass under the heat flow does not move.(b) From (ii) one can easily conclude some of the a.s.path properties of X.
For example, since the p-variation (p > 0) is invariant under changing the parametrization, we know that any segment of the path has infinite total variation, but the whole path has finite quadratic variation a.s.⋄

The mass center stabilizes
Notice that and so the net attraction pulls the particle towards the center of mass (net repulsion pushes it away from the center of mass).Thus the following lemma is relevant: that is, Z is the center of mass for Z.Then lim t→∞ Z t = N , where N ∼ N d (0, 1).
Proof.For t ∈ [m, m + 1), there are 2 m particles moving around.If t, t + ∆t ∈ [m, m + 1), then Z i t moves as a Brownian motion plus a vector , where the {B i 0 (s), s ≥ 0; i = 1, 2, ..., 2 m } are independent Brownian motions starting at the origin.Thus Using induction, we obtain that (it is easy to check that, as the notation suggests, the summands are independent) which, using Kolmogorov's Three Series Theorem converges almost surely.(We will denote the a.s.limit by N .)On the other hand, since B (m) is a Brownian motion, we can apply Brownian scaling and get 2 −m/2 B (m) (τ ) , where W (m) , m ≥ 1 are independent Brownian motions.We have 2 m , and so, where τ := t − n t .Since the summands are independent on the right hand side, N has the same distribution as a Brownian motion at t = 2, that is, N ∼ N (0, 2).
For another proof see the remark after Lemma 8.
Remark 6.It is interesting to note that Z is in fact a Markov process.Indeed, the distribution of Z t (t = m + τ, 0 ≤ τ < 1) conditional on F m is the same as conditional on Z m , because Z itself is a Markov process.But the distribution of Z t only depends on Z m through Z m , as We will also need the following fact later.
Lemma 7. The coordinate processes of Z are independent.
We leave the simple proof to the reader.

Normality via decomposition
We will need the following result.The decomposition appearing in the proof will also be useful.Recall that m = ⌊t⌋.
) is joint normal for all t ≥ 0. Proof of lemma.By Lemma 13, we may assume that d = 1.We prove the statement by induction.
For m = 1 it is trivial.Suppose that the statement is true for ) is normal, we can consider it just as well as a 2 m dimensional degenerate normal at the instant of the fission of the particles.Indeed, the vector ) has the same distribution on the 2 m−1 dimensional subspace (The reader can easily visualize this for m = 2: the distribution of (Z 1  1 , Z 1 1 ) is clearly √ 2 times the distribution of a Brownian particle at time 1, i.e.N (0, √ 2) on the line Since the convolution of normals is normal, therefore, by the Markov branching property, it is enough to prove the statement when the 2 m particles start at the origin and the clock is reset: t ∈ [0, 1).
Define the 2 m dimensional process Z * on the time interval t ∈ [0, 1) by ), starting at the origin.Because of the interaction between the particles attracts the particles towards the center of mass, Z * is a Brownian motion with drift Notice that this drift is orthogonal to the vector 2 v := (1, 1, ..., 1), that is, the vector (Z t , Z t , ..., Z t ) is nothing but the orthogonal projection of (Z 1 t , Z 2 t , ..., Z 2 m t ) to the line of v.This observation immediately leads to the following decomposition.The process Z * can be decomposed into two components: • the component in the direction of v is a Brownian motion • in the ortho-complement of v, it is an independent Ornstein-Uhlenbeck process with parameter γ.It follows from this decomposition that Z * is Gaussian.
Remark 9. Consider the Brownian component in the decomposition appearing in the proof.Since, on the other hand, this coordinate is 2 m/2 Z t , using Brownian scaling, one obtains another way of seeing that Z t stabilizes at a position which is distributed as the time 1+2 −1 +2 −2 +...+2 −m +... = 2 value of a Brownian motion.(The decomposition shows this for d = 1 and then it is immediately upgraded to general d by independence.)⋄ Corollary 10 (Asymptotics for finite subsystem).Let k ≥ 1 and consider the subsystem (Z 1 t , Z 2 t , ..., Z k t ), t ≥ m 0 for m 0 := ⌊log k⌋ + 1. (This means that at time m 0 we pick k particles and at every fission replace the parent particle by randomly picking one of its two descendants.)Let the real numbers c 1 , ..., c k satisfy t and note that Ψ t is invariant under the translations of the coordinate system.Let L t denote its law.
For every k ≥ 1 and c 1 , ..., c k satisfying (3.1), Ψ (c1,...,c k ) is the same d-dimensional Ornstein Uhlenbeck process corresponding to the operator 1/2∆−γ∇•x, and in particular, For example, taking we obtain that when viewed from a tagged particle's position, any given other particle moves as √ 2 times the above Ornstein Uhlenbeck process.Proof.By independence (Lemma 13) it is enough to consider d = 1.For m fixed, consider the decomposition appearing in the proof of Lemma 8 and recall the 2 For simplicity, we use row vectors in this proof.notation.By (3.1), whatever m ≥ m 0 is, the 2 m dimensional unit vector (c 1 , c 2 , ..., c k , 0, 0, ..., 0) is orthogonal to the 2 m dimensional vector v.This means that Ψ (c1,...,c k ) is a one dimensional projection of the Ornstein Uhlenbeck component of Z * , and thus it is itself a one dimensional Ornstein Uhlenbeck process (with parameter γ) on the unit time interval.Now, although as m grows, the Ornstein Uhlenbeck components of Z * are defined on larger and larger spaces (S ⊂ R 2 m is a 2 m−1 dimensional linear subspace), the projection onto the direction of (c 1 , c 2 , ..., c k , 0, 0, ..., 0) is always the same one dimensional Ornstein Uhlenbeck process, i.e. the different unit time 'pieces' of Ψ (c1,...,c k ) obtained by those projections may be concatenated.

The interacting system as viewed from the center of mass
Recall that by (2.2) the interaction has no effect on the motion of Z.Let us see now how the interacting system looks like when viewed from Z.

4.1.
The description of a single particle.Using our usual notation, assume that t ∈ [m, m + 1) and let τ := t − ⌊t⌋.Also recall that ⊕ denotes independent sum.When viewed from Z, the time τ relocation 3 of a particle can be described as follows: Clearly, and thus the right hand side is a Brownian motion with mean zero and variance We have thus obtained that (for fixed m) Y 1 corresponds to the operator which is an Ornstein-Uhlenbeck process on the time interval [m, m + 1).Since for m large σ m is close to one, the relocation viewed from the center of mass is asymptotically governed by an O-U process corresponding to 1 2 ∆ − γx • ∇.
3 I.e. the relocation between time m and time t. 4 Here ∆ is the Laplace operator unlike in previous lines where it denoted difference.
Remark 11 (Asymptotically vanishing correlation between driving BM's).Let Hence the pairwise correlation decays to zero as t → ∞ (recall that m = ⌊t⌋ and And of course, for the variances we have The description of the system; the 'degree of freedom'.Fix m ≥ 1 and T , where () T denotes transposed.(This is a vector of length 2 m where each component itself is a d dimensional vector; one can actually look at it as a 2 m × d matrix too.)We then have where are mean zero Brownian motions with correlation structure given by (4.1)-(4.2).Just like at the end of subsection 1.1, we can consider Y as a single 2 m ddimensional diffusion.Each of its components is an Ornstein-Uhlenbeck process with asymptotically unit diffusion coefficient.
By independence, it is enough to consider the d = 1 case, and so from now on, in this subsection we assume that d = 1.
Let us first describe the distribution of W t for t ≥ 0 fixed.Recall that {B i 0 (s), s ≥ 0; i = 1, 2, ..., 2 m } are independent Brownian motions starting at the origin.By definition, W t is a 2 m -dimensional multivariate normal: However, since we are viewing the system from the center of mass, it is a singular multivariate normal.Its 'true' dimension is the rank of the matrix A (m) .
Proof.We will simply write A instead of A (m) .Since the columns of A add up to zero, the matrix A is not of full rank: r(A) ≤ 2 m − 1.On the other hand, where I is the 2 m -dimensional unit matrix, and so by subadditivity, By Lemma 12, W t is concentrated on the (2 m −1)-dimensional linear subspace given by the orthogonal complement of the vector (1, 1, ..., 1) T ; in this 2 m − 1 dimensional subspace W t has non-singular multivariate normal distribution.What this means is that even though W 1 , W 2 , ..., W 2 m are not independent, their 'degree of freedom' is 2 m − 1, i. e. the 2 m -dimensional vector W is determined by 2 m − 1 independent components (corresponding to 2 m − 1 principal axes).

Asymptotic behavior
How can we put together that Z t tends to a random final position a.s. with the description of the system 'as viewed from Z t ?' Lemma 13.For t ≥ 0, the random vector Y t is independent of the path {Z s } s≥t .
Proof.First, for any t > 0, Y t is independent of Z t , because (assuming d = 1) the vector is normal (since it is a linear transformation of the vector (Z 1 t , Z 2 t , . . ., Z 2 m t ) T , which is normal by Lemma 8), and so it is sufficient to show that Z t and Z i t − Z t are uncorrelated for 1 ≤ i ≤ 2 m .But this is obvious, because the random variables Z 1 t , Z 2 t , ..., Z 2 m t are exchangeable and thus, denoting To complete the proof of the lemma, recall Lemma 5 and its proof and notice that the distribution of {Z s } s≥t only depends on its starting point Z t , as it is that of a Brownian path appropriately slowed down, whatever Y t (or even Z t ) is. Since, as we have seen, Y t is independent of Z t , we are done.
We have the following result on the asymptotic behavior of the system.
Theorem 14.The asymptotic behavior of Z is that of a branching Ornstein-Uhlenbeck process, but with the origin shifted by a random, normally distributed vector.More precisely, for almost all x, conditionally on lim t→∞ Z t = x, one has where Y is the branching Ornstein-Uhlenbeck process of section 4.1.
Proof.We already know from Lemma 5 that the almost sure limit N := lim t→∞ Z t exists.Let By Lemma 13, Y t is independent of  Explanation of the conjecture: Once we have Theorem 14, we can try to put it together with the Strong Law of Large Numbers for the local mass from [1] for the process Y .So N is the final position of the center of mass.First, let γ > 0. If the components of Y were independent and the branching rate were exponential, Theorem 6 of [1] would be readily applicable.However, since the 2 m components of Y are not independent (as we have seen, their degree of freedom is 2 m − 1) and since, unlike in [1], we now have unit time branching, the method of [1] must be adapted to our setting.This adaption would require some extra work.
Once this is done, however, it immediately follows that there exists a random variable N ∼ N d (0, 2), such that, conditional on Thus the distribution of lim n→∞ 2 −n Z n , g is the convolution of N (0, 2I d ) and N (0, (2γ) −2 I d ), yielding the second statement in the first part.The γ > 0 case is similar but since the limiting density is translation invariant, i.e.Lebesgue (see Example 11 in [1]), the final position of the center of mass plays no role.
6. Proof of Theorem 3 (i) Since α, β are constant, the branching is independent of the motion, and therefore N defined by N t := e −βt X t is a nonnegative martingale (positive on S) tending to a limit almost surely.It is straightforward to check that it is uniformly bounded in L 2 and is therefore uniformly integrable (UI).Write We now claim that N ∞ > 0 a.s. on S. Let A := {N ∞ = 0}.Clearly S∁ ⊂ A, and so if we show that P (A) = P (S∁), then we are done.As is well known, P (S∁) = e −β/α .On the other hand, a standard martingale argument (see the argument after formula (20) in [2]) shows that 0 ≤ u(x) := − log P δx (A) must solve the equation 1 2 ∆u + βu − αu 2 = 0, but since P δx (A) = P (A) constant, therefore − log P δx (A) solves βu − αu 2 = 0. Since N is UI, no mass is lost in the limit, giving P (A) < 1.So u > 0, which in turn implies that − log P δx (A) = β/α.Once we know that N ∞ > 0 a.s. on S , it is enough to focus on the term e −βt x, X t .Let H(t) := e −βt .Then X H is a ( 1 2 ∆, 0, e −βt α; R d )-superdiffusion, that is, a critical super-Brownian motion with a clock that is slowing down.One can write e −βt x, X t = x, X H t .Define T s = S H s := e −βs S s ; then the semigroup {T s } s≥0 corresponds to Brownian motion.In particular then (6.1) T s [id] = id, where id(x) = x.Therefore x, X H t is a martingale. 5If we show that the martingale is UI, we are done.It is enough to show that it is uniformly bounded in L 2 .To achieve this, define g n by g n (x) = |x| • 1 {|x|<n} .Then we have and by the monotone convergence theorem we can continue with Since g n is compactly supported, there is no problem to use the moment formula and continue with where W is Brownian motion.Since g n (x) ≤ |x|, therefore we can trivially upper estimate the last expression by Since this upper estimate is independent of t, we are done: (ii) Keeping in mind that we are on the survival set S := {ω ∈ Ω | X t (ω) > 0, ∀t > 0}, we first claim that X is an (S, P, {F t } t≥0 )-martingale, where F t := σ({X t ; t ≥ 0}).
Let  We now show that E(X t | F s ) = X s for 0 ≤ s < t.By the Markov branching property, where X δx are independent copies of super-Brownian motions starting at δ x .Since X is mean zero, shifting by x and using Fubini's Theorem, we can continue the last displayed formula with Next, we show that X has continuous paths.)t} , where ǫ > 0 is fixed for the rest of the argument.By [5] it follows that there exists an a.s.finite random time T = T (ω) such that for all t > T , x, X t = u(•, t), X t .

lim n→∞ 2
−n E Z n , g = E M, g a.s.(HereI d denotes the d-dimensional unit matrix.)(ii) If γ < 0 (repulsion), then lim n→∞ 2 −n Z n , g = 1, g a.s.Remark 16.The function 2 + 1 4γ 2 in Conjecture 15(i) is monotone decreasing.The intuitive meaning is that stronger attraction results in smaller variance of the limiting distribution.⋄
Let u(x, t) := x1 {|x|≤( √ 2β+ǫ)t} + ( 2β + ǫ)t1 {|x|>( √ 2β+ǫ N , and thus P x (Y t ∈ •) = P (Y t ∈ •) for almost all x.By definition, Z t = Z t + Y t , and by the above discussion, we can in fact write Z t = Z t ⊕ Y t under P x .Putting this together with the description of Y t in section 4.1, we are done.We close this section with a conjecture on the local behavior of the system.If γ > 0 (attraction), then there exists a random variable N ∼ N d (0, 2), such that, conditional on N = x 0 , us start with showing that E|X t | < ∞.To see this, first recall that |X t |; S = E N −1 t | x, X H t |; S , and since we have seen in part (i) that x, X H t is (uniformly) bounded in L 2 , by Cauchy-Schwartz, it is enough to see thatE N −2 t ; S = E[ X H t −2 ; S] < ∞, ∀t > 0, i.e. that E[ X t −2 ; S] < ∞, ∀t > 0. This is true because X is a one-dimensional diffusion on [0, ∞) with generator x(α d 2 dx 2 +β d dx ), which, on S, tends to infinity, and therefore, by Fatou's Lemma, lim t→∞ E[ X t −2 ; S] = 0. Hence E[ X t −2 ; S] < ∞ for large t's, but then by continuity, E[ X t −2 ; S] < ∞, ∀t > 0. Next, since E|X t | < ∞, in fact EX t = 0 for t ≥ 0, because X t is symmetrically * , ∀B ∈ M 1 (R d