Branching diffusion with particle interactions

A d-dimensional branching diffusion, Z, is investigated, where the linear attraction or repulsion between particles is competing with an Ornstein-Uhlenbeck drift, with parameter b (we take b > 0 for inward O-U and b < 0 for outward O-U). This work has been motivated by [4], where a similar model was studied, but without the drift component. We show that the large time behavior of the system depends on the interaction and the drift in a nontrivial way. Our method provides, inter alia, the SLLN for the non-interactive branching (inward) O-U process. First, regardless of attraction (γ > 0) or repulsion (γ < 0), a.s., as t → ∞, the center of mass of Zt converges to the origin when b > 0, while escapes to infinity exponentially fast (rate |b|) when b < 0. We then analyze Z as viewed from the center of mass, and finally, for the system as a whole, we provide a number of results and conjectures regarding the long term behavior of the system; some of these are scaling limits, while some others concern local extinction.

(Note the negative sign of the drift. It is somewhat unusual, but it fits our setup better, because of the sign of the interaction parameter γ, introduced below.) If b > 0, then we have an 'inward' O-U process; if b < 0, then we have an 'outward' O-U process. If b = 0, then it is a Brownian motion.
As far as the aforementioned interaction is concerned, let us fix the interaction parameter γ = 0. We assume that the i th particle Z i t , on the time interval [m, m + 1), 'feels' a drift caused by attraction/repulsion of all other particles as and so Z i t satisfies the following stochastic differential equation: If γ > 0, then this means that particles attract each other, whereas if γ < 0, then this means that they repel each other.
In the stochastic differential equation above, the {W i,m t } 1≤i≤2 m are independent Brownian motions on [m, m + 1). In other words, the infinitesimal generator of the ith particle in the time interval is Notation. Throughout the paper, the symbol w ⇒ (or just ⇒) will denote weak convergence of finite measures; the symbol v ⇒ will denote vague convergence. By a bounded rational rectangle we will mean a set B ⊂ R d of the form B = I 1 × I 2 × · · · × I d , where I i is a bounded interval with rational endpoints for each 1 ≤ i ≤ d. The family of all bounded rational rectangles will be denoted by R. The symbol X ⊕ Y will denote the independent sum of the random variables X and Y . As usual, N (µ, σ 2 ) will denote the normal distribution with mean µ and variance σ 2 ; Leb will denote d-dimensional Lebesgue measure. Finally, for z ∈ R, z will denote the largest integer which is less than or equal to z.
Branching diffusion with interactions is a Brownian motion, being slowed down such that it tends to a 'terminal position' N almost surely, and N is a d-dimensional, normally distributed random variable, with mean zero. If P x denotes the probability conditioned on N = x, x ∈ R d , then the following theorem was demonstrated for γ > 0 (attraction): exp −γ|y − x| 2 dy, P x − a.s., as n → ∞, for almost all x ∈ R d , where Z(dy) denotes the discrete measure-valued process corresponding to the interacting branching particle system. For γ < 0, a conjecture was stated. A similar model for superdiffusions has been introduced and studied by H. Gill recently [6] and results analogous to those in [4], were obtained. The toolsets used in the two papers are very different though. Gill's paper utilizes the so-called historical calculus of E. Perkins.
It should be mentioned that, although our original motivation was to analyze the compound effect of the drift and the interaction, it turns out that our method yields an elementary proof for the Strong Law of Large Numbers for the case of a non-interactive branching (inward) Ornstein-Uhlenbeck process as well. See Example 7.5.
Finally, for classical results on limit theorems for branching particle systems (without interaction), see the fundamental monograph [1], and the more recent article [5].

Existence and uniqueness
In this section, we show the existence and uniqueness for this system (process). Actually, it is easy to see 2 that we only need to show that, given the initial value in time interval [m, m + 1), the system exists and is unique. Now, in the time interval [m, m + 1), we can look at the 2 m interacting particles (diffusions) as a single 2 m d−dimensional Brownian motion with a drift d : R 2 m d → R 2 m d defined as d(x 1 , x 2 , · · · , x d−1 , x d , · · · , x 2 m d ) := γ(β 1 , β 2 , · · · , β d−1 , β d , · · · , β 2 m d ) T , where β k = 2 −m γ(x k + x d+k + · · · + x (2 m −1)d+k ) − (γ + b)x k . Here k − k is a multiple of d and 0 < k ≤ d. As the drift d is Lipschitz, the existence and uniqueness of our system follows from the uniqueness/existence theorem for stochastic differential equations in several dimensions.

Summary of our results
In this paper, we will provide the following results/conjectures.

Branching diffusion with interactions
In this section we are going to show that as t → ∞: (1) if b > 0, then the center of mass converges to the origin, no matter if attraction or repulsion holds; (2) if b < 0, then it will tend to infinity with 'speed' e −bt .
The significance of this result is that the attraction/repulsion for Z i t is given by where Z t is as above. Hence, one can replace the interaction between particles by the interaction with the center of mass. Therefore, as a first step, we will study the large time behavior of the center of mass Z t .
Before stating our first result, we note that in this section, we will be interested in a.s. and L 2 convergence of the COM. Since it is easy to see that these limits can be verified coordinate-wise, we assume d = 1 for this section. (The reader should keep in mind that the results work for any d ≥ 1.) Our main results here will concern the behavior of the COM in the attractive/repulsive case. But we need some preliminary lemmas first. Below we give two results regarding a general, one-dimensional stochastic differential equation where we assume that b > 0 and that β(·) > 0 is locally Lipschitz. Here β(t) can be considered a time change of the Brownian part. We assume that β(t) converges to 0 as t → ∞, that is that the Brownian motion is slowing down completely. We then want to determine the limiting distribution.
Lemma 2.2 (Convergence to zero). Let X be the solution of (2.1).
Proof. (a) Assume, that X t = X(t) is of the form X(t) = X 1 (t)X 2 (t), with X 1 (0) = 0 and X 1 , X 2 being of finite variation. Keeping the product rule for dX(t) in mind, set We obtain X 2 (t) = Ce −bt , C = 0, and thus dX 1 (t) = C −1 β(t)e bt dW t . Then X(t) := e −bt t 0 β(s)e bs dW s satisfies (2.1); by uniqueness, it is in fact the solution to the equation.
Since X t is centered Gaussian, the claim is tantamount to (b) We need to show that for any > 0, we have Then, by the Borel-Cantelli lemma, it is sufficient to show that Denote Y t := e bt X t = t 0 β(s)e bs dW s , and note that Y t is a martingale as it is an Itô integral. Thus |Y t | is a submartingale. We have P (A m ) < P (sup{|Y t | : m ≤ t < m + 1} > e bm ). By Doob's inequality, and thus  We show now that both As β decreases to 0, there is a constant C such that β(s) < C for all s ≥ 0. Then  Note that (m + 1)/2 ≤ 2 (m + 1)/2 for m ≥ 1, and, since β is a decreasing function, one Since, by assumption, Next, we describe the evolution of the COM using a stochastic differential equation.
For each i, the particle Z i,m t 's motion satisfies the stochastic differential equation where γ is the interaction coefficient, and b is the drift part of the Brownian motion. In our case, we consider b > 0. Taking averages on both sides, the center of mass Z t will thus satisfy As the Brownian components of different particles are independent, Brownian scaling yields that where W t is standard Brownian motion in the time interval [m, m + 1). We thus have Hence, in the time interval [m, m + 1), we have an Ornstein-Uhlenbeck process, while on [0, ∞), the process Z satisfies the general stochastic differential equation in the statement.
After these preparations, we now turn to the attractive case.
Proof. Recall the stochastic differential equation satisfied by COM. In order to prove the theorem, it is sufficient to check that the conditions of Lemma 2.2 are satisfied.
For the repulsive case (b < 0), we have the following theorem. Proof. By the independence of the coordinate processes, it is enough to consider d = 1.
In order to show the existence of the limit and to identify it, we are going to utilize the Dambis-Dubins-Schwarz Theorem. We will use the shorthand X t := e bt Z t . More precisely, we are going to show that there exists a Brownian motion B on the same probability space where Z is defined, .
Let U denote the increasing process for a process U . It then follows that To achieve all these, recall first that by (2.3), X t = t 0 β(s)e bs dW s , and thus, it is a continuous martingale. Therefore by the Dambis-Dubins-Schwarz Theorem (see e.g. Theorem V.1.6 in [8]), X t is a time-changed Brownian motion: Since the increasing process is deterministic in this case, we have that where β(s) := 2 −m/2 for s ∈ [m, m + 1). Thus, X t = B s(t) , almost surely, and furthermore, To evaluate the infinite sum, one can use Abel's (summation by part) formula, which leads to: completing the proof.
We note that without the Dambis-Dubins-Schwarz Theorem, much more elementary, standard arguments still prove the existence of the almost sure limit, but only along certain 'discrete time skeletons.' Remark 2.6 (Exponential speed of COM). As lim t→∞ e bt Z t exists a.s. and e |b|t → ∞, the point Z t will tend to infinity, almost surely, with 'speed' e |b|t in the sense that Z t ≈ e |b|t · N . Furthermore, even in higher dimensions, it is clear by symmetry considerations that the angular component of Z t will be uniformly distributed.
Finally, it is easy to see that lim b→0 T (b) = 2, in accordance with the already studied drift-less case.

The system as viewed from the center ('relative system')
Having described the motion of the center of mass Z t , in order to study the whole system, we need to investigate the 'relative system', that is the system as viewed from Z t .
will be called the relative system, or the system, as viewed from the center of mass.
We focus on the behavior of the relative system in this section. We will use the shorthand σ 2 m := 1 − 2 −m . First, we want to determine the stochastic differential equation for Y i t . It can be obtained by direct computation, as follows. Fixing the time interval [m, m + 1), recall that for each i, the particle Z i,m t 's motion satisfies Subtracting the second equation from the first, one has As a next step, we need to study the correlation between the particles of {Y i t } for t > 0. As i W i t = 0, they are obviously not independent. First we determine the 'degree of freedom' of { W i t }. Similarly to [4], one can show that the degree of freedom of are mean zero, correlated Brownian motions.
Just like in subsection 1.3, here we can also consider Y as a single 2 m d-dimensional diffusion. Each of its components is an Ornstein-Uhlenbeck process with asymptotically unit diffusion coefficient.
Since we are viewing the system from the center of mass, W (m) t is a singular multivariate normal and thus Y is a degenerate diffusion. The 'true' dimension of W (m) t is rank(A (m) ). Then the same argument as in [4], yields that rank(A (m) ) = 2 m − 1, and the above comment about the degrees of freedom should be understood in this sense.
Moreover, the driving Brownian motions { W i t } will be exactly the same as in [4], and thus they have asymptotically vanishing correlation (see Remark 12 in [4]).
The relative system thus coincides with the drift-less one in [4], if γ is replaced by γ + b.

A useful transformation: the process Z ∆
We first make an important observation, making the last sentence of the previous section more general: we notice that γ and b are 'interchangeable' in the following sense. Lemma 4.1 (Interchangeable coefficients). Suppose that we have two branching particle systems, and Y and Y represent the relative systems for them. Denote by b 1 , γ 1 and b 2 , γ 2 the corresponding coefficients of Y and Y. (Here γ i = 0 is allowed.) Assume that b 1 + γ 1 = b 2 + γ 2 . Then the laws of Y and Y are the same.
Proof. Fix m ≥ 0 and i = 1, 2, · · · , 2 m . Then Y i and Y i satisfy the same stochastic differential equation Thus, the fact that single particles have the same law in the two systems, follows by induction, along with the EJP 21 (2016), paper 67. existence and uniqueness of the solution for stochastic differential equations. (We know that for m = 0 they start with the same initial value.) The fact that even the joint distributions of the two particle systems agree, follows the same way as we proved existence and uniqueness for the model in subsection 1.3, except that now the independent driving Brownian motions must be replaced by ). Since the piecewise Lipschitz-ness of the coefficients is preserved, the existence and uniqueness theorem is still in force.
We now define a transformation which plays a crucial role in this paper.
Z has the same law as the process t → Z t (· − D t ), The behavior of D t for large times depends on the signs of b and b + γ. E.g. if they are both positive, then D t tends to the origin almost surely.
In particular, with appropriate transformations we can 'knock out' either the interaction or the O-U drift: (a) Representation with non-interactive system: pick ∆ := γ. That is, let the non-interactive process Z correspond to γ ∆ = 0 and b ∆ := b + γ. Then (4.2) gives a remarkable link between the interactive and the non-interactive models. This connection is reminiscent of the one found in [6] (see Remark 3.2 there). is just a Brownian motion, similarly to [4]. Then (4.2) gives a link between our model and the drift-less one studied in [4].

Outline of the strategy of the rest of the proofs
In light of the previous section, we could choose to base the analysis of the relative system on the corresponding result in [4] when b + γ > 0 (by eliminating the drift -see part (b) in the previous section), or on the results on the global system in Example 11 of [5] when b + γ < 0 (by eliminating the interaction -see part (a) in the previous section).
In the second case, we should handle the problem that the setting is different in [5] in that the branching is not unit time but rather exponential.
On top of that, the method of the proof in both [4] and [5] requires the introduction of two non-trivial auxiliary functions, related to the model.
Instead of choosing one of the paths alluded to above, we decided to give a completely elementary proof in the next section for the Strong Law for the relative system in our case, when b + γ > 0. It does not use the complicated machinery of [4] or [5], and it is done in unit time. The proof only uses some calculations involving the most recent common ancestors of particles and some covariance estimates.
In particular, it gives a new, elementary proof for the Strong Law for the global system, for the case of a non-interactive branching (inward) Ornstein-Uhlenbeck process. See Example 7.5.

General comments
Recall that σ 2 m := 1 − 2 −m , and that for the relative system Y , Assuming b + γ > 0, our goal is to find lim m→∞ (Here, we consider Y t as a random measure, and that is why we may write Y i t (B) = 1 B (Y i t ), i = 1, 2, 3, · · · , 2 m .) For d = 1, this will be achieved in Theorem 6.11 and subsequently, it will be upgraded to higher dimensions; before these, we will prove several preparatory results.
Remark 6.1 (Integer vs. continuous times). When taking the limit, we will just consider integer times. This is somewhat weaker than considering continuous times, however, since the model is about unit time branching, we did not have sufficient motivation to go into the technical details as to how one upgrades the limit along integer times to a limit along continuous times. (There are existing techniques though, going back to the work of Asmussen and Hering, see [1,5].) As mentioned above, for simplicity we will first treat d = 1, and then show that the main result we got also works for higher dimensions.
Next, let us sketch our strategy of the investigation: (1) Find the correlation between the particle positions Y i m , 1 ≤ i ≤ 2 m ; m ≥ 1. In this section, when studying the relative system, we will assume that b = 0 and γ > 0. We can do this without the loss of generality, since given γ and b, one may apply a ∆-transformation with ∆ := −b, that is, eliminate the O-U drift. Then b ∆ = 0 and γ ∆ = b + γ, and Y and Y have the same law by Lemma 4.1.

Crucial estimates
Now let us focus on the distribution of the random variables and since for the general differential equation is a normal variable with distribution N 0, σ 2 m (2γ) −1 e 2γ − 1 , and Y i 0 = 0. Note that Y i m and X i m (or more generally, X m and Y m ) are independent, as X m is defined by a stochastic integral on [m, m + 1]. Proposition 6.3 (Variance; single particle). For any integer m ≥ 0, Proof. Since Y m+1 = e −γ (Y m X m ) and Y 0 = 0, we can compute the variance of Y m+1 using recursion, due to the independence of Y m and X m . First, we have and σ 2 (Y 0 ) = 0. Thus, for convenience, we denote a m := σ 2 (Y m ) and b m := σ 2 (X m ) = 1 2γ 1 − 1 2 m e 2γ − 1 .
Using the definition of a m+1 , the proof is complete.
We now need to analyze the covariance between Y i m and Y j m for 1 ≤ i, j ≤ 2 m and m ≥ 1. Here the notion of the most recent common ancestor (MRCA) becomes important.  As different pairs of particles (Y i m , Y j m ) with different MRCA's may have different covariances, we need to take into account the MRCA of (Y i m , Y j m ). Hence, with splitting time a > 0, we will write (Y i m,a , Y j m,a ) for (Y i m , Y j m ).
where C is a constant which only depends on γ.
Proof. Since Y i m+1 = e −γ (Y i m X i m ) for any 'relative particle,' it follows by recursion Here X i n is the process as in (6.1) on the unit time interval [n, n + 1]. From the preceding formula we can easily compute Cov(Y i m , Y j m ), once we know the corresponding Cov(X i n , X j n ) for n = a, a + 1, ..., m − 1. For the ith particle (i can be replaced by j, of course), where σ n W i s = − 1 2 n k =i W k s ⊕ σ 2 n W i s and {W k s , k = 1, 2, 3, · · · , 2 n }, as well as W i are standard independent Brownian motions on the unit time interval [n, n + 1].
Using that the {W k s , k = 1, 2, · · · , 2 n } are independent, Cov(X i n , X j n ) can now be computed. Indeed, the stochastic integrals X i n , X j n are just linear combinations of the stochastic integrals A k := n+1 n e γ(s−n) dW k s , k = 1, 2, 3, · · · , that is, Note that, by the Itô-isometry, Therefore, using the independence of the A k 's, EJP 21 (2016), paper 67.
According to Proposition 6.3, Using this along with (6.2) and (6.3), one obtains that It then follows that were C 2 is also a constant which only depends on γ.
where C is a constant that only depends on γ. 3 In this formula, i > 2 k may occur. Then X i k stands for the normal variable associated with the ancestor of i at time k. The definition is similar to that of X i n .
Recall that our goal is to prove the existence of lim for Borel sets B ⊂ R d and identify them. To achieve this, we will use the Borel-Cantelli Lemma in conjunction with the Chebyshev inequality, and so we need to estimate Var 2 −m Clearly, ) for i = j, and in order to do that, we need to analyze the splitting time (still denoted by a) for the particles Y i m and Y j m . The following lemma on joint normal distribution will be useful. Lemma 6.6 (Covariance for indicators). Let (X, Y ) be a joint normal vector such that its marginals X and Y are standard normal, and denote ρ := Cov(X, Y ). Then there exists an absolute constant C > 0 such that with k(x, y, ρ) := 2xy2(1 − ρ 2 ) + 4ρ(x 2 + y 2 + 2ρxy). For |ρ| ≤ 1 2 , we have x 2 +y 2 2 ≤ x 2 + y 2 + 2ρxy, i.e. x 2 + y 2 + 4ρxy ≥ 0, and thus, EJP 21 (2016), paper 67.

4
(1 + x 2 + y 2 ), ∀|ρ| ≤ 1/2. Consequently, Remark 6.7. More generally, let both X and Y be N (0, σ 2 )-distributed. If (X, Y ) is joint normal with Cov(X, Y ) = ρ, then we can scale X and Y to use Lemma 6.6. Indeed, X σ and Y σ are then standard normal variables, and Cov( X σ , Y σ ) = ρ σ 2 . From Lemma 6.6, if Returning to the question of covariances with splitting time a, note that {Y i m } are linear combinations of a number of underlying Brownian motions; hence (Y i m , Y j m ) will be joint normal. From Theorem 6.5, we know that Cov(Y i m , Y j m ) ≤ C(γ)m(e −2γ(m−a) + 2 −m ).
From (6.1), we then have the following corollary.
Corollary 6.8. With some constant C > 0 (that depends only on γ), .) Remark 6.9. From Proposition 6.3, it follows that lim 2γ . Therefore, if we write a(m) in place of a, and lim m→∞ (m − a(m)) = ∞, then the condition in Corollary 6.8 will always be true for large enough m's.

Strong Law for the relative system
First we need a technical (but very simple) lemma that shows that it will be sufficient to handle the one dimensional case.

Proof. One has
Using the assumption, Using the shorthands a i : and from the computation above, Therefore the statement becomes Use that 0 ≤ a i ≤ 1 and 0 ≤ b i ≤ 1 and induction on d as follows. The statement is true for d = 1, and if it is true for some d ≥ 1, then and so it is also true for d + 1. Now we are ready to prove the strong law.
Note: Here, of course lim m→∞ P (Y 1 m ∈ B) exists. That is, similarly to [4], Proof. First, because of Assumption 6.2 and Lemma 6.10, we may and will assume that d = 1, b = 0 and γ > 0. Now, we need to show that for any > 0, a.s., only finitely many of the events will occur. By the Borel-Cantelli Lemma and Chebyshev's inequality, it is enough to show that To this end, note that for a fixed i, since the roles of i and j are symmetric.
Hence, one needs to analyze is satisfied. Now, the important point is that the majority of particle-pairs have a ≤ m/2, that is, they are not 'close' relatives. Indeed, the number of pairs with a > m/2 (close relatives) is 2 m (1 + 2 + 4 + · · · + 2 m−1 2 ), which is bounded from above by 2 m+ m 2 +1 . As we know, for any pair, we have Cov Thus, for all of those pairs with a > m/2, the total covariance will be controlled by 2 m+ m 2 +1 · 1 4 = 2 m+ m 2 −1 .
Moreover, as discussed above, for the pairs with a ≤ m/2, we may apply Lemma 6.6 and the remark following it, yielding where C only depends on N, γ, B. Consequently, Here C 0 is a constant which only depends on N, γ, B. Given that γ > 0 and the three terms on the righthand side are all summable in m, (6.2) holds and the proof is complete.

The distribution of the particle system
Now we have collected enough information to describe the large time behavior the system as a whole.

Preparation
Below we describe the system's behavior as it depends on the parameters γ, b. The statements about the large time behavior of the branching particle system will follow from the behavior of the center of mass (Theorem 2.4 and Theorem 2.5), that of the relative system (Theorem 6.11), and finally, from the following proposition. Proposition 7.1 (Independence). The tail σ-algebra T of Z is independent of the relative system Y .
Proof. The proof of this proposition is exactly the same as the corresponding proof of Lemma 14 in [4].
Recall that Z(dy) denotes the discrete measure-valued process corresponding to the interacting branching system. The following notion will be important. Definition 7.2 (Local extinction). We say that Z suffers local extinction, if Z n (dy) v ⇒ 0, as n → ∞, a.s. Since Z is a discrete particle system, (7.1) is tantamount to the property that for all B ⊂ R d ball, there exists an almost surely finite random time T B such that P (Z n (B) = 0, ∀n ≥ T B ) = 1.

Inward drift in the motion component
We now turn to the first results about the behavior of Z, starting with the case of an inward drift (b > 0). In fact, we distinguish between three further sub-cases.
As we have demonstrated, the center of mass converges to zero as t → ∞, and the relative system will become an inward O-U process with parameter γ + b. Putting Theorem 6.11 together with the a.s. converge of the COM (Theorem 2.4), and finally, with Proposition 7.1, we arrive at the following conclusion. complementing the exponential-clock results in [5].
Proof. The proof is based on an argument which 'switches off' the interaction. In order to accomplish this, we are going to utilize the lemma on interchangeability (Lemma 4.1). Namely, we match the relative system with that of another system without interaction.
This other system is the one with b = γ = 0 (branching Brownian motion without interaction). As far as the behavior of this second system is concerned, it is well known (see [9,2]), that (7.2) holds. Even though in [9,2], the decomposition into COM and a relative system was not considered, we do that now. Doing so is useful, because by Lemma 4.1, the relative system is the same for the two processes, even though the behavior of the COM is not: for the original system it converges to the origin almost surely (Theorem 2.4), and for the non-interacting BBM it has an almost sure (Gaussian) limit (see [4]).
Now use the fact that Leb is translation invariant. In both systems, one has EJP 21 (2016), paper 67. hence Cn d/2 2 −n Z n (B) = Cn d/2 2 −n Y n (B + Z n ).
By conditioning on the almost sure limit of Z n , and using Proposition 7.1, the translation invariance of the Lebesgue measure yields (7.2) for the relative system in the b = γ = 0 case.
But then, by Lemma 4.1, the same holds for the relative system in the original model. Since the COM converges to the origin almost surely for the original model, using Proposition 7.1, we conclude that the scaling limit (7.2) also holds for the original system.
As the relative system behaves asymptotically like an outward O-U process, we have a conjecture similar to the one in [4]. In our case, however, the center of mass tends to 0 as t → ∞. Thus, the conjecture will take the following form: Conjecture 7.7 (Inward drift, strong repulsion). The following dichotomy holds for the long term behavior of the process: 1. If log 2 d ≤ |b + γ|, then Z suffers local extinction.
2. If log 2 d > |b + γ|, then almost surely, Remark 7.8. The intuitive explanation of the dichotomy in the conjecture is as follows. Even though the motion has a strong inward component (forcing the center of mass to tend to the origin, according to Theorem 2.4), this is offset by the even stronger repulsion term.
This combined effect is then competing with the mass creation (the 'rate' of mass creation in this case can be taken log 2): if mass creation is stronger, then the Law of Large Numbers is still in force; otherwise the mass creation is no longer able to compensate the fact that particles are 'being pushed away.'

Outward drift in the motion component
This case is more difficult than the case of the inward drift. The result below is quite natural once the decomposition of the process (COM plus relative system) is established, however, it may be somewhat surprising if one is just given the definition of the model with the pairwise particle interactions.
In this case, according to Theorem 2.5, the center of mass converges to infinity a.s. as t → ∞, and so the question is, intuitively, whether this effect will be compensated by the large number of particles.
The next result says that for outward drift and attraction, the system always exhibits local extinction, irrespective of the relationship between the drift size |b| and the attraction parameter γ. Z n (dy) v ⇒ 0 a.s. Remark 7.10. As far as the relative system's behavior is concerned, that is of course given by Theorem 6.11.
Proof. The proof is based on comparing the speed of the COM with that of the relative system. Let M rel t denote the radius of the smallest ball containing the support of Y t for t ≥ 0. It is clear that the relation lim t→∞ M rel t |Z t | = 0 a.s., (7.3) proven below, implies local extinction; we will verify it by distinguishing between three sub-cases.
(a) When b = γ = 0, the non-interactive branching Brownian motion will be denoted by Z (b = γ = 0). Let Y non-int be the relative system for Z and let M non-int t be the radius of the smallest ball containing the support of Y non-int .
Indeed, this follows from the fact that n 2 n i=1 P (Z i n (R d \ B(0, cn)) > 0) < ∞, which, in turn, follows from the estimate 4 P (Z i n (R d \ B(0, cn)) > 0) = exp(−c 2 n/2 + o(n)), where {Z i n ; 1 ≤ i ≤ 2 n } are the particles forming Z n . (The asymptotic 'speed of the support' of Z is actually exactly √ 2 log 2, but we don't need this fact here. Cf. [7].) We now claim that b + γ = 0 implies M rel t = O(t), t → ∞, a.s. (7.4) Indeed, recall that by ∆-transform invariance, Y d = Y non-int . But M non-int t = O(t), as t → ∞ a.s., because it is well known that max{|z| | z ∈ supp(Z t )} = O(t) a.s., and because of the existence of lim t→∞ Z t a.s.
Turning back to Z, if b + γ = 0, then (7.4) and the fact that Z t escapes to infinity exponentially fast (according to our Theorem 2.5) yields (7.3).
(b) If b + γ > 0, then by ∆-transform invariance, for the relative system, we may assume that it is non-interacting and the particles are performing inward Ornstein-Uhlenbeck motions with parameter b + γ > 0. The proof is then the same as in (a), given that P (Z i n (R d \ B(0, cn)) > 0) is even smaller now. (c) Finally, for the case when b + γ < 0 and γ > 0, we still have (7.3). To see this, recall first that the logarithmic escape rate of Z t is −bt. On the other hand, a calculation similar 5 to the one in Example 11 in [5] reveals that the logarithmic rate of spread of the relative system is −(b + γ)t, which yields (7.3). (Since the probability that at least one particle is present in a set B is trivially dominated by the expected particle number in that set, that is, P (Z n (B) > 0) ≤ E(Z n (B)), the calculation reduces to computing certain expectations. In our case it is even easier than in [5], as the total population size is deterministic.)