Extinction of Fleming-Viot-type particle systems with strong drift

We consider a Fleming-Viot-type particle system consisting of independently moving particles that are killed on the boundary of a domain. At the time of death of a particle, another particle branches. If there are only two particles and the underlying motion is a Bessel process on $(0,\infty)$, both particles converge to 0 at a finite time if and only if the dimension of the Bessel process is less than 0. If the underlying diffusion is Brownian motion with a drift stronger than (but arbitrarily close to, in a suitable sense) the drift of a Bessel process, all particles converge to 0 at a finite time, for any number of particles.


Introduction
Our paper is motivated by an open problem concerning extinction in a finite time of a branching particle system. We prove two results that are related to the original problem and might shed some light on the still unanswered question.
The following Fleming-Viot-type particle system was studied in [5]. Consider an open bounded set D ⊂ R d and an integer N ≥ 2. Let X t = (X 1 t , . . . , X N t ) be a process with values in D N defined as follows. Let X 0 = (x 1 , . . . , x N ) ∈ D N . Then the processes X 1 t , . . . , X N t evolve as independent Brownian motions until the time τ 1 when one of them, say, X j hits the boundary of D. At this time one of the remaining particles is chosen uniformly, say, X k , and the process X j jumps at time τ 1 to X k τ 1 . The processes X 1 t , . . . , X N t continue evolving as independent Brownian motions after time τ 1 until the first time τ 2 > τ 1 when one of them hits the boundary of D. Again at the time τ 2 the particle which approaches the boundary jumps to the current location of a particle chosen uniformly at random from amongst the ones strictly inside D. The subsequent evolution of X proceeds in the same way. We will say that X constructed above is driven by Brownian motion.
The main results in this paper are concerned with Fleming-Viot particle systems driven by other processes.
The above recipe defines the process X t only for t < τ ∞ , where There is no natural way to define the process X t for t ≥ τ ∞ , and, therefore, it is of interest to investigate what conditions ensure that τ ∞ = ∞. In Theorem 1.1 of [5] the authors claim that in every domain D ⊂ R d and every N ≥ 2, we have τ ∞ = ∞, so the Fleming-Viot process is always well-defined. However, the proof of Theorem 1.1 in [5] contains an error which is irreparable in the following sense. That proof is based on only two properties of Brownian motion-the strong Markov property and the fact the the hitting time distribution of a compact set has no atoms (assuming that the starting point lies outside the set). Hence, if some version of that argument were true, it would apply to almost all non-trivial examples of Markov processes with continuous time, and in particular to all diffusions. However, in [3], the authors provided an example of a diffusion X on D = (0, ∞) (a Bessel process with dimension ν = −4), such that τ ∞ < ∞ for the Fleming-Viot process driven by this diffusion with N = 2.
It is not known whether Theorem 1.1 of [5] is correct in full generality. It was proved in [3,10] that the theorem holds in domains which do not have thin channels.
1.1. Main results. We will prove two theorems. The first theorem is concerned with Bessel processes but it is motivated by the original model based on Brownian motion in an open bounded subset of R d . Recall that for any real ν, a ν-dimensional Bessel process on (0, ∞) killed at 0 may be defined as a solution to the stochastic differential equation ∞ and a C ∞ function ρ : D → (0, ∞) with the following properties, The above estimates and the Itô formula Brownian motion and Z t = ρ(B t ) then where the functions a k ( · ) and b( · ) are bounded. This shows that the dynamics of Z resembles that of a Bessel process. Note that if τ ∞ < ∞ for the Fleming-Viot process driven by Brownian motion in a domain D then the distances of all particles to ∂D go to 0 as t ↑ τ ∞ , by Lemma 5.2 of [3]. Hence, it is of some interest to see whether a Fleming-Viot process based on a Bessel process can become extinct in a finite time. We have a complete answer only for N = 2, i.e., a two-particle process.
Theorem 1. Let X be a Fleming-Viot process with N particles on (0, ∞) driven by Bessel process of dimension ν ∈ R.
Our second main result is also motivated by some results presented in [5]. Several theorems in [5,3] are concerned with limits when N → ∞. To formulate rigorously any of these theorems it would suffice that τ ∞ = ∞, a.s., for all sufficiently large N. In other words, it is not necessary to know whether τ ∞ = ∞ for small values of N. One may wonder whether it is necessarily the case that τ ∞ = ∞ for any Fleming-Viot-type process and sufficiently large N. Our next result shows that once the drift of the diffusion is slightly stronger than the drift of any Bessel process then τ ∞ < ∞ for the Fleming-Viot process driven by this diffusion and every N.
Consider the following SDE for a diffusion on (0, 2], where x 0 ∈ (0, 2], β > 2, W is Brownian motion, T 0 is the first hitting time of 0 by X, and L t is the local time of X at 2, i.e., L t is a CAF of X such that ∞ 0 1 {Xs =2} dL s = 0, a.s. It is well known that (1.2) has a unique pathwise solution (X, L) (see, e.g., [2], Theorem I.12.1). We will analyze a Fleming-Viot process on (0, 2] driven by the diffusion defined in (1.2). The role of the boundary is played by the point 0, and only this point. In other words, the particles jump only when they approach 0. Let P x denote the distribution of the Fleming-Viot particle system starting from X 0 = x.
Remark 1. (i) If we take β = 2 in (1.2) then the diffusion is a Bessel process (locally near 0). Hence, we may say that Theorem 2 is concerned with a diffusion with a drift "slightly stronger" than the drift of any Bessel process. with no reflection but with very strong negative drift far away from 0.
We end this section with two open problems.
Problem 1. Find necessary and sufficient conditions, in terms of N and ν, for nonextinction in a finite time of an N-particle Fleming-Viot process driven by ν-dimensional Bessel process.

Problem 2.
Does there exist a Fleming-Viot-type process, not necessarily driven by Brownian motion, such that τ ∞ = ∞, a.s., for the N-particle system, but τ ∞ < ∞ with positive probability for the (N + 1)-particle system, for some N ≥ 2?
The rest of the paper contains the proofs of the two main theorems.
2. Proof of Theorem 1 2.1. Bessel processes. We start with a review of some facts about Bessel processes and Gamma distributions. Let Z t , t ≥ 0, be a square of Bessel process of dimension ν ∈ R starting at x ≥ 0, (Z ∼ BESQ ν (x), for short), i.e., Z is the unique strong solution to stochastic differential equation where W is a one-dimensional Brownian motion (see [12,Chapter 11] for the case ν ≥ 0 and [8] for the general case).
Squares of Bessel processes have the following scaling property: if Z t ∼ BESQ ν (x) and for some c > 0 and all x > 0, and T 0 denotes the first hitting time of 0, then T 0 = ∞, a.s., if ν ≥ 2, and T 0 < ∞, a.s., if ν < 2. Moreover, in the latter case we have Here and in what follows we say that a random variable is Γ(α)-distributed if it has the density denotes the standard gamma function. Note that we consider only a one-parameter family of gamma densities, unlike the traditional two-parameter family.
In [12], Bessel process X of dimension ν ≥ 0 starting at x ≥ 0 (X ∼ Bes ν (x)), is defined as the square root of BESQ ν (x 2 ) process Z. If ν ≥ 0, then by so called comparison theorems, the paths of Z t are defined for all t ≥ 0, so X t is well defined for all t ∈ [0, ∞).
We define Bessel process X of dimension ν < 0 starting at x ≥ 0 as the square root of a BESQ ν (x 2 ) process Z, i.e., X t = √ Z t for t ≤ T 0 . For any real ν, these definitions are equivalent to the definition given in (1.1) by the Itô formula.
Processes Bes ν (x) with ν ∈ R scale as follows. If X ∼ Bes ν (x) is a Bessel process on [0, T 0 ), then for all c > 0, cX c −2 t is a Bes ν (cx) process on [0, c 2 T 0 ]. This follows easily from the scaling property of BESQ ν (x) processes.

2.2.
Proof of Theorem 1 (i). We start with an alternative construction of the Fleming- It is easily seen that σ 1 may be represented as is an independent copy of T 0 , and that (σ i , i = 1, 2, . . . ) is a sequence of independent and identically distributed random variables.
We construct a two-particle Fleming-Viot type process X t = (X 1 t , X 2 t ) as follows. First let τ 1 = σ 1 and set X t = Y 1 t for t ∈ [0, τ 1 ). At τ 1 one of the particles hits the boundary of D = (0, ∞), and it jumps to ξ 1 = α 1 . To continue the process we use the scaling property . At τ 2 , one of the particles hits the boundary and jumps, this time to ξ 2 = α 2 ξ 1 . We continue the process in the same way by setting It is easy to see that the construction of X given above is equivalent to that given in the Introduction, except that the driving process is a ν-dimensional Bessel process. The process X t is well defined up until τ ∞ , and we will show now that τ ∞ < ∞ almost surely if and only if ν < 0.
Note that X 0 = (1, 1) for the process constructed above. However, it is easy to see that for any two starting points X 0 = (x 1 0 , x 2 0 ) and X 0 = (z 1 0 , z 2 0 ) with x 1 0 , x 2 0 , z 1 0 , z 2 0 > 0, the distributions of X τ 1 are mutually absolutely continuous. This implies that the argument given below proves the theorem for any initial value of X.
Proof of (i). Note that, in view of (2.1), where ψ is well known digamma function ([1, Section 6.3]) defined as Proof of (ii). By Theorem 11 of [11] we get that the density of α 2 1 is given by we have where for x > −1 and k ∈ N, is a generalized binomial coefficient. Therefore, as for a ∈ R ∞ n=0 n + a n z n = (1 − z) −a−1 .
and therefore for y ≥ 0 So, to prove (ii) we need to study the sign of the integral Recall the Student's t-distribution with a > 0 degrees of freedom ([1, section 26.7]). The density for this distribution is given by Changing the variable y = 2x √ 2−ν in I(ν) we get where X is a random variable with t-distribution with (2 − ν)-degrees of freedom.

Proof of Theorem 2
3.1. Preliminaries. We will give new meanings to some symbols used in the previous section. Constants denoted by c with subscripts will be tacitly assumed to be strictly positive and finite; in addition, they may be assumed to satisfy some other conditions.
(i) Let W be one-dimensional Brownian motion and let b be a Lipschitz function defined on an interval in R, i.e., |b(x 1 ) − b(x 2 )| ≤ L|x 1 − x 2 | for some L < ∞ and all x 1 and x 2 in the domain of b. Consider a diffusion X t , t ∈ [s, u], satisfying the following stochastic differential equation, Let y t be the solution to the ordinary differential equation We will later write y ′ = b(y) instead of d dt y t = b(y t ). The following inequality appears in Ch. 3, Sect. 1 of the book by Freidlin and Wentzell [7]. For every δ > 0, where L is a Lipschitz constant of b. It follows that where c 0 is an absolute constant.
(ii) Recall that β > 2 and consider the function We need the assumption that β > 2 for the main part of the argument but many calculations given below hold for a larger family of β's. It is easy to check that y s,a (t) := a β + s − t 1/β , s ≤ t ≤ s + a β , (3.4) is the solution to the ordinary differential equation with the initial condition y s,a (s) = a, where s ∈ R, a > 0. Note that the function y s,a (t) approaches 0 vertically at t = s + a β .
(iii) Fix any γ ∈ (0, 1) and let L be the Lipschitz constant of b on the interval a(γ/2) 1/β /2, 2a . Then and, therefore, (3.6) Let X be the solution to (3.1) with b defined in (3.3). Assume that δ > 0 is so small that Hence, we can apply (3.2) with L given by (3.6) to obtain the following estimate P sup where X t satisfies (3.1) and c 1 depends on β and γ, but it does not depend on δ and a.
Note that two processes X i and X j can meet either when their paths intersect at a time when both processes are continuous or when one of the processes jumps onto the other.
Let j 2 be the smallest index in J 1 such that the equality in the definition of S 1 holds with j = j 2 . Let I 2 = {j 1 , j 2 } and J 2 = [N] \ I 2 .
Next we proceed by induction. Assume that, for some n < N, the sets I 1 , . . . , I n , J 1 , . . . , J n , and stopping times S 1 < S 2 < . . . < S n−1 are defined. Then we let where j n+1 is the smallest index in J n such that the equality in the definition of S n holds with j = j n+1 .
The set I n has n elements which are indices of particles which are "descendants" of the particle X j 1 that was the highest at time 0. By convention, we let I n = I N and S n = u for n ≥ N.
Step 2. Write a = x j 1 and u = (1 − γ)a β . Then x ∈ (0, a] N . Recallδ and M defined in Note that for every t, max j∈In X j t ≥ max j∈Jn X j t . Hence, and, therefore, where we adopted the convention P x (F 1 | F c 0 ) = P x (F 1 ). Suppose that F c n−1 holds and j ∈ I n . Then |X j S n−1 −y 0,a (S n−1 )| ≤δ n−1 . Let y j t , t ≥ S n−1 , be a solution to y ′ = b(y) with y j S n−1 = X j S n−1 . By (3.15), It follows from (3.16) that we can apply (3.8) (with an appropriate shift of the time scale) to X j , assuming that We combine this estimate with (3.18) to see that Step 3. We will prove that there exist v < ∞ and r ∈ (0, 2) such that if x ∈ (0, r] N then (3.20) Consider an r ∈ (0, 2) and for x = (x 1 , . . . , x N ), let A 0 = max j x j , U 0 = 0, and for k = 0, 1, 2, . . . , let Let Y k t denote the solution to ODE (3.5) with the initial condition Y k U k = A k . Recall ε from (3.10) and let ∆ k = A k 1 − (1 − εγ) 1/β . For k = 0, 1, 2, . . . define events Note that Y k U k+1 = γ 1/β A k . Suppose that ∞ k=0 Γ c k holds. Then for all k, where c 4 = γ 1/β + 1 − (1 − εγ) 1/β < 1, by (3.11). Hence, A k ≤ c k 4 A 0 and, therefore, ≤ v and lim sup t↑v max 1≤j≤N X j t = 0. This implies easily that τ ∞ ≤ v. Thus, to prove (3.20), it will suffice to show that there exists r ∈ (0, 2) such that if By (3.19) and the strong Markov property applied at U k , where c 4 < 1. So by (3.22), if max j x j ≤ r, then , which is convergent. Since 2 − β < 0, we can choose r > 0 so small that the above sum is less than 1/2, proving (3.21).
Step 4. Let r ∈ (0, 2) and v be as in Step 3. Partition the set (0, 2] N into two sets A = (0, r] N and A c . First we will show that the time when process X enters the set A has a distribution with an exponentially decreasing tail. So assume that X 0 ∈ A c and let Consider p 1 (x) = P x ∀ j∈I 1 ∀ 0≤t≤1/2 X j t ∈ r 2 , 2 ; η < 1/2; ∀ i∈I 2 ∀ τ j 1 <t≤1/2 X j t ∈ r 4 , 2 .
We will argue that for x ∈ A c we have Indeed, with probability at least q 1 > 0 any particle from I 1 stays in the interval [r /2, 2] up to time t = 1/2. With probability at least q 2 > 0 any particle from I 2 hits 0 before time t = 1/2; with probability at least 1/N it jumps onto a particle in I 1 ; and then with probability at least q 3 > 0 it stays in the interval [r/4, 2] up to time t = 1/2. Therefore (3.23) holds with p 1 = (q 1 q 2 q 3 /N) N . Obviously q 1 , q 2 , q 3 and p 1 depend on r.