aa r X i v : . [ m a t h . P R ] S e p INFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES
ANDREY SARANTSEV
Abstract.
Consider a system of infinitely many Brownian particles on the real line. At anymoment, these particles can be ranked from the bottom upward. Each particle moves as a Brownianmotion with drift and diffusion coefficients depending on its current rank. The gaps betweenconsecutive particles form the (infinite-dimensional) gap process. We find a stationary distributionfor the gap process. We also show that if the initial value of the gap process is stochastically largerthan this stationary distribution, this process converges back to this distribution as time goes toinfinity. This continues the work by Pal and Pitman (2008). Also, this includes infinite systemswith asymmetric collisions, similar to the finite ones from Karatzas, Pal and Shkolnikov (2016). Introduction
Consider the standard setting: a filtered probability space (Ω , F , ( F t ) t ≥ , P ), with the filtrationsatisfying the usual conditions. Take i.i.d. ( F t ) t ≥ -Brownian motions W i = ( W i ( t ) , t ≥ i =1 , , . . . Consider an infinite system X = ( X i ) i ≥ of real-valued adapted processes X i = ( X i ( t ) , t ≥ , i = 1 , , . . . , with P -a.s. continuous trajectories. Suppose we can rank them in the increasingorder at every time t ≥ X (1) ( t ) ≤ X (2) ( t ) ≤ . . . If there is a tie : X i ( t ) = X j ( t ) for some i < j and t ≥
0, we assign a lower rank to X i and higherrank to X j . Now, fix coefficients g , g , . . . ∈ R and σ , σ , . . . >
0. Assume each process X i (wecall it a particle ) moves according to the following rule: if at time t X i has rank k , then it evolvesas a one-dimensional Brownian motion with drift coefficient g k and diffusion coefficient σ k . Letting1( A ) be the indicator function of an event A , we can write this as the following system of SDEs:(1) d X i ( t ) = ∞ X k =1 X i has rank k at time t ) ( g k d t + σ d W i ( t )) , i = 1 , , . . . The gaps Z k ( t ) = X ( k +1) ( t ) − X ( k ) ( t ) for k = 1 , , . . . form the gap process Z = ( Z ( t ) , t ≥ Z ( t ) = ( Z k ( t )) k ≥ . Then X is called an infinite system of competing Brownian particles . A moreprecise definition is given in Definitions 6 and 7 later in this article.This system was studied in [35, 18]. For g = 1 , g = g = . . . = 0 and σ = σ = . . . = 1, this iscalled the infinite Atlas model , which was studied in [27, 8]. The term Atlas stands for the bottomparticle, which moves as a Brownian motion with drift 1 (as long as it does not collide with otherparticles) and “supports other particles on its shoulders”. This system is, in fact, a generalizationof a similar finite system X = ( X , . . . , X N ) ′ , which is defined analogously to the equation (1).Finite systems of competing Brownian particles were originally introduced in [2] as a model inStochastic Portfolio Theory, see [10, 12]. They also serve as scaling limits for exclusion processes Date : September 3, 2016. Version 47.2010
Mathematics Subject Classification.
Key words and phrases.
Reflected Brownian motion, competing Brownian particles, asymmetric collisions, in-teracting particle systems, weak convergence, stochastic comparison, triple collisions, stationary distribution. on Z , see [22, Section 3], and as a discrete analogue of McKean-Vlasov equation, which governs anonlinear diffusion process, [36, 7, 21]. Finite systems were thoroughly studied recently. We canask the following questions about them:(a) Does this system exist in the weak or strong sense? Is it unique in law or pathwise?(b) Do we have triple collisions between particles, when three or more particles occupy the sameposition at the same time?(c) Does the gap process have a stationary distribution? Is it unique?(d) What is the exact form of this stationary distribution?(e) Does Z ( t ) converge weakly to this stationary distribution as t → ∞ ?For finite systems, these questions have been to a large extent answered.(a) The system exists in the weak sense and is unique in law, [4]. Until the first moment of atriple collision, it exists in the strong sense and is pathwise unique, [18]. It is not known whetherit exists in the strong sense after this first triple collision.(b) It was shown in [17, 18, 32] that there are a.s. no triple collisions if and only if the sequence( σ , . . . , σ N ) is concave :(2) σ k ≥ (cid:0) σ k − + σ k +1 (cid:1) , k = 2 , . . . , N − . (c) The gap process has a stationary distribution if and only if(3) g k > g N , k = 1 , . . . , N − , where g k := 1 k ( g + . . . + g k ) for k = 1 , . . . , N. In this case, this stationary distribution is unique, see [2, 3].(d) If, in addition to (3), the sequence ( σ , . . . , σ N ) is linear :(4) σ k +1 − σ k = σ k − σ k − for k = 2 , . . . , N − , then this stationary distribution has a product-of-exponentials form, see [2, 3].(e) The answer is affirmative, under the condition (3), see [3, 39, 6].Before surveying the answers for infinite systems, let us define some notation. Let N ∈{ , , . . . } ∪ {∞} . Introduce a componentwise (partial) order on R N . Namely, take x = ( x i )and y = ( y i ) from R N . For M ≤ N , we let [ x ] M := ( x i ) i ≤ M . For a distribution π on R N , we let[ π ] M be the marginal distribution on R M , corresponding to the first M components. For a matrix C = ( c ij ) i,j ≤ N , we let [ C ] M = ( c ij ) i,j ≤ M . We say that x ≤ y if x i ≤ y i for all i ≥
1. For x ∈ R N ,we let [ x, ∞ ) := { y ∈ R N | y ≥ x } . We say that two probability measures ν and ν on R N satisfy ν (cid:22) ν , or, equivalently, ν (cid:23) ν , if for every y ∈ R ∞ we have: ν [ y, ∞ ) ≤ ν [ y, ∞ ). In thiscase, we say that ν is stochastically dominated by ν , and ν stochastically dominates ν , or ν is stochastically smaller than ν , or ν is stochastically larger than ν . We denote weak convergenceof probability measures by ν n ⇒ ν . We denote by I k the k × k -identity matrix. For a vector x = ( x , . . . , x d ) ′ ∈ R d , let k x k := ( x + . . . + x d ) / be its Euclidean norm. For any two vectors x, y ∈ R d , their dot product is denoted by x · y = x y + . . . + x d y d . The Lebesgue measure isdenoted by mes. A one-dimensional Brownian motion with zero drift and unit diffusion, startingfrom 0, is called a standard Brownian motion . LetΨ( u ) := 1 √ π Z ∞ u e − v / d v, u ∈ R , NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 3 be the tail of the standard normal distribution.For infinite systems, the answers to questions (a) - (e) are quite different.(a) For infinite systems, it seems that a necessary condition for weak existence is that initialpositions X i (0) = x i , i = 1 , , . . . of the particles should be “far apart”. Indeed, it is an easyexercise to show that a system of i.i.d standard Brownian motions starting from the same pointis not rankable from bottom to top at any fixed time t >
0. Some sufficient conditions for weakexistence and uniqueness in law are found in [35, 18]. We restate them in Theorem 3.1 in a slightlydifferent form:(5) lim i →∞ x i = ∞ and ∞ X i =1 e − αx i < ∞ , α > . We also prove a few other similar results: Theorem 3.2 and Theorem 3.3, under slightly differentconditions. Strong existence and pathwise uniqueness for finite systems are known from [18] tohold until the first triple collision , when three or more particles simltaneously occupy the sameposition. It is not known whether these hold after this first triple collision.(b) In this paper, we continue on the research in [18] and prove essentially the same result asfor finite systems. There are a.s. no triple collisions if and only if the sequence ( σ k ) k ≥ is concave:see Theorem 5.1 and Remark 7.(c) In this paper, see Theorem 4.4, we prove that there exists a certain stationary distribution π under the condition which is very similar to (3):(6) g k > g l , ≤ k < l. Actually, we can even relax this condition (6) a bit, see (25). The question whether it is uniqueor not is still open.(d) The exact form of this distribution π is found in (26) for a special case (4); it is also aproduct of exponentials, as in the finite case.(e) We prove a partial convergence result in Theorem 4.6 and Theorem 4.7: if Z (0) stochasticallydominates this stationary distribution π : Z (0) (cid:23) π , then Z ( t ) ⇒ π as t → ∞ . However, we donot know whether Z ( t ) weakly converges as t → ∞ for other initial distributions. Since we do notknow whether a stationary distribution is unique, this means that we do not know what are the“domains of attraction”.Let us give a preview of results for a special case:(7) g = g = . . . = g M = 1 , g M +1 = g M +2 = . . . = 0 , σ = σ = . . . = 1 . The following theorem is a corollary of more general results (which are enumerated above) fromthis paper; see Example 4.2 below.
Theorem 1.1.
Under conditions (7) , the system (1) exists in the strong sense, is pathwise unique,there are a.s. no triple and simultaneous collisions, and the stationary distribution π for the gapprocess is given by (8) π M := Exp(2) ⊗ Exp(4) ⊗ . . . ⊗ Exp(2 M ) ⊗ Exp(2 M ) ⊗ . . . ANDREY SARANTSEV
For M = 1, this is the infinite Atlas model, and the stationary distribution π M = π = ⊗ ∞ k =1 Exp(2) is already known from [27, Theorem 14]. It is worth noting that the
Harris systemof Brownian particles (independent Brownian motions B n , n ∈ Z , starting from B n (0) = x n ), infact, has infinitely many stationary distributions for its gap process, [16]. Indeed, a Poisson pointprocess with constant intensity λ is invariant with respect to this system for any λ >
0. Therefore,the product ⊗ n ∈ Z Exp( λ ) is a stationary distribution for this system, for all λ > a >
0, the following is astationary distribution for the gap process:(9) π M ( a ) := ∞ O k =1 Exp (2( k ∧ M ) + ka ) . In particular, for the infinite Atlas model we have: π ( a ) := ∞ O k =1 Exp (2 + ka ) . Note that the distribution (8) can also be included in the family (9), if we let a = 0.Other ordered particle systems derived from independent driftless Brownian motions were stud-ied by Arratia in [1], and by
Sznitman in [38]. Several other papers study connections betweensystems of queues and one-dimensional interacting particle systems: [24, 14, 15, 34]. Links tothe directed percolation and the directed polymer models, as well as the GUE random matrixensemble, can be found in [26].An important generalization of a finite system of competing Brownian particles is a system withasymmetric collisions , when, roughly speaking, ranked particles Y k , have “different mass”, andwhen they collide, they “fly apart” with “different speed”. This generalization was introducedin [22] for finite systems. We carry out this generalization for infinite systems, and prove weakexistence (but not uniqueness) in Section 3. All results answering the questions (a) - (e) aboveare stated also for this general case of asymmetric collisions.There are other generalizations of competing Brownian particles: competing L´evy particles,[35]; a second-order stock market model , when the drift and diffusion coefficients depend on thename as well as the rank of the particle, [11, 3]; competing Brownian particles with values in thepositive orthant R N + , see [13]. Two-sided infinite systems ( X i ) i ∈ Z of competing Brownian particlesare studied in [33].The proofs in this article rely heavily on comparison techniques for systems of competing Brow-nian particles, developed in [30].1.1. Organization of the paper.
Section 2 is devoted to the necessary background: finite sys-tems of competing Brownian particles. It does not contain any new results, just an outline ofalready known results. Section 3 introduces infinite systems of competing Brownian particles andstates existence and uniqueness results (including Theorem 3.7). In this section, we also generalizethese comparison techniques for infinite systems. Section 4 deals with the gap process: stationarydistributions and the questions of weak convergence as t → ∞ . In particular, we state Theo-rems 4.4 and 4.6 and in this section. Section 5 contains results about triple collisions. Section 6is devoted to proofs for most of the results. The Appendix contains some technical lemmas. NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 5 Background: finite systems of competing Brownian particles
In this section, we recall definitions and results which are already known. First, as in [2, 32],we rigorously define finite systems of competing Brownian particles for the case of symmetriccollisions, when the k th ranked particles moves as a Brownian motion with drift coefficient g k anddiffusion coefficient σ k . This gives us a system of named particles; we shall call them classicalsystems of competing Brownian particles . Then we find an equation for corresponding ranked particles, following [2, 3]. This gives us a motivation to introduce systems of ranked competingBrownian particles with asymmetric collisions, as in [22]. Finally, we state results about the gapprocess: stationary distribution and convergence.2.1. Classical systems of competing Brownian particles.
In this subsection, we use defini-tions from [2]. Assume the usual setting: a filtered probability space (Ω , F , ( F t ) t ≥ , P ) with thefiltration satisfying the usual conditions. Let N ≥ g , . . . , g N ∈ R ; σ , . . . , σ N > . We wish to define a system of N Brownian particles in which the k th smallest particle movesaccording to a Brownian motion with drift g k and diffusion σ k . We resolve ties in the lexicographicorder, as described in the Introduction. Definition 1.
Take i.i.d. standard ( F t ) t ≥ -Brownian motions W , . . . , W N . For a continuous R N -valued process X = ( X ( t ) , t ≥ X ( t ) = ( X ( t ) , . . . , X N ( t )) ′ , let us define p t , t ≥
0, the rankingpermutation for the vector X ( t ): this is the permutation on { , . . . , N } such that:(i) X p t ( i ) ( t ) ≤ X p t ( j ) ( t ) for 1 ≤ i < j ≤ N ;(ii) if 1 ≤ i < j ≤ N and X p t ( i ) ( t ) = X p t ( j ) ( t ), then p t ( i ) < p t ( j ).Suppose the process X satisfies the following SDE:(10) dX i ( t ) = N X k =1 p t ( k ) = i ) [ g k d t + σ k d W i ( t )] , i = 1 , . . . , N. Then this process X is called a classical system of N competing Brownian particles with driftcoefficients g , . . . , g N and diffusion coefficients σ , . . . , σ N . For i = 1 , . . . , N , the component X i = ( X i ( t ) , t ≥
0) is called the i th named particle . For k = 1 , . . . , N , the process Y k = ( Y k ( t ) , t ≥ , Y k ( t ) := X p t ( k ) ( t ) ≡ X ( k ) ( t ) , is called the k th ranked particle . They satisfy Y ( t ) ≤ Y ( t ) ≤ . . . ≤ Y N ( t ), t ≥
0. If p t ( k ) = i ,then we say that the particle X i ( t ) = Y k ( t ) at time t has name i and rank k .The coefficients of the SDE (10) are piecewise constant functions of X ( t ) , . . . , X N ( t ); therefore,weak existence and uniqueness in law for such systems follow from [4].2.2. Asymmetric collisions.
In this subsection, we consider the model defined in [22]: finitesystems of competing Brownian particles with asymmetric collisions . For k = 2 , . . . , N , let theprocess L ( k − ,k ) = ( L ( k − ,k ) ( t ) , t ≥
0) be the semimartingale local time at zero of the nonnegativesemimartingale Y k − Y k − . For notational convenience, we let L (0 , ( t ) ≡ L ( N,N +1) ( t ) ≡ B , . . . , B N , the ranked particles Y , . . . , Y N satisfy the following equation:(11) Y k ( t ) = Y k (0) + g k t + σ k B k ( t ) + 12 L ( k − ,k ) ( t ) − L ( k,k +1) ( t ) , k = 1 , . . . , N. ANDREY SARANTSEV
This was proved in [3, Lemma 1]; see also [2, Section 3]. The process L ( k − ,k ) is called the local timeof collision between the particles Y k − and Y k . The local time process L ( k − ,k ) has the followingproperties: L ( k − ,k ) (0) = 0, L ( k − ,k ) is nondecreasing, and(12) Z ∞ Y k ( t ) = Y k − ( t ))d L ( k − ,k ) ( t ) = 0 . If we change coefficients 1 / asymmetriccollisions from the paper [22]. The local times in this new model are split unevenly between thetwo colliding particles, as if these particles have different mass.Let us now formally define this model with asymmetric collisions. Let N ≥ g , . . . , g N and positive real numbers σ , . . . , σ N , as before. In addition,fix real numbers q +1 , q − , . . . , q + N , q − N , which satisfy the following conditions: q + k +1 + q − k = 1 , k = 1 , . . . , N −
1; 0 < q ± k < , k = 1 , . . . , N. Definition 2.
Take i.i.d. standard ( F t ) t ≥ -Brownian motions B , . . . , B N . Consider a continuousadapted R N -valued process Y = ( Y ( t ) , t ≥ , Y ( t ) = ( Y ( t ) , . . . , Y N ( t )) ′ , and N − L ( k − ,k ) = ( L ( k − ,k ) ( t ) , t ≥ , k = 2 , . . . , N, with the following properties:(i) Y ( t ) ≤ . . . ≤ Y N ( t ) , t ≥ Y satisfies the following system of equations:(13) Y k ( t ) = Y k (0) + g k t + σ k B k ( t ) + q + k L ( k − ,k ) ( t ) − q − k L ( k,k +1) ( t ) , k = 1 , . . . , N (we let L (0 , ( t ) ≡ L ( N,N +1) ( t ) ≡ k = 2 , . . . , N , the process L ( k − ,k ) = ( L ( k − ,k ) ( t ) , t ≥
0) has the properties men-tioned above: L ( k − ,k ) (0) = 0, L ( k − ,k ) is nondecreasing and satisfies (12).Then the process Y is called a system of N competing Brownian particles with asymmetriccollisions , with drift coefficients g , . . . , g N , diffusion coefficients σ , . . . , σ N , and parameters ofcollision q ± , . . . , q ± N . For each k = 1 , . . . , N , the process Y k = ( Y k ( t ) , t ≥
0) is called the k th rankedparticle . For k = 2 , . . . , N , the process L ( k − ,k ) is called the local time of collision between theparticles Y k − and Y k . The Brownian motions B , . . . , B N are called driving Brownian motions forthis system Y . The process L = (cid:0) L (1 , , . . . , L ( N − ,N ) (cid:1) ′ is called the vector of local times .The state space of the process Y is W N := { y = ( y , . . . , y N ) ′ ∈ R N | y ≤ y ≤ . . . ≤ y N } .Strong existence and pathwise uniqueness for Y and L are proved in [22, Section 2.1].2.3. The gap process for finite systems.
The results of this subsection are taken from [2, 3,22, 39]. However, we present an outline of proofs in Section 6 for completeness.
Definition 3.
Consider a finite system (classical or ranked) of N competing Brownian particles.Let Z k ( t ) = Y k +1 ( t ) − Y k ( t ) , k = 1 , . . . , N − , t ≥ . Then the process Z = ( Z ( t ) , t ≥ Z ( t ) = ( Z ( t ) , . . . , Z N − ( t )) ′ is called the gap process . Thecomponent Z k = ( Z k ( t ) , t ≥
0) is called the gap between the k th and k + 1 st ranked particles . NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 7
The following propositions about the gap process are already known. We present them in aslightly different form than that from the sources cited above; for the sake of completeness, wepresent short outlines of their proofs in Section 6. Let(14) R = − q − . . . − q +2 − q − . . . − q +3 − q − . . . . . . − q − N − . . . − q + N − , (15) µ = ( g − g , g − g , . . . , g N − g N − ) ′ . Proposition 2.1. (i) The matrix R is invertible, and R − ≥ , with strictly positive diagonalelements ( R − ) kk , k = 1 , . . . , N − .(ii) The family of random variables Z ( t ) , t ≥ , is tight in R N − , if and only if R − µ < . Inthis case, for every initial distribution of Y (0) we have: Z ( t ) ⇒ π as t → ∞ , where π is theunique stationary distribution of Z .(iii) If, in addition, the skew-symmetry condition holds: (16) ( q − k − + q + k +1 ) σ k = q − k σ k +1 + q + k σ k − , k = 2 , . . . , N − , then π = N − O k =1 Exp( λ k ) , λ k = 2 σ k + σ k +1 (cid:0) − R − µ (cid:1) k , k = 1 , . . . , N − . For symmetric collisions, we can refine Proposition 2.1. Recall the notation from (3): g k := g + . . . + g k k , k = 1 , . . . , N. Proposition 2.2.
For the case of symmetric collisions q ± k = 1 / , k = 1 , . . . , N , we have:(i) − R − µ = 2 ( g − g N , g + g − g N , . . . , g + g + . . . + g N − − ( N − g N ) ′ ;(ii) the tightness condition from Proposition can be written as g k > g N , k = 1 , . . . , N − (iii) the skew-symmety condition can be equivalently written as σ k +1 − σ k = σ k − σ k − , k = 2 , . . . , N − in other words, σ k must linearly depend on k ;(iv) if both the tightness condition and the skew-symmetry condition are true, then π = N − O k =1 Exp( λ k ) , λ k := 4 kσ k + σ k +1 ( g k − g N ) . ANDREY SARANTSEV
Example . If g = 1 , g = g = . . . = g N = 0, and σ = σ = . . . = σ N = 1 (the finite Atlas model with N particles), then π = N − O k =1 Exp (cid:18) · N − kN (cid:19) . The following is a technical lemma, with a (very short) proof in Section 6.
Lemma 2.3.
Take a finite system of competing Brownian particles (either classical or ranked).For every t > , the probability that there is a tie at time t is zero. Existence and Uniqueness Results for Infinite Systems
In this section, we first state existence results for classical infinite systems of competing Brow-nian particles (recall that classical means particles with individual names rather than ranks):Theorem 3.1, Theorem 3.2, and Theorem 3.3. Then we define infinite ranked systems with asym-metric collisions. We prove an existence theorem: Theorem 3.7 for these systems. Unfortunately,we could not prove uniqueness: we just construct a copy of an infinite ranked system using ap-proximation by finite ranked systems. This copy is called an approximative version of the infiniteranked system. We also develop comparison techniques for infinite systems, which parallel similartechniques for finite systems from [30].Assume the usual setting: (Ω , F , ( F t ) t ≥ , P ), with the filtration satisfying the usual conditions.3.1. Infinite classical systems.
Fix parameters g , g , . . . ∈ R and σ , σ , . . . >
0. We say thata sequence ( x n ) n ≥ of real numbers is rankable if there exists a one-to-one mapping (permutation) p : { , , , . . . } → { , , , . . . } which ranks the components of x : x p ( i ) ≤ x p ( j ) for i, j = 1 , , . . . , i < j. As in the case of finite systems, we resolve ties (when x i = x j for i = j ) in the lexicographicorder: we take a permutation p which ranks the components of x , and, in addition, if i < j and x p ( i ) = x p ( j ) , then p ( i ) < p ( j ). There exists a unique such permutation p , which is called theranking permutation and is denoted by p x . For example, if x = (2 , , , , , , , . . . ) ′ (that is, x ( i ) = i for i ≥ p x (1) = 3 , p x (2) = 1 , p x (3) = 2 , p x ( n ) = n, n ≥
4. Not all sequencesof real numbers are rankable: for example, x = ( x i = i − , i ≥ Definition 4.
Consider an R ∞ -valued process X = ( X ( t ) , t ≥ , X ( t ) = ( X n ( t )) n ≥ , with continuous adapted components, such that for every t ≥
0, the sequence X ( t ) = ( X n ( t )) n ≥ is rankable. Let p t be the ranking permutation of X ( t ). Let W , W , . . . be i.i.d. standard ( F t ) t ≥ -Brownian motions. Assume that the process X satisfies an SDE dX i ( t ) = ∞ X k =1 p t ( k ) = i ) ( g k d t + σ k d W i ( t )) , i = 1 , , . . . Then the process X is called an infinite classical system of competing Brownian particles with drift coefficients ( g k ) k ≥ and diffusion coefficients ( σ k ) k ≥ . For each i = 1 , , . . . , the component X i = ( X i ( t ) , t ≥
0) is called the i th named particle . If we define Y k ( t ) ≡ X p t ( k ) ( t ) for t ≥ k = 1 , , . . . , then the process Y k = ( Y k ( t ) , t ≥
0) is called the k th ranked particle . The R ∞ + -valuedprocess Z = ( Z ( t ) , t ≥ , Z ( t ) = ( Z k ( t )) k ≥ , NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 9 defined by Z k ( t ) = Y k +1 ( t ) − Y k ( t ) , k = 1 , , . . . , t ≥ , is called the gap process . If X (0) = x ∈ R ∞ , then we say that the system X starts from x . Thissystem is called locally finite if for any u ∈ R and T > i ≥ [0 ,T ] X i ( t ) ≤ u .The following existence and uniqueness theorem was partially proved in [18] and [35]. We restateit here in a different form. Theorem 3.1.
Suppose x ∈ R ∞ is a vector which satisfies the condition (5) . Assume also thatthere exists n ≥ for which g n +1 = g n +2 = . . . and σ n +1 = σ n +2 = . . . > . Then, in a weak sense there exists an infinite classical system of competing Brownian particleswith drift coefficients ( g k ) k ≥ and diffusion coefficients ( σ k ) k ≥ , starting from x , and it is uniquein law. Let us also show a different existence and uniqueness result, analogous to [27, Lemma 11].
Theorem 3.2.
Suppose x ∈ R ∞ is a vector which satisfies the condition (5) . Assume also that σ n = 1 , n ≥ and G := ∞ X n =1 g n < ∞ . Then in a weak sense there exists an infinite classical system of competing Brownian particles withdrift coefficients ( g k ) k ≥ and diffusion coefficients ( σ k ) k ≥ , starting from x , and it is unique in law. Now, let us define an approximative version of an infinite classical system. Fix parameters( g n ) n ≥ and ( σ n ) n ≥ and an initial condition x = ( x i ) i ≥ . For each N ≥
1, consider a finite systemof N competing Brownian particles X ( N ) = (cid:16) X ( N )1 , . . . , X ( N ) N (cid:17) ′ with drift coefficients ( g n ) ≤ n ≤ N and diffusion coefficients ( σ n ) ≤ n ≤ N , starting from [ x ] N . Let Y ( N ) = (cid:16) Y ( N )1 , . . . , Y ( N ) N (cid:17) ′ be the ranked version of this system. Take an increasing sequence ( N j ) j ≥ . Definition 5.
Consider a version of the infinite classical system X = ( X i ) i ≥ of competing Brown-ian particles with parameters ( g n ) n ≥ , ( σ n ) n ≥ , starting from x . Let Y k be the k th ranked particle.Take an increasing sequence ( N j ) j ≥ of positive integers. Assume for every T > M ≥ C ([0 , T ] , R M ), we have: (cid:16) X ( N j )1 , . . . , X ( N j ) M , Y ( N j )1 , . . . , Y ( N j ) M (cid:17) ′ ⇒ ( X , . . . , X M , Y , . . . , Y M ) ′ . Then X is called an approximative version of this infinite classical system, corresponding to the approximation sequence ( N j ) j ≥ .We prove weak existence (but not uniqueness in law) under the following conditions, which areslightly more general than the ones in Theorem 3.1 and Theorem 3.2. Theorem 3.3.
Consider parameters ( g n ) n ≥ and ( σ n ) n ≥ which satisfy (17) g := sup n ≥ | g n | < ∞ , and σ := sup n ≥ σ n < ∞ . Take initial conditions x = ( x i ) i ≥ satisfying the conditions (5) . Fix an increasing sequence ( N j ) j ≥ . Then there exists a subsequence ( N ′ j ) j ≥ which serves as an approximation sequencefor an approximative version X of the infinite classical system of competing Brownian particleswith parameters ( g n ) n ≥ , ( σ n ) n ≥ , starting from x . This infinite classical system has the following properties.
Lemma 3.4.
Consider any infinite classical system X = ( X i ) i ≥ , of competing Brownian particleswith parameters ( g n ) n ≥ , ( σ n ) n ≥ , satisfying the condition (17) . Assume the initial condition X (0) = x satisfies (5) . Then this system is locally finite. Also, the following set is the state spacefor X = ( X ( t ) , t ≥ : V := (cid:8) x = ( x i ) i ≥ ∈ R ∞ | lim i →∞ x i = ∞ and ∞ X i =1 e − αx i < ∞ for all α > (cid:9) . . Now, let us describe the dynamics of the ranked particles Y k . Denote by L ( k,k +1) the local timeprocess at zero of Z k , k = 1 , , . . . For notational convenience, let L (0 , ( t ) ≡
0. For k = 1 , , . . . and t ≥
0, let B k ( t ) = ∞ X i =1 Z t p s ( k ) = i )d W i ( s ) . Lemma 3.5.
Take a version of an infinite classical system of competing Brownian particles withparameters ( g n ) n ≥ and ( σ n ) n ≥ . Assume this version is locally finite. Then the processes B k =( B k ( t ) , t ≥ , k = 1 , , . . . are i.i.d. standard Brownian motions. For t ≥ and k = 1 , , . . . , wehave: (18) Y k ( t ) = Y k (0) + g k t + σ k B k ( t ) − L ( k,k +1) ( t ) + 12 L ( k − ,k ) ( t ) . Lemma 3.6.
Under conditions of Lemma , for every t > there is a.s. no tie at time t > . Infinite systems with asymmetric collisions.
Lemma 3.5 provides motivation to intro-duce infinite systems of competing Brownian particles with asymmetric collisions , when we havecoefficients other than 1 / approximativeversion of the infinite ranked system. Definition 6.
Fix parameters g , g , . . . ∈ R , σ , σ , . . . > q ± n ) n ≥ such that q + n +1 + q − n = 1 , < q ± n < , n = 1 , , . . . Take a sequence of i.i.d. standard ( F t ) t ≥ -Brownian motions B , B , . . . Consider an R ∞ -valuedprocess Y = ( Y ( t ) , t ≥
0) with continuous adapted components and continuous adapted real-valued processes L ( k,k +1) = ( L ( k,k +1) ( t ) , t ≥ , k = 1 , , . . . (for convenience, let L (0 , ≡ Y ( t ) ≤ Y ( t ) ≤ Y ( t ) ≤ . . . for t ≥ NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 11 (ii) for k = 1 , , . . . , t ≥
0, we have: Y k ( t ) = Y k (0) + g k t + σ k B k ( t ) + q + k L ( k − ,k ) ( t ) − q − k L ( k,k +1) ( t );(iii) each process L ( k,k +1) is nondecreasing, L ( k,k +1) (0) = 0 and Z ∞ ( Y k +1 ( t ) − Y k ( t )) d L ( k,k +1) ( t ) = 0 , k = 1 , , . . . The last equation means that L ( k,k +1) can increase only when Y k ( t ) = Y k +1 ( t ).Then the process Y is called an infinite ranked system of competing Brownian particles with drift coefficients ( g k ) k ≥ , diffusion coefficients ( σ k ) k ≥ , and parameters of collisions ( q ± k ) k ≥ . Theprocess Y k = ( Y k ( t ) , t ≥
0) is called the k th ranked particle . The R ∞ + -valued process Z = ( Z ( t ) , t ≥ , Z ( t ) = ( Z k ( t )) k ≥ , defined by Z k ( t ) = Y k +1 ( t ) − Y k ( t ) , k = 1 , , . . . , t ≥ , is called the gap process . The process L ( k,k +1) is called the local time of collision between Y k and Y k +1 . If Y (0) = y , then we say that this system Y starts from y . The processes B , B , . . . arecalled driving Brownian motions . The system Y = ( Y k ) k ≥ is called locally finite if for all u ∈ R and T > k such that min [0 ,T ] Y k ( t ) ≤ u . Remark . We can reformulate Lemma 3.5 as follows: take an infinite classical system X = ( X i ) i ≥ of competing Brownian particles with drift coefficients ( g n ) n ≥ and diffusion coefficients ( σ n ) n ≥ .Rank this system X ; in other words, switch from named particles X , X , . . . , to ranked particles Y , Y , . . . . The resulting system Y = ( Y k ) k ≥ is an infinite ranked system of competing Brownianparticles with drift coefficients ( g n ) n ≥ , diffusion coefficients ( σ n ) n ≥ , and parameters of collision q ± n = 1 /
2, for n ≥ Definition 7.
Using the notation from Definition 6, for every N ≥
2, let Y ( N ) = (cid:16) Y ( N )1 , . . . , Y ( N ) N (cid:17) ′ be the system of N ranked competing Brownian particles with drift coefficients g , . . . , g N , diffu-sion coefficients σ , . . . , σ N and parameters of collision ( q ± n ) ≤ n ≤ N , driven by Brownian motions B , . . . , B N . Suppose there exist limitslim N →∞ Y ( N ) k ( t ) =: Y k ( t ) , which are uniform on every [0 , T ], for every k = 1 , , . . . Assume that Y = ( Y k ) k ≥ turns out tobe an infinite system of competing Brownian particles with parameters ( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ .Then we say that Y is an approximative version of this system. Remark . From Theorem 3.3, Lemma 3.4, and Lemma 3.5, we know that if we take an approxi-mative version of an infinite classical system of competing Brownian particles and rank it, we getthe approximative version of an infinite ranked system. This allows us to use subsequent resultsof Sections 3, 4, and 5 for approximative versions of infinite classical systems. In particular, if(under conditions of Theorem 3.1 or Theorem 3.2) there is a unique in law version of an infiniteclassical system, then this only version is necessarily the approximative version, and we can applyresults of Sections 3, 4, and 5 to this system.
Now comes the main result of this subsection.
Theorem 3.7.
Take a sequence of drift coefficients ( g n ) n ≥ , a sequence of diffusion coefficients ( σ n ) n ≥ , and a sequence of parameters of collision ( q ± n ) n ≥ . Suppose that the initial conditions y ∈ R ∞ are such that y ≤ y ≤ . . . , and ∞ X n =1 e − αy n < ∞ for all α > . Assume that (19) inf n ≥ g n =: g > −∞ , sup n ≥ σ n =: σ < ∞ , and there exists n ≥ such that (20) q + n ≥ for n ≥ n . Take any i.i.d. standard Brownian motions B , B , . . . Then there exists the approximative versionof the infinite ranked system of competing Brownian particles with parameters ( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ , starting from y , with driving Brownian motions B , B , . . . Remark . We have not proved uniqueness for infinite ranked system from Theorem 3.7. Wecan so far only claim uniqueness for infinite classical systems. Now, suppose we take the infiniteranked system from Theorem 3.7 with symmetric collisions, when q ± n = 1 / n . Under theadditional assumption that this system must be the result of ranking a classical system, we alsoget uniqueness. But without this special condition, it is not known whether this ranked system isunique.Let us now present some additional properties of this newly constructed approximative versionof an infinite system of competing Brownian particles. These are analogous to the properties ofan infinite classical system of competing Brownian particles, stated in Lemma 3.4 and Lemma 3.6above. Lemma 3.8.
An approximative version of an infinite ranked system from Theorem is locallyfinite. The process Y = ( Y ( t ) , t ≥ has the state space W := (cid:8) y = ( y k ) k ≥ ∈ R ∞ | y ≤ y ≤ y ≤ . . . , lim k →∞ y k = ∞ , ∞ X k =1 e − αy k < ∞ , for all α > (cid:9) . Lemma 3.9.
Consider an infinite system from Definition , which is locally finite. Then for every t > a.s. the vector Y ( t ) = ( Y k ( t )) k ≥ has no ties. Comparison techniques for infinite systems.
We developed comparison techniques forfinite systems of competing Brownian particles in [30]. These techniques also work for approxi-mative versions of infinite ranked systems. By taking limits as the number N of particles goes toinfinity, we can formulate the same comparison results for these two infinite systems. Let us givea few examples. The proofs trivially follow from the corresponding results for finite systems from[30, Section 3]. These techniques are used later in Section 4 of this article, as well as in proofs ofstatements from Section 3. NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 13
Corollary 3.10.
Take two approximative versions Y and Y of an infinite system of competingBrownian particles with the same parameters ( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ , with the same driving Brownian motions, but starting from different initial conditions Y (0) and Y (0) . Let Z and Z be the corresponding gap processes, and let L and L be the correspondingvectors of local time terms. Then the following inequalities hold a.s.:(i) If Y (0) ≤ Y (0) , then Y ( t ) ≤ Y ( t ) , t ≥ .(ii) If Z (0) ≤ Z (0) , then Z ( t ) ≤ Z ( t ) , t ≥ , and L ( t ) − L ( s ) ≥ L ( t ) − L ( s ) , ≤ s ≤ t . Corollary 3.11.
Fix M ≥ . Take two approximative versions Y = ( Y n ) n ≥ M and Y = ( Y n ) n ≥ of an infinite system of competing Brownian particles with parameters ( g n ) n ≥ M , ( σ n ) n ≥ M , ( q ± n ) n ≥ M ;( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ . Assume that Y k (0) = Y k (0) for k ≥ M . If B , B , . . . are driving Brownian motions for Y , thenlet B M , B M +1 , . . . be the driving Brownian motions for Y . Let Z = ( Z k ) k ≥ M and Z = ( Z k ) k ≥ bethe corresponding gap processes, and let L = ( L ( k,k +1) ) k ≥ M and L = ( L ( k,k +1) ) k ≥ be the vectors ofboundary terms. Then a.s. the following inequalities hold: Y k ( t ) ≤ Y k ( t ) , k ≥ M, t ≥ L ( k,k +1) ( t ) − L ( k,k +1) ( s ) ≤ L ( k,k +1) ( t ) − L ( k,k +1) ( s ) , ≤ s ≤ t, k ≥ M ; Z k ( t ) ≥ Z k ( t ) , t ≥ , k ≥ M. Corollary 3.12.
Take two approximative versions Y and Y of an infinite system of competingBrownian particles with parameters ( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ ;( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ , with the same driving Brownian motions, starting from the same initial conditions. Let Z and Z be the corresponding gap processes. Then:(i) If q ± n = q ± n , but g n ≤ g n for n = 1 , , . . . , then Y ( t ) ≤ Y ( t ) , t ≥ ;(ii) If q ± n = q ± n , but g n +1 − g n ≤ g n +1 − g n for n = 1 , , . . . , then Z ( t ) ≤ Z ( t ) , t ≥ ;(iii) If g n = g n , but q + n ≤ q + n for n = 1 , , . . . , then Y ( t ) ≤ Y ( t ) , t ≥ .Remark . Suppose that in each of these three corollaries, we remove the requirement that thetwo infinite systems have the same driving Brownian motions. Then we get stochastic orderinginstead of pathwise ordering. The same applies to Corollary 3.10 if we switch from a.s. comparisonto stochastic comparison in the inequalities Y (0) ≤ Y (0) and Z (0) ≤ Z (0), respectively.4. The Gap Process: Stationary Distributions and Weak Convergence
In this section, we construct a stationary distribution π for the gap process Z = ( Z ( t ) , t ≥ Z ( t ) as t → ∞ . Construction of a stationary distribution.
Consider again an infinite system Y of com-peting Brownian particles with parameters ( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ . Let Z be its gap process.Let us recall a definition from the Introduction. Definition 8.
Let π be a probability measure on R ∞ + . We say that π is a stationary distribution for the gap process for the system above if there exists a version Y of this system such that forevery t ≥
0, we have: Z ( t ) ∼ π .Let us emphasize that in this section, we do not study uniqueness and Markov property. Wesimply construct a copy of the system with required properties. Assumption 1.
Consider, for each N ≥
2, the ranked system of N competing Brownian particleswith parameters ( g n ) ≤ n ≤ N , ( σ n ) ≤ n ≤ N , ( q ± n ) ≤ n ≤ N . There exists a sequence ( N j ) j ≥ such that N j → ∞ and for every j ≥
1, the system of N = N j particles is such that its gap process has astationary distribution. Let π ( N j ) be this stationary distribution on R N j − .Define an ( N − × ( N − R ( N ) and a vector µ ( N ) from R N − , as in (14) and (15). ByProposition 2.1, Assumption 1 holds if and only if[ R ( N j ) ] − µ ( N j ) < . Let B , B , . . . be i.i.d. standard Brownian motions. Let z ( N j ) ∼ π ( N j ) be an F -measurablerandom variable. Consider the system Y ( N j ) of N j ranked competing Brownian particles withparameters ( g n ) ≤ n ≤ N j , ( σ n ) ≤ n ≤ N j , ( q ± n ) ≤ n ≤ N j , starting from (0 , z ( N j )1 , . . . , z ( N j )1 + . . . + z ( N j ) N j − ) ′ , with driving Brownian motions B , . . . , B N j . The following statement, which we state separatelyas a lemma, is a direct corollary of [30, Corollary 3.14]. Lemma 4.1. [ π ( N j +1 ) ] N j − (cid:22) π ( N j ) . Without loss of generality, by changing the probability space we can take z ( N j ) ∼ π ( N j ) suchthat a.s. [ z ( N j +1 ) ] N j − ≤ z ( N j ) , for j ≥
1. In other words, for all j = 1 , , . . . and k = 1 , . . . , N j − ≤ z ( N j +1 ) k ≤ z ( N j ) k . A bounded monotone sequence has a limit: z k = lim j →∞ z ( N j ) k , k ≥ . Denote by π the distribution of ( z , z , . . . ) on R ∞ + . Then π becomes a prospective stationary dis-tribution for the gap process for the infinite system of competing Brownian particles. Equivalently,we can define π as follows: for every M ≥
1, let[ π ( N j ) ] M ⇒ ρ ( M ) , j → ∞ . These finite-dimensional distributions ρ ( M ) are consistent :[ ρ ( M +1) ] M = ρ ( M ) , M ≥ . By Kolmogorov’s theorem there exists a unique distribution π on R ∞ + such that [ π ] M = ρ ( M ) for all M ≥
1. Note that this limiting distribution does not depend on the sequence ( N j ) j ≥ , as shownin the next lemma. NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 15
Lemma 4.2.
If there exist two sequences ( N j ) j ≥ and ( ˜ N j ) j ≥ which satisfy Assumption 1, and if π and ˜ π are the resulting limiting distributions, then π = ˜ π . The next lemma allows us to rewrite the condition (5) in terms of the gap process.
Lemma 4.3.
For a sequence y = ( y n ) n ≥ ∈ R ∞ such that y n ≤ y n +1 , n ≥ , let z = ( z n ) n ≥ ∈ R ∞ be defined by z n = y n +1 − y n , n ≥ . Then y satisfies (5) if and only if z satisfies (21) ∞ X n =1 exp (cid:0) − α ( z + . . . + z n ) (cid:1) < ∞ for all α > . Now, let us state one of the two main results of this section.
Theorem 4.4.
Consider an infinite system of competing Brownian particles with parameters ( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ . (i) Let the Assumption 1 and (19) , (20) be true. Then we can construct the distribution π .(ii) Assume, in addition, that if a R ∞ + -valued random variable z is distributed according to π ,then z = ( z , z , . . . ) a.s. satisfies (21) . Then we can construct an approximative version of theinfinite system of competing Brownian particles with parameters ( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ , such that π is a stationary distribution for the gap process.Remark . As mentioned in the Introduction, if a stationary distribution for the gap process of finite systems exists, it is unique. This was proved in [3]. For infinite systems, this is an openquestion.In this subsection, we apply Theorem 4.4 to the case of the skew-symmetry condition , similarto (16):(22) ( q − k − + q + k +1 ) σ k = q − k σ k +1 + q + k σ k − , k = 2 , , . . . Under this condition, by Proposition 2.1, π ( N j ) = N j − O k =1 Exp( λ ( N j ) k ) , where we define for k = 1 , . . . , N j − λ ( N j ) k = 2 σ k + σ k +1 (cid:0) − [ R ( N j ) ] − µ ( N j ) (cid:1) k . Consider the following marginal of this stationary distribution:[ π ( N j +1 ) ] N j − = N j − O k =1 Exp( λ ( N j +1 ) k ) . By Lemma 4.1, we can compare:[ π ( N j +1 ) ] N j − (cid:22) π ( N j ) = N j − O k =1 Exp( λ ( N j ) k ) . But Exp( λ ′ ) (cid:22) Exp( λ ′′ ) is equivalent to λ ′ ≥ λ ′′ . Therefore, λ ( N j ) k ≤ λ ( N j +1 ) k , for k = 1 , . . . , N j − k , the sequence ( λ ( N j ) k ) is nondecreasing. There exists a limit (possiblyinfinite) λ k := lim j →∞ λ ( N j ) k , k = 1 , , . . . Assume that λ k < ∞ for all k = 1 , , . . . Then(23) π = ∞ O k =1 Exp( λ k ) . If λ k = ∞ for some k , then we can also write (23), understanding that Exp( ∞ ) = δ is the Diracpoint mass at zero. This π is a candidate for a stationary distribution. If the condition (21)is satisfied π -a.s., then π is, indeed, a stationary distribution. Let us give a sufficient conditionfor (21). Lemma 4.5.
Consider a distribution π as in (23) . Let Λ n := P nk =1 λ − k .(i) If sup n ≥ λ n < ∞ , then π -a.s. (21) is satisfied.(ii) If P ∞ n =1 λ − n < ∞ , then π -a.s. (21) is satisfied if and only if (24) ∞ X n =1 e − α Λ n < ∞ for all α > . The case of symmetric collisions.
Assume now that the collisions are symmetric: q ± n =1 / n = 1 , , . . . Then the skew-symmetry condition (22) takes the form σ k +1 − σ k = σ k − σ k − ,for k = 2 , , . . . . In other words, σ k must linearly depend on k . If, in addition, (19) holds, then σ k = σ , k = 1 , , . . . Recall the definition of g k from (3). It was shown in Proposition 2.2 that inthis case, [ R ( N j ) ] − µ ( N j ) < g k > g N j , k = 1 , . . . , N j − . If the inequality (25) is true for j = 1 , , . . . , then π ( N j ) = N j − O k =1 Exp (cid:16) λ ( N j ) k (cid:17) , λ ( N j ) k := 2 kσ (cid:16) g k − g N j (cid:17) . Assume the sequence ( g n ) n ≥ is bounded from below, as in (19). Then the sequence ( g N j ) j ≥ isalso bounded below. From (25), we get: g N j > g N j +1 for j = 1 , , . . . . Therefore, there exists thelimit lim j →∞ g N j =: g ∞ . Then, as j → ∞ , we get: λ ( N j ) k → λ k := 2 kσ ( g k − g ∞ ) . Thus, the distribution π has the following product-of-exponentials form:(26) π = ∞ O k =1 Exp( λ k ) = ∞ O k =1 Exp (cid:18) kσ ( g k − g ∞ ) (cid:19) . If λ k , k = 1 , , . . . , satisfy Lemma 4.5, then π is a stationary distribution. NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 17
Example . Consider an infinite system with symmetric collisions, with drift and diffusion coeffi-cients g , g , . . . , g M > , g M +1 = g M +2 = . . . = 0 , σ = σ = . . . = 1 . Then g k = g + . . . + g M k , k > M. Therefore, g ∞ = lim k →∞ g k = 0, and the parameters λ k from (26) are equal to λ k = ( g + . . . + g k ) , ≤ k ≤ M ;2( g + . . . + g M ) , k > M. These parameters satisfy Lemma 4.5 (i). Therefore, the conclusions of this section are valid. Inparticular, if g = . . . = g M = 1, as in Theorem 1.1 from the Introduction, then π = Exp(2) ⊗ Exp(4) ⊗ . . . ⊗ Exp(2 M ) ⊗ Exp(2 M ) ⊗ . . . Convergence Results.
Now, consider questions of convergence of the gap process as t →∞ to the stationary distribution π constructed above. Let us outline the facts proved in thissubsection (omitting the required conditions for now).(a) The family of random variables Z ( t ) , t ≥
0, is tight in R ∞ + with respect to the componentwiseconvergence (which is metrizable by a certain metric). Any weak limit point of Z ( t ) as t → ∞ isstochastically dominated by π .(b) If we start the approximative version of the infinite system Y with gaps stochastically largerthan π , then the gap process converges weakly to π .(c) Any other stationary distribution for the gap process (if it exists) must be stochasticallysmaller than π .The rest of this subsection is devoted to precise statements of these results. Theorem 4.6.
Consider any version (not necessarily approximative) of the infinite system ofcompeting Brownian particles with parameters ( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ . Suppose Assumption 1 holds.(i) Then the family of R ∞ + -valued random variables Z ( t ) , t ≥ is tight in R ∞ + .(ii) Suppose for some sequence t j ↑ ∞ we have: Z ( t j ) ⇒ ν as j → ∞ , where ν is someprobability measure on R ∞ + . Then ν (cid:22) π : the measure ν is stochastically dominated by π .(iii) Under conditions of Theorem (ii), every stationary distribution π ′ for the gap processis stochastically dominated by π : π ′ (cid:22) π .Remark . Let us stress: we do not need Y to be an approximative version of the system, and wedo not need the initial conditions Y (0) = y to satisfy (5). Theorem 4.7.
Consider an approximative version Y of the infinite system of competing Brownianparticles with parameters ( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ . Let Z be the corresponding gap process.Suppose it satisfies conditions of Theorem (ii). Then we can construct the distribution π , andit is a stationary distribution for the gap process. If Z (0) (cid:23) π , then Z ( t ) ⇒ π, t → ∞ . Proof.
Let us show that for each t ≥ Z ( t ) (cid:23) π . (Together with Theorem 4.6 (i), (ii), thiscompletes the proof.) Consider another system Y : an approximative version of the system withthe gap process Z having stationary distribution π . Then Z (0) (cid:23) Z (0) ∼ π . By Corollary 3.10(ii) above, Z ( t ) (cid:23) Z ( t ) ∼ π, t ≥ (cid:3) Triple Collisions for Infinite Systems
Let us define triple and simultaneous collisions for an infinite ranked system Y = ( Y n ) n ≥ ofcompeting Brownian particles. Definition 9.
We say that a triple collision between particles Y k − , Y k and Y k +1 occurs at time t ≥ Y k − ( t ) = Y k ( t ) = Y k +1 ( t ) . We say that a simultaneous collision occurs at time t ≥ ≤ k < l , we have: Y k ( t ) = Y k +1 ( t ) and Y l ( t ) = Y l +1 ( t ) . A triple collision is a particular case of a simultaneous collision. For finite systems of competingBrownian particles (both classical and ranked), the question of a.s. absence of triple collisions wasstudied in [17, 18, 22]. A necessary and sufficient condition for a.s. absence of any triple collisionswas found in [32]; see also [5] for related work. This condition also happens to be sufficient fora.s. absence of any simultaneous collisions. In general, triple collisions are undesirable, becausestrong existence and pathwise uniqueness for classical systems of competing Brownian particleswas shown in [18] only up to the first moment of a triple collision. Some results about triplecollisions for infinite classical systems were obtained in the paper [18]. Here, we strengthen thema bit and also prove results for asymmetric collisions.It turns out that the same necessary and sufficient condition works for infinite systems as wellas for finite systems.
Theorem 5.1.
Consider a version of the infinite ranked system of competing Brownian particles Y = ( Y n ) n ≥ with parameters ( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n ) n ≥ . (i) Assume this version is locally finite. If (27) ( q − k − + q + k +1 ) σ k ≥ q − k σ k +1 + q + k σ k − , k = 2 , , . . . Then a.s. for any t > there are no triple and no simultaneous collisions at time t .(ii) If the condition (27) is violated for some k = 2 , , . . . , then with positive probability thereexists a moment t > such that there is a triple collision between particles with ranks k − , k ,and k + 1 at time t . An interesting corollary of [32, Theorem 1.2] for finite systems is that if there are a.s. no triplecollisions, then there are also a.s. no simultaneous collisions. This is also true for infinite systemsconstructed in Theorem 3.7.
Remark . For symmetric collisions: q ± n = 1 / , n = 1 , , . . . , this result takes the following form.There are a.s. no triple collisions if and only if the sequence ( σ k ) k ≥ is concave. In this case, thereare also a.s. no simultaneous collisions. If for some k ≥ σ k +1 < (cid:0) σ k + σ k +2 (cid:1) , then with positive probability there exists t > Y k ( t ) = Y k +1 ( t ) = Y k +2 ( t ). NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 19
Remark . Let us restate the main result of [18]: for a infinite classical systems of competingBrownian particles which satisfies conditions of Theorem 3.1, there exists a unique strong versionup to the first triple collision. In particular, if the sequence of diffusion coefficients ( σ k ) k ≥ isconcave, then there exists a unique strong solution on the infinite time horizon. Remark . Partial results of [18] for infinite classical systems of competing Brownian particles areworth mentioning: if there are a.s. no triple collisions, then ( σ k ) k ≥ is concave; if the sequence(0 , σ , σ , . . . ) is concave, then there are a.s. no triple collisions. In particular, it was alreadyshown in [18] that the model (7), as any model with σ = σ = . . . = 1, a.s. does not have triplecollisions. 6. Proofs
Proof of Proposition 2.1.
The concept of a semimartingale reflected Brownian motion (SRBM) in the positive orthant R d + is discussed in the survey [39]; we refer the reader to thisarticle for definition and main known results about this process. Here, we informally introducethe concept. Take a d × d -matrix R with diagonal elements equal to 1, and denote by r i the i thcolumn of R . Next, take a symmetric positive definite d × d -matrix A , as well as µ ∈ R d . A semimartingale reflected Brownian motion (SRBM) in the orthant with drift vector µ , covariancematrix A , and reflection matrix R is a Markov process in R d + such that:(i) when it is in the interior of the orthant, it behaves as a d -dimensional Brownian motion withdrift vector µ and covariance matrix A ;(ii) at each face { x ∈ R d + | x i = 0 } of the boundary of this orthant, it is reflected instantaneouslyaccording to the vector r i (if r i = e i , which is the i th standard unit vector in R d , this is normal reflection).It turns out that Z is an SRBM in the orthant R N − with reflection matrix R given by (14),drift vector µ as in (15), and covariance matrix(28) A = σ + σ − σ . . . − σ σ + σ − σ . . . − σ σ + σ − σ . . . . . . σ N − + σ N − − σ N − . . . − σ N − σ N − + σ N See [22, subsection 2.1], [32, 3]. The results of Proposition 2.1 follow from the properties of anSRBM. Property (i) of the matrix R was proved in [22, subsection 2.1]; see also [32, Lemma 2.9].The skew-symmetry condition for an SRBM is written in the form RD + DR ′ = 2 A, where D = diag( A ) is the ( N − × ( N − A .As mentioned in [39, Theorem 3.5], this is a necessary and sufficient condition for the stationarydistribution to have product-of-exponentials form. This condition can be rewritten for R and A from (14) and (28) as (16). Proof of Lemma 2.3.
There is a tie for a system of competing Brownian particles at time t > t hits the boundary of the orthant R N − . But the gapprocess is an SRBM Z = ( Z ( t ) , t ≥
0) in R N − , with the property from [28]: P ( Z ( t ) ∈ ∂ R N − ) = 0for every t > Proof of Theorem 3.1.
Because of the results of [18], we need only to prove the followingcondition. Fix
T > x ∈ R . Let Ξ be the set of all progressively measurable real-valuedprocesses ζ = ( ζ ( t )) ≤ t ≤ T with values in [min i ≥ σ i , max i ≥ σ i ]. Then for every ζ ∈ Ξ,(29) ∞ X i =1 sup ξ ∈ Ξ P (cid:18) x i − gT − max ≤ t ≤ T Z t ζ ( s )d W i ( s ) < x (cid:19) < ∞ , But this follows from Lemma 7.2 and Lemma 7.1.6.4.
Proof of Theorem 3.2.
The proof closely follows that of [27, Lemma 11]. Assume withoutloss of generality that initially, the particles are ranked, that is, x k ≤ x k +1 for k ≥
1. Consideri.i.d. standard Brownian motions W , W , . . . , and let X i ( t ) = x i + W i ( t ) , i ≥ Lemma 6.1.
For every t ≥ , the sequence X ( t ) = ( X n ( t )) n ≥ is rankable.Proof. It suffices to show that the system X is locally finite. This statement follows from Lem-mata 7.2, 7.1, the Borel-Cantelli lemma and the fact that the initial condition x satisfies (5). (cid:3) Recall our standard setting: (Ω , F , ( F t ) t ≥ , P ). Let p t be the ranking permutation of the se-quence X ( t ). Fix T > X = ( X n ) n ≥ on F T . We construct thenew measure Q | F t = D ( t ) · P | F t , where D ( t ) := exp (cid:16) M ∞ ( t ) − h M ∞ i t (cid:17) , t ≥ , and(30) M ∞ ( t ) = ∞ X i =1 ∞ X k =1 Z t g k p s ( k ) = i ) d W i ( s ) . It suffices to show that the process M ∞ exists and is a continuous square-integrable martingale,with h M ∞ i t = Gt for all t ≥
0. Indeed, the rest follows from Girsanov theorem. Fix
T > M of continuous square-integrable martingales M = ( M ( t ) , ≤ t ≤ T ),starting from M (0) = 0. This is a Hilbert space with the following inner product and norm:( M ′ , M ′′ ) := E h M ′ , M ′′ i T , and k M k := [ E h M i T ] / . For each i, k = 1 , , . . . , define M i,k ( t ) := Z t g k p s ( k ) = i ) d W i ( s ) , t ≥ . Then the process M ∞ from (30) can be represented as(31) M ∞ ( t ) = ∞ X i =1 ∞ X k =1 M i,k ( t ) , t ≥ . Lemma 6.2.
All processes M i,k , i, k = 1 , , . . . , are elements of the space M and are orthogonalin this space. NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 21
Proof.
That each of these processes is a continuous square-integrable martingale is straightforward.Let us show that ( M i ′ ,k ′ , M i ′′ ,k ′′ ) = 0 when i ′ = i ′′ or k ′ = k ′′ . Indeed, for i ′ = i ′′ , this follows fromthe fact that the Brownian motions W i ′ and W i ′′ are independent, and therefore, h W i ′ , W i ′′ i s ≡ i ′ = i ′′ = i and k ′ = k ′′ , this follows from an observation that the mapping p s : { , , . . . } →{ , , . . . } is one-to-one for every s ≥
0, and therefore1 ( p s ( k ′ ) = i ) 1 ( p s ( k ′′ ) = i ) ≡ . (cid:3) It is easy to see that(32) ∞ X i =1 ∞ X k =1 k M i,k k = ∞ X i =1 ∞ X k =1 Z T g k p s ( k ) = i ) d s = T ∞ X k =1 g k = T G.
From (32) and Lemma 6.2, we get that the series (31) converges in the space M , which provesthat M ∞ is a continuous square-integrable martingale. The calculation similar to the one in (32)with t instead of T shows that h M ∞ i t ≡ Gt . This completes the proof of Theorem 3.2.6.5. Proof of Lemma 3.4.
Parts of this result were already proved in [18] for (slightly morerestrictive) conditions of Theorem 3.1. We can write each X i in the form X i ( t ) = x i + Z t β i ( s )d s + Z t ρ i ( s )d W i ( s ) , t ≥ , where the drift and diffusion coefficients β i ( t ) = ∞ X k =1 p t ( k ) = i ) g k , ρ i ( t ) = ∞ X k =1 p t ( k ) = i ) σ k satisfy the following inequalities: | β i ( t ) | ≤ g, | ρ i ( t ) | ≤ σ, ≤ t ≤ T. There exists a random but a.s. finite i such that for i ≥ i we have: x i > gT + u . For these i , byLemma 7.2 we have: P (cid:18) min ≤ t ≤ T X i ( t ) ≤ u (cid:19) ≤ (cid:18) x i − gT − uσ √ T (cid:19) . Apply Lemma 7.1 and the Borel-Cantelli lemma and finish the proof of the local finiteness. Now,let us show that a.s. there exist only finitely many i such that X i ( t ) ≤ x i /
2. There exists arandom but a.s. finite i such that for i ≥ i we have: x i / > gT . Then x i > x i / gT for these i . For i ≥ i ∨ i , by Lemma 7.2 we have: P ( X i ( t ) ≤ x i / ≤ P (cid:18) min ≤ s ≤ t X i ( s ) ≤ x i / (cid:19) ≤ (cid:18) x i − x i / − gTσ √ T (cid:19) . Apply Lemma 7.1 and the Borel-Cantelli lemma. This proves that there exists a random but a.s.finite i ≥ i ∨ i such that X i ( t ) ≥ x i / i ≥ i . Thus, for i ≥ i , we have: X i ( t ) ≥ x i / ≥ ∞ X i = i e − αX i ( t ) ≤ ∞ X i = i e − α ( x i / < ∞ . Because i is a.s. finite, this completes the proof. Proof of Lemma 3.5.
This statement follows from similar statement for finite systems(see (11)). Indeed, take the k th ranked particle Y k and let u := max [0 ,T ] Y k + 1. Let us show thatfor every t ∈ [0 , T ] there exists a (possibly random) neighborhood of t in [0 , T ] such that (18)holds. The statement of Lemma 3.5 would then follow from compactness of [0 , T ] and the factthat T > i such that min [0 ,T ] X i > u for i > i . Take the minimal such i . Then,take m > k and assume the event { i ≤ m } happened. Fix time t ∈ [0 , T ]. We claim that if Y k doesnot collide at time t with other particles, then there exists a (random) neighborhood when Y k doesnot collide with other particles. Indeed, particles X i , i > m , cannot collide with Y k , by definitionof u and i . And for every particle X i , i = 1 , . . . , m , other than Y k (say Y k has name j at time t ), there exists a (random) open time neighborhood of t such that this particle does not collidewith Y k = X j in this neighborhood. Take the finite intersection of these m − L ( k − ,k ) and L ( k,k +1) are constant in this neighborhood.Now, if Y k ( t ) does collide with particles X i , i ∈ I , then I ⊆ { , . . . , m } . We claim that thereexists a neighborhood of t such that, in this neighborhood, the particles X i , i ∈ I , do not collidewith any other particles. Indeed, for every i ∈ I , we have: X i ( t ) = Y k ( t ) ≤ u −
1. There exists aneighborhood of t in which X i does not collide with any particles X l , l ∈ { , . . . , m } \ I . Thereexists another neighborhood in which X i ( t ) < u . Therefore, X i does not collide with any particles X l , l > m . Intersect all these neighborhoods (there are 2 | I | of them) and complete the proofof this claim. In this neighborhood, the system ( X i ) i ∈ I behaves as a finite system of competingBrownian particles. It suffices to refer to (11).6.7. Proof of Theorem 3.7. Step 1. q + n ≥ / for all n ≥ . For N ≥
2, consider a rankedsystem Y ( N ) = (cid:16) Y ( N )1 , . . . , Y ( N ) N (cid:17) ′ , of N competing Brownian particles, with parameters( g n ) ≤ n ≤ N , ( σ n ) ≤ n ≤ N , ( q ± n ) ≤ n ≤ N , starting from Y ( N ) k (0) = y k , k = 1 , . . . , N , with driving Brownian motions B , B , . . . , B N . Definethe new parameters of collision q ± n = 12 , n ≥ . Consider another ranked system Y ( N ) = (cid:16) Y ( N )1 , . . . , Y ( N ) N (cid:17) ′ , of N competing Brownian particles, with parameters( g n ) ≤ n ≤ N , ( σ n ) ≤ n ≤ N , (cid:0) q ± n (cid:1) ≤ n ≤ N , starting from the same initial conditions Y ( N ) k (0) = Y ( N ) k (0) = y k , k = 1 , . . . , N , with the samedriving Brownian motions B , B , . . . , B N . We can construct such a system in the strong sense,by result o f Section 2 and [22] so that the sequences of driving Brownian motions ( B , . . . , B N )for each N are nested into each other. By [30, Corollary 3.9], for k = 1 , . . . , N and t ≥
0, we have:(33) Y ( N +1) k ( t ) ≤ Y ( N ) k ( t ) , Y ( N +1) k ( t ) ≤ Y ( N ) k ( t ) . NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 23
Since q + n ≥ q + n = 1 / n = 1 , . . . , N , by [30, Corollary 3.12], we have:(34) Y ( N ) k ( t ) ≤ Y ( N ) k ( t ) , t ≥ , k = 1 , . . . , N. Lemma 6.3.
For every
T > , we have a.s. lim N →∞ min ≤ t ≤ T Y ( N )1 ( t ) = inf N ≥ min ≤ t ≤ T Y ( N )1 ( t ) > −∞ . The proof of Lemma 6.3 is postponed until the end of the proof of Theorem 3.7. This lemma isused for the pathwise lower bound of the sequence ( Y ( N )1 ) N ≥ of processes. Assuming we provedthis lemma, let us continue the proof of Theorem 3.7. Step 2.
Note that for all s ∈ [0 , T ],min ≤ t ≤ T Y ( N )1 ( t ) ≤ Y ( N )1 ( s ) . Therefore, by Lemma 6.3, for every k ≥ t ≥ N ≥ k , we have: Y ( N ) k ( t ) ≥ Y ( N ) k ( t ) ≥ Y ( N )1 ( t ) ≥ lim N →∞ min ≤ t ≤ T Y ( N )1 ( t ) . By (33), there exists a finite pointwise limit(35) Y k ( t ) := lim N →∞ Y ( N ) k ( t ) . Now, let L ( N ) = (cid:16) L ( N )(1 , , . . . , L ( N )( N − ,N ) (cid:17) ′ be the vector of local times for the system Y ( N ) . Lemma 6.4.
There exist a.s. continuous limits L ( k,k +1) ( t ) := lim N →∞ L ( N )( k,k +1) ( t ) , for each k ≥ , uniform on every [0 , T ] . The limit Y k ( t ) from (35) is also continuous and uniformon every [0 , T ] for every k ≥ . The proof of Lemma 6.4 is also postponed until the end of the proof of Theorem 3.7. Assumingwe proved this lemma, let us complete the proof of Theorem 3.7 for the case when q + n ≥ / n ≥
1. For k = 1 , , . . . and t ≥
0, we have: Y ( N ) k ( t ) = y k + g k t + σ k B k ( t ) + q + k L ( N )( k − ,k ) ( t ) − q − k L ( N )( k,k +1) ( t ) . Letting N → ∞ , we have: Y k ( t ) = y k + g k t + σ k B k ( t ) + q + k L ( k − ,k ) ( t ) − q − k L ( k,k +1) ( t ) . Finally, let us show that L ( k,k +1) and Y k satisfy the properties (i) - (iii) of Definition 6. Some ofthese properties follow directly from the uniform covergence and the corresponding properties forfinite systems Y ( N ) . The nontrivial part is to prove that L ( k,k +1) can increase only when Y k = Y k +1 .Suppose that for some k ≥ Y k ( t ) < Y k +1 ( t ) for t ∈ [ α, β ] ⊆ R + . By continuity, thereexists ε > Y k +1 ( t ) − Y k ( t ) ≥ ε for t ∈ [ α, β ]. By uniform convergence, there exists an(a.s. finite) N such that for N ≥ N we have: Y ( N ) k +1 ( t ) − Y ( N ) k ( t ) ≥ ε , t ∈ [ α, β ] . Therefore, L ( N )( k,k +1) is constant on [ α, β ]: L ( N )( k,k +1) ( α ) = L ( N )( k,k +1) ( β ). This is true for all N ≥ N .Letting N → ∞ , we get: L ( k,k +1) ( α ) = L ( k,k +1) ( β ). Therefore, L ( k,k +1) is also constant on [ α, β ]. Step 3.
Now, consider the case when q + n ≥ / n ≥ n . It suffices to show thatthe sequence ( Y ( N ) k ( t )) N ≥ k is bounded from below, since this is the crucial part of the proof. For N ≥ n + 2, consider the system ˜ Y ( N ) = (cid:16) ˜ Y ( N ) n +1 , . . . , ˜ Y ( N ) N (cid:17) ′ of N − n competing Brownian particles with parameters( g n ) n
Proof of Lemma . Applying [30, Corollary 3.9], we have: for 0 ≤ s ≤ t and 1 ≤ k < N < M ,(41) L ( N )( k,k +1) ( t ) − L ( N )( k,k +1) ( s ) ≤ L ( M )( k,k +1) ( t ) − L ( M )( k,k +1) ( s ) . By construction of these systems, the initial conditions y k = Y ( N ) k (0) , N ≥ k , do not depend on N . Therefore, Y ( N )1 ( t ) = y + g t + σ B ( t ) − q − L ( N )(1 , ( t ) . Since Y ( N )1 ( t ) → Y ( t ) and q − >
0: the sequence ( L ( N )(1 , ( t )) N ≥ has a limit L (1 , ( t ) := lim N →∞ L ( N )(1 , ( t ) , for every t ≥ . Letting M → ∞ in (41), we get: for t ≥ s ≥ L (1 , ( t ) − L (1 , ( s ) ≥ L ( N )(1 , ( t ) − L ( N )(1 , ( s ) . We can equivalently rewrite this as(42) L (1 , ( t ) − L ( N )(1 , ( t ) ≥ L (1 , ( s ) − L ( N )(1 , ( s ) . But we also have: ( L ( N )(1 , ( t )) N ≥ is nondecreasing. Therefore,(43) L (1 , ( s ) − L ( N )(1 , ( s ) ≥ . In addition, we get the following convergence:(44) L ( N )(1 , ( t ) → L (1 , ( t ) as N → ∞ . Combining (42), (43), (44), we get:lim N →∞ L ( N )(1 , ( s ) = L (1 , ( s ) uniformly on every [0 , t ] . Therefore, letting N → ∞ in (33), we get: Y ( t ) = y + g t + σ B ( t ) − q − L (1 , ( t ) , t ≥ , and Y ( N )1 ( s ) → Y ( s ) uniformly on every [0 , t ]. Since Y ( N )1 and L ( N )(1 , are continuous for every N ≥
2, and the uniform limit of continuous functions is continuous, we conclude that the functions Y and L (1 , are also continuous. Now, Y ( N )2 ( t ) = y + g t + σ B ( t ) + q +2 L ( N )(1 , ( t ) − q − L ( N )(2 , ( t ) , t ≥ . But Y ( N )2 ( t ) → Y ( t ) and L ( N )(1 , ( t ) → L (1 , ( t ) as N → ∞ . Since q − >
0, we have: there exists a limit L (2 , ( t ) := lim N →∞ L ( N )(2 , ( t ). Similarly, we prove thatthis convergence is uniform on every [0 , T ]. Therefore, lim N →∞ Y ( N )2 = Y uniformly on every[0 , T ]. Thus Y and L (2 , are continuous. Analogously, we can prove that for every k ≥
1, thelimits L ( k,k +1) ( t ) = lim N →∞ L ( N )( k,k +1) ( t ) and Y k ( t ) = lim N →∞ Y ( N ) k ( t )exist and are uniform on every [0 , T ]. This completes the proof of Lemma 6.4, and with it theproof of Theorem 3.7.6.8. Proof of Lemma 3.8. Step 1.
First, consider the case q + n ≥ / all n ≥
1. Take anapproximative version ˜ Y = ( ˜ Y , ˜ Y , . . . ) of the infinite classical system with parameters ( g k ) k ≥ and ( σ k ) k ≥ , with symmetric collisions, and with the same initial conditions. By comparisontechniques, Corollary 3.12 (iii), we have the stochastic domination:(45) Y k ( t ) (cid:23) ˜ Y k ( t ) , k = 1 , , . . . , ≤ t ≤ T. Step 2.
Now, let us prove the two statements for the general case. Consider the approximativeversion ˜ Y = ( ˜ Y k ) k>n of the infinite ranked system of competing Brownian particles with parame-ters ( g n ) n>n , ( σ n ) n>n , ( q ± n ) n>n . But q + n ≥ / n > n , and therefore the system ˜ Y satisfiesthe statements of Lemma 3.8. By comparison techniques for infinite systems, see Corollary 3.11,we get: Y k ( t ) ≥ ˜ Y k ( t ) , t ∈ [0 , T ] , n < k ≤ N. Therefore, the system ( Y k ) k ≥ also satisfies the statements of Lemma 3.8. NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 27
Proof of Lemma 3.9.
Let D = { Y ( t ) has a tie } . Assume ω ∈ D , that is, the vector Y hasa tie:(46) Y k − ( t ) < Y k ( t ) = Y k +1 ( t ) = . . . = Y l ( t ) < Y l +1 ( t ) . This tie cannot contain infinitely many particles, because this would contradict Lemma 3.8. Fixa rational q ∈ ( Y l ( t ) , Y l +1 ( t )). By continuity of Y l and Y l +1 , there exists M ≥ s ∈ [ t − /M, t + 1 /M ] we have: Y l ( s ) < q < Y l +1 ( s ). Let C ( k, l, q, M ) = (cid:26) Y k − ( t ) < Y k ( t ) = Y k +1 ( t ) = . . . = Y l ( t ) < Y l +1 ( t ) , and Y l ( s ) < q < Y l +1 ( s ) for all s ∈ (cid:20) t − M , t + 1 M (cid:21)(cid:27) . We just proved that(47) P D \ ∞ [ M =1 [ q ∈ Q [ k Let p ( N ) t be the ranking permutation for the vector X ( N ) ( t ) ∈ R N .Then for 1 ≤ i ≤ N we have:(49) X ( N ) i ( t ) = x i + Z t β N,i ( s )d s + Z t ρ N,i ( s )d W N,i ( s ) , t ≥ , where W N, , . . . , W N,N are i.i.d. standard Brownian motions, β N,i ( t ) = N X k =1 p ( N ) t ( k ) = i ) g k , and ρ N,i ( s ) = N X k =1 p ( N ) t ( k ) = i ) σ k . Note that (cid:12)(cid:12) β N,i ( t ) (cid:12)(cid:12) ≤ max k ≥ | g k | =: g, and (cid:12)(cid:12) ρ N,i ( t ) (cid:12)(cid:12) ≤ max k ≥ σ k =: σ. Fix T > 0. It follows from the Arzela-Ascoli criterion and Lemma 7.4 that the sequence ( X ( N ) i ) N ≥ i is tight in C [0 , T ]. Now, let us show that the following sequence is also tight in C (cid:0) [0 , T ] , R k (cid:1) , foreach k ≥ X ( N ) i , Y ( N ) i , W N,i , i = 1 , . . . , k ) N ≥ k . For the components Y ( N ) i , this follows from Theorem 4.4: as N → ∞ , Y ( N ) i ⇒ Y i , where Y =( Y i ) i ≥ is an approximative version of the infinite system of competing Brownian particles withparameters ( g n ) n ≥ , ( σ n ) n ≥ , ( q ± n = 1 / n ≥ . For the components W N,i , this is immediate, becauseall these elements have the same law in C ([0 , T ] , R d ) (the law of the d -dimensional Brownianmotion starting from the origin). By the diagonal argument, for every subsequence ( N m ) m ≥ thereexists a sub-subsequence ( N ′ m ) m ≥ such that for every k ≥ 1, the following subsequence of (50)( X ( N ′ m )1 , . . . , X ( N ′ m ) k , Y ( N ′ m )1 , . . . , Y ( N ′ m ) k , W N ′ m , , . . . , W N ′ m ,k ) m ≥ converges weakly in C (cid:0) [0 , T ] , R k (cid:1) . By Skorohod theorem, we can assume that the convergenceis, in fact, a.s. Let X i := lim m →∞ X ( N ′ m ) i , Y i := lim m →∞ Y ( N ′ m ) i , W i := lim m →∞ W N ′ m ,i , i ≥ , T ]. As mentioned earlier, Y = ( Y i ) i ≥ is an approximativeversion of the infinite system of competing Brownian particles with parameters ( g n ) n ≥ , ( σ n ) n ≥ ,( q ± n = 1 / n ≥ . Also, W i are i.i.d. standard Brownian motions.Next, it suffices to show that X is a version of the infinite classical system, because the sub-sequence ( N m ) m ≥ is arbitrary, and the tightness is established above. Take the (random) set N ( ω ) of times t ∈ [0 , T ] when the system Y or a system Y ( N ′ m ) for some m ≥ ∗ ⊆ Ω of measure P (Ω ∗ ) = 1 such that for all ω ∈ Ω ∗ ,the set N ( ω ) has Lebesgue measure zero. Therefore, for every ε > ω ∈ Ω ∗ , thereexists an open subset U ε ( ω ) ⊆ [0 , T ] with measure mes( U ε ( ω )) < ε such that N ( ω ) ⊆ U ε ( ω ). Lemma 6.5. Fix i ≥ . Then for every ω ∈ Ω ∗ , there exists an m ( ω ) such that for m ≥ m ( ω ) and k ≥ , { t ∈ [0 , T ] \ U ε ( ω ) | X i ( t ) = Y k ( t ) } ⊆ n t ∈ [0 , T ] \ U ε ( ω ) | X ( N ′ m ) i ( t ) = Y ( N ′ m ) k ( t ) o . Proof. Assume the converse. Then there exists a sequence ( t j ) j ≥ in [0 , T ] ⊆ U ε ( ω ) and a sequence( m j ) j ≥ such that m j → ∞ and X i ( t j ) = Y k ( t j ) , X ( N ′ mj ) i ( t j ) = Y ( N ′ mj ) k ( t j ) . Therefore, the particle with name i in the system X ( N ′ mj ) has rank other than k : either larger than k , in which case we have:(51) X ( N ′ mj ) i ( t j ) ≥ Y ( N ′ mj ) k +1 ( t j ) , or smaller than k , in which case(52) X ( N ′ mj ) i ( t j ) ≤ Y ( N ′ mj ) k − ( t j ) . By the pigeonhole principle, at least one of these inequalities is true for infinitely many j . Withoutloss of generality, we can assume that (51) holds for infinitely many j ≥ 1; the case when (52)holds for infinitely many j ≥ j ≥ 1. There exists a convergent subsequence of ( t j ) j ≥ , because [0 , T ] is compact.Without loss of generality, we can assume t j → t . We shall use the principle: if f n → f uniformlyon [0 , T ] and s n → s , then f n ( s n ) → f ( s ). Since X ( N ′ mj ) i ( t j ) → X i ( t ) and Y ( N ′ mj ) k +1 ( t j ) → Y k +1 ( t ) NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 29 uniformly on [0 , T ], we have after letting j → ∞ : X i ( t ) ≥ Y k +1 ( t ). But we can also let j → ∞ in X i ( t j ) = Y k ( t j ). We get: X i ( t ) = Y k ( t ). Thus, Y k +1 ( t ) ≤ Y k ( t ). The reverse inequality alwaysholds true. Therefore, there is a tie at the point t . But the set [0 , T ] \ U ε is closed; therefore, t ∈ [0 , T ] \ U ε . This contradiction completes the proof. (cid:3) Lemma 6.6. For ω ∈ Ω ∗ , t ∈ [0 , T ] \ N ( ω ) , and i ≥ , as m → ∞ , we have: β N ′ m ,i ( t ) → β i ( t ) := ∞ X k =1 Y k ( t ) = X i ( t )) g k , and ρ N ′ m ,i ( t ) → ρ i ( t ) := ∞ X k =1 Y k ( t ) = X i ( t )) σ k . Proof. Let us prove the first convergence statement; the second statement is proved similarly. ByLemma 6.5, we have: β N ′ m ,i ( t ) = β i ( t ) and ρ N ′ m ,i ( t ) = ρ i ( t ) , t ∈ [0 , T ] \ U ε , m > m . This proves that β N ′ m ,i ( t ) → β i ( t ) and ρ N ′ m ,i ( t ) → ρ i ( t ) for t ∈ [0 , T ] \ U ε as m → ∞ . Since the set mes( U ε ) < ε and ε is arbitrarily small, this proves Lemma 6.6. (cid:3) Now, let us return to the proof of Theorem 3.3. Fix t ∈ [0 , T ]. Apply [31, Lemma 7.1] to showthat in L (Ω , F , P ), we have:(53) Z t ρ N ′ m ,i ( s )d W N ′ m ,i ( s ) → Z t ρ i ( s )d W i ( s ) . Also, by Lebesgue dominated convergence theorem (because mes( N ( ω )) = 0 for ω ∈ Ω ∗ ),(54) Z t β N ′ m ,i ( s )d s → Z t β i ( s )d s a.s. for all t ∈ [0 , T ] . Finally, we have a.s.(55) X ( N ′ m ) i ( t ) = x i + Z t β N ′ m ,i ( s )d s + Z t ρ N ′ m ,i ( s )d W N ′ m ,i ( s ) → X i ( t ) . From (55) and (54) we have that(56) Z t ρ N ′ m ,i ( s )d W N ′ m ,i ( s ) → X i ( t ) − x i − Z t β i ( s )d s. But if a sequence of random variables converges to one limit in L and to another limit a.s., thenthere limits coincide a.s. Comparing (53) and (56), we get: X i ( t ) = x i + Z t β i ( s )d s + Z t ρ i ( s )d W i ( s ) , which is another way to write the SDE governing the infinite classical system. We have found asequence ( N ′ m ) m ≥ which corresponds to convergence on [0 , T ]. By taking a sequence T j → ∞ andusing the standard diagonal argument, we can finish the proof. Proof of Lemma 4.2. Because of symmetry of π and ˜ π , it suffices to show that π (cid:22) ˜ π .Next, it suffices to show that for every fixed M ≥ π ] M (cid:22) [˜ π ] M . Recall that we have the following weak convergence:[ π ( ˜ N j ) ] M ⇒ [˜ π ] M , j → ∞ , and the stochastic comparison is preserved under weak limits. Therefore, to show (57), it sufficesto prove that(58) [ π ] M (cid:22) [ π ( ˜ N j ) ] M . Now, take J large enough so that N J > ˜ N j . By [30, Corollary 3.14], we have:(59) [ π ( ˜ N j ) ] M (cid:23) [ π ( N J ) ] M . By construction of π , we get:(60) [ π ] M (cid:22) [ π ( N J ) ] M . From (59) and (60), we get (58).6.12. Proof of Theorem 4.4. Using the notation of Theorem 3.7, we have: Y ( N j ) k → Y k , j → ∞ , for every k ≥ 1, uniformly on every [0 , T ]. Now, let Y ( N j ) = (cid:16) Y ( N j )1 , . . . , Y ( N j ) N j (cid:17) ′ be the ranked system of N j competing Brownian particles, which has the same parameters anddriving Brownian motions as Y ( N j ) = (cid:16) Y ( N j )1 , . . . , Y ( N j ) N j (cid:17) ′ , but starts from (0 , z ( N j )1 , z ( N j )1 + z ( N j )2 , . . . , z ( N j )1 + z ( N j )2 + . . . + z ( N j ) N j − ) ′ , rather than (0 , z , z + z , . . . , z + z + . . . + z N j − ) ′ . In other words, the gap process Z ( N j ) of thesystem Y ( N j ) is in its stationary regime: Z ( N j ) ( t ) ∼ π ( N j ) , t ≥ 0. Now, let us state an auxillarylemma; its proof is postponed until the end of the proof of Theorem 4.4. Lemma 6.7. Almost surely, as j → ∞ , for all t ≥ and k ≥ , we have: (61) Y k ( t ) = lim j →∞ Y ( N j ) k ( t ) . Assuming that we have already shown Lemma 6.7, we can finish the proof. For every t ≥ k = 1 , , . . . , a.s. Z ( N j ) k ( t ) = Y ( N j ) k +1 ( t ) − Y ( N j ) k ( t ) → Z k ( t ) = Y k +1 ( t ) − Y k ( t ) , j → ∞ . Therefore, for every t ≥ M ≥ 1, a.s. we have: (cid:16) Z ( N j )1 ( t ) , . . . , Z ( N j ) M ( t ) (cid:17) ′ → ( Z ( t ) , . . . , Z M ( t )) ′ , j → ∞ . But Z ( N j ) ( t ) = (cid:16) Z ( N j )1 ( t ) , . . . , Z ( N j ) N j − ( t ) (cid:17) ′ ∼ π ( N j ) NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 31 for j ≥ t ≥ 0. Moreover, as j → ∞ , we have the following weak convergence:[ π ( N j ) ] M ⇒ [ π ] M . Therefore, for M ≥ t ≥ 0, we get:( Z ( t ) , . . . , Z M ( t )) ′ ∼ [ π ] M . Thus, for Z ( t ) := ( Z ( t ) , Z ( t ) , . . . ), we have: Z ( t ) ∼ π, t ≥ . Proof of Lemma . First, since z ≤ z ( N j )1 , . . . , z N j − ≤ z ( N j ) N j − , we have: Y ( N j ) (0) =(0 , z , z + z , . . . , z + z + . . . + z N j − ) ′ ≤ Y ( N j ) (0) = (0 , z ( N j )1 , z ( N j )1 + z ( N j )2 , . . . , z ( N j )1 + z ( N j )2 + . . . + z ( N j ) N j − ) ′ . By [30, Corollary 3.11(i)],(62) Y ( N j ) k ( t ) ≤ Y ( N j ) k ( t ) , t ≥ , j ≥ . As shown in the proof of Theorem 3.7,(63) Y ( N j ) k ( t ) ≥ Y k ( t ) , k = 1 , . . . , N j , t ≥ . Combining (62) and (63), we get:(64) Y k ( t ) ≤ Y ( N j ) k ( t ) , k = 1 , . . . , N j , t ≥ . On the other hand, fix ε > j ≥ 1. Then lim l →∞ z ( N l ) k = z k , for k = 1 , . . . , N j − 1. There existsan l ( j, ε ) such that for l > l ( j, ε ) and k = 1 , . . . , N j − z ( N l )1 + . . . + z ( N l ) k ≤ z + . . . + z k + ε. For such l , let ˇ Y = ( ˇ Y , . . . , ˇ Y N j ) ′ , be another system of N j competing Brownian particles, withthe same parameters and driving Brownian motions, as Y ( N j ) , but starting from (0 , z ( N l )1 , z ( N l )1 + z ( N l )2 , . . . , z ( N l )1 + z ( N l )2 + . . . + z ( N l ) N j − ) ′ . By [30, Corollary 3.9],(65) ˇ Y k ( t ) ≥ Y ( N l ) k ( t ) , k = 1 , . . . , N j , t ≥ , since ˇ Y is obtained from Y ( N l ) by removing the top N l − N j particles. However, Y ( N j ) + ε N j := ( Y ( N j )1 + ε, . . . , Y ( N j ) N j + ε ) ′ , is also a system of N j competing Brownian particles, with the same parameters and drivingBrownian motions as Y ( N j ) , but starting from ( ε, z + ε, . . . , z + . . . + z N j − + ε ) ′ . Since Y ( N j ) (0)+ ε ≥ ˇ Y (0), because of (6.12), by [30, Corollary 3.11(i)], we have:(66) ˇ Y k ( t ) ≤ Y ( N j ) k ( t ) + ε, k = 1 , . . . , N j , t ≥ . Combining (65) and (66), we get: Y ( N l ) k ( t ) ≤ Y ( N j ) k ( t ) + ε , for k = 1 , . . . , N j , and t ≥ 0. Butfor every fixed k = 1 , , . . . , lim j →∞ Y ( N j ) k ( t ) = Y k ( t ). Therefore, there exists j ( k ) ≥ Y ( N j k ) ) k ( t ) ≤ Y k ( t ) + ε . Meanwhile, for l > l ( j ( k ) , k ) we get:(67) Y ( N l ) k ( t ) ≤ Y k ( t ) + 2 ε. We also have from (64) that(68) Y ( N l ) k ( t ) ≥ Y k ( t ) . Since ε > Proof of Lemma 4.5. (i) Define λ := sup n ≥ λ n and z ′ k = λ k λ − z k ∼ Exp( λ ). We have: z + . . . + z n ≥ z ′ + . . . + z ′ n . By the Law of Large Numbers, z ′ + . . . + z ′ n = nλ − (1 + o (1)) as n → ∞ . Therefore, we can estimate the infinite series as ∞ X n =1 e − α ( z + ... + z n ) ≤ ∞ X n =1 e − α ( z ′ + ... + z ′ n ) ≤ ∞ X n =1 e − α ( λ − (1+ o (1)) n < ∞ . (ii) Recall that Var z n = λ − n . For S n := z + . . . + z n , n ≥ 1, we have: E S n = Λ n . By [37,Theorem 1.4.1], we have: S n − Λ n is bounded. The rest is trivial.6.14. Proof of Theorem 4.6. (i) It suffices to show that for every k = 1 , , . . . , the family ofreal-valued random variables Z k = ( Z k ( t ) , t ≥ R + . Find an N j > k such that [ R ( N j ) ] − µ ( N j ) < 0. Consider a finite system of N j competing Brownian particles with parameters( g n ) ≤ n ≤ N j , ( σ n ) ≤ n ≤ N j , ( q ± n ) ≤ n ≤ N j . Denote this system by Y ( N j ) , as in the proof of Theorem 3.7. Let Z ( N j ) = ( Z ( N j )1 , . . . , Z ( N j ) N j − ) ′ be the corresponding gap process. By Proposition 2.1, the family of R N j − -valued random variables Z ( N j ) ( t ) , t ≥ 0, is tight in R N j − . By [30, Corollary 3.9, Remark 9], Z ( N j ) k ( t ) ≥ Z k ( t ) ≥ , k = 1 , . . . , N j − . Since the collection of real-valued random variables Z ( N j ) k ( t ), t ≥ 0, is tight, then the collection Z k ( t ) , t ≥ 0, is also tight.(ii) Fix M ≥ 2. It suffices to show that [ ν ] M (cid:22) [ π ] M . Since [ π ( N j ) ] M ⇒ [ π ] M , as j → ∞ , itsuffices to show that for N j > M , we have: [ ν ] M (cid:22) [ π ( N j ) ] M . Consider the system Y ( N j ) = (cid:16) Y ( N j )1 , . . . , Y ( N j ) N j (cid:17) ′ , which is defined in Definition 7. Let Z ( N j ) be the corresponding gap process. Then Z ( N j ) ( t ) ⇒ π ( N j ) , t → ∞ . But by [30, Corollary 3.9, Remark 9], Z ( N j ) k ( t ) ≥ Z k ( t ), k = 1 , . . . , N j − 1. Therefore, [ Z ( N j ) ( t )] M ≥ [ Z ( t )] M , for t ≥ 0. And [ Z ( t j )] M ⇒ [ ν ] M , as j → ∞ . Thus, [ π ( N j ) ] M (cid:23) [ ν ] M .(iii) Follows directly from (i). NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 33 Proof of Theorem 5.1. The proof resembles that of Lemma 3.9 and uses Lemma 3.8.(i) Define the following events: D = {∃ t > ∃ k < l : Y k ( t ) = Y k +1 ( t ) , Y l ( t ) = Y l +1 ( t ) } ; D k,l = {∃ t > Y k ( t ) = Y k +1 ( t ) , Y l ( t ) = Y l +1 ( t ) } for k < l. Then it is easy to see that D = [ k Lemma 7.1. Assume that ( y n ) n ≥ is a sequence of real numbers such that y n → ∞ and ∞ X n =1 e − αy n < ∞ for α > . Then for every v ∈ R and β > we have: ∞ X n =1 Ψ (cid:18) y n + vβ (cid:19) < ∞ . Proof. By [9, Chapter 7, Lemma 2], we have for v ≥ v ) ≤ √ πv e − v / ≤ √ π e − v / . But y n → ∞ as n → ∞ , and there exists n such that for n ≥ n we have: ( y n + v ) /β ≥ n ≥ n , we have:Ψ (cid:18) y n + vβ (cid:19) ≤ √ π exp (cid:18) − β ( y n + v ) (cid:19) . Using an elementary inequality ( c + d ) ≥ c / − d for all c, d ∈ R , we get:12 β ( y n + v ) ≥ β y n − β v . Thus, X n>n Ψ (cid:18) y n + vβ (cid:19) ≤ √ π X n>n exp (cid:18) − y n β + v β (cid:19) = 1 √ π exp (cid:18) v β (cid:19) X n>n exp (cid:18) − y n β (cid:19) < ∞ . (cid:3) Lemma 7.2. Take an Itˆo process V ( t ) = v + Z t β ( s )d s + Z t ρ ( s )d W ( s ) , t ≥ , where v ∈ R , W = ( W ( t ) , t ≥ , is a standard Brownian motion, β = ( β ( t ) , t ≥ and ρ = ( ρ ( t ) , t ≥ , are adapted processes such that a.s. for all t ≥ we have the following estimates: β ( t ) ≥ g , | ρ ( t ) | ≤ σ . If x ≤ v + gT , then we have the following estimate: P (cid:18) min ≤ t ≤ T V ( t ) ≤ x (cid:19) ≤ (cid:18) v − x − ( gT ) − σ √ T (cid:19) . Proof. Let M ( t ) = R t ρ ( s )d W ( s ) , t ≥ 0. Then M = ( M ( t ) , t ≥ 0) is a continuous square-integrable martingale with h M i t = R t ρ ( s )d s . There exists a standard Brownian motion B = NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 35 ( B ( t ) , t ≥ 0) so that we can make a time-change: M ( t ) ≡ B ( h M i t ). Then (cid:26) min ≤ t ≤ T V ( t ) ≤ x (cid:27) ⊆ (cid:26) min ≤ t ≤ T M ( t ) − ( gT ) − + v ≤ x (cid:27) ⊆ (cid:26) min ≤ t ≤ T B ( h M i t ) ≤ x − v + ( gT ) − (cid:27) . Because h M i t ≤ σ T for t ∈ [0 , T ], we have: (cid:26) min ≤ t ≤ T B ( h M i t ) ≤ x − v + ( gT ) − (cid:27) ⊆ (cid:26) min ≤ t ≤ σ T B ( t ) ≤ x − v + ( gT ) − (cid:27) . Finally, P (cid:18) min ≤ t ≤ σ T B ( t ) ≤ x − v + ( gT ) − (cid:19) = 2 P (cid:0) B ( σ T ) ≤ x − v + ( gT ) − (cid:1) = 2Ψ (cid:18) v − x − ( gT ) − σ √ T (cid:19) . (cid:3) Lemma 7.3. Assume that in the setting of Lemma , we have | β ( t ) | ≤ g and | ρ ( t ) | ≤ σ for t ≥ a.s. If x ≥ | v | + gT , then P (cid:18) max ≤ t ≤ T | V ( t ) | ≤ x (cid:19) ≤ (cid:18) v − x − gTσ √ T (cid:19) . Proof. This follows from applying Lemma 7.2 twice: once for the minimum and once for themaximum of the process V . (We can adjust Lemma 7.2 to work for maximum of V in an obviousway.) (cid:3) Lemma 7.4. Take a sequence ( M n ) n ≥ of continuous local martingales on [0 , T ] , such that M n (0) =0 , and h M n i t is differentiable for all n , and sup n ≥ sup t ∈ [0 ,T ] d h M n i t d t = C < ∞ . Then the sequence ( M n ) n ≥ is tight in C [0 , T ] .Proof. Use [23, Chapter 2, Problem 4.11] (with obvious adjustments, because the statement inthis problem is for R + instead of [0 , T ]). We need only to show that(69) sup n ≥ E ( M n ( t ) − M n ( s )) ≤ C ( t − s ) for all 0 ≤ s ≤ t ≤ T and for some constant C , depending only on C and T . By the Burkholder-Davis-Gundy inequality, see [23, Chapter 3, Theorem 3.28], for some absolute constant C > E ( M n ( t ) − M n ( s )) ≤ C E ( h M n i t − h M n i s ) ≤ C ( C ( t − s )) = C C ( t − s ) . (cid:3) Acknoweldgements I would like to thank Ioannis Karatzas , Soumik Pal , Xinwei Feng , Amir Dembo , and Vladas Sidoravicius for help and useful discussion. I am also thankful to anonymous refereesfor meticulously reviewing this article, which helped to significantly improve it. This researchwas partially supported by NSF grants DMS 1007563, DMS 1308340, DMS 1405210, and DMS1409434. References [1] Richard Arratia (1983). The Motion of a Tagged Particle in the Simple Symmetric Exclusion system on Z . Ann. Probab. (2), 362-373.[2] Adrian D. Banner, E. Robert Fernholz, Ioannis Karatzas (2005). Atlas Models of Equity Markets. Ann. Appl. Probab. (4), 2996-2330.[3] Adrian D. Banner, E. Robert Fernholz, Tomoyuki Ichiba, Ioannis Karatzas, Vassilios Pap-athanakos (2011). Hybrid Atlas Models. Ann. Appl. Probab. (2), 609-644.[4] Richard Bass, E. Pardoux (1987). Uniqueness for Diffusions with Piecewise Constant Coefficients. Probab.Th. Rel. Fields (4), 557-572.[5] Cameron Bruggeman, Andrey Sarantsev (2016). Multiple Collisions in Systems of Competing BrownianParticles. To appear in Bernoulli . Available at arXiv:1309.2621.[6] Hong Chen (1996). A Sufficient Condition for the Positive Recurrence of a Semimartingale Reflecting Brow-nian Motion in an Orthant. Ann. Appl. Probab. (3), 758-765.[7] Amir Dembo, Mykhaylo Shkolnikov, S.R. Srinivasa Varahna, Ofer Zeitouni (2016). Large Devia-tions for Diffusions Interacting Through Their Ranks. Comm. Pure Appl. Math. (7), 1259-1313.[8] Amir Dembo, Li-Cheng Tsai (2015). Equilibrium Fluctuations of the Atlas Model. Available atarXiv:1503.03581.[9] William Feller (1968). An Introduction to Probability Theory and Its Applications , Vol. 1. Third edition,Wiley.[10] E. Robert Fernholz (2002). Stochastic Portfolio Theory. Applications of Mathematics . Springer.[11] E. Robert Fernholz, Tomoyuki Ichiba, Ioannis Karatzas (2013). A Second-Order Stock MarketModel. Ann. Finance (3), 439-454.[12] E. Robert Fernholz, Ioannis Karatzas (2009). Stochastic Portfolio Theory: an Overview. Handbook ofNumerical Analysis: Mathematical Modeling and Numerical Methods in Finance , 89-168. Elsevier.[13] E. Robert Fernholz, Tomoyuki Ichiba, Ioannis Karatzas, Vilmos Prokaj (2013). Planar Diffusionswith Rank-Based Characteristics and Perturbed Tanaka Equations. Probab. Th. Rel. Fields , (1-2), 343-374.[14] Patrik L. Ferrari (1996). Limit Theorems for Tagged Particles. Markov Proc. Rel. Fields (1), 17-40.[15] Patrik L. Ferrari, Luiz Renato G. Fontes (1994). The Net Output Process of a System with InfinitelyMany Queues. Ann. Appl. Probab. (4), 1129-1144.[16] Theodore E. Harris (1965). Diffusions with Collisions Between Particles. J. Appl. Probab. , 323-338.[17] Tomoyuki Ichiba, Ioannis Karatzas (2010). On Collisions of Brownian Particles. Ann. Appl. Probab. (3), 951-977.[18] Tomoyuki Ichiba, Ioannis Karatzas, Mykhaylo Shkolnikov (2013). Strong Solutions of StochasticEquations with Rank-Based Coefficients. Probab. Th. Rel. Fields , 229-248.[19] Tomoyuki Ichiba, Soumik Pal, Mykhaylo Shkolnikov (2013). Convergence Rates for Rank-Based Mod-els with Applications to Portfolio Theory. Probab. Th. Rel. Fields , 415-448.[20] Benjamin Jourdain, Florent Malrieu (2008). Propagation of Chaos and Poincare Inequalities for aSystem of Particles Interacting Through Their Cdf. Ann. Appl. Probab. (5), 1706-1736.[21] Benjamin Jourdain, Julien Reygner (2013). Propagation of Chaos for Rank-Based Interacting Diffusionsand Long Time Behaviour of a Scalar Quasilinear Parabolic Equation. SPDE Anal. Comp. (3), 455-506.[22] Ioannis Karatzas, Soumik Pal, Mykhaylo Shkolnikov (2016). Systems of Brownian Particles withAsymmetric Collisions. Ann. Inst. H. Poincare (1), 323-354.[23] Ioannis Karatzas, Steven E. Shreve (1991). Brownian Motion and Stochastic Calculus . Graduate Textsin Mathematics . Second edition, Springer. NFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES 37 [24] Claude Kipnis (1986). Central Limit Theorems for Infinite Series of Queues and applications to SimpleExclusion Processes. Stoch. Proc. Appl. (2), 397-408.[25] Li-Cheng Tsai, Andrey Sarantsev (2016). Stationary Gap Distributions for Infinite Systems of CompetingBrownian Particles. Available at arXiv:1608.00628.[26] Neil O’Connell, Marc Yor (2001). Brownian Analogues of Burke’s Theorem. Stoch. Proc. Appl. (2),285-304.[27] Soumik Pal, Jim Pitman (2008). One-Dimensional Brownian Particle Systems with Rank-Dependent Drifts. Ann. Appl. Probab. (6), 2179-2207.[28] I. Martin Reiman, Ruth J. Williams (1988). A Boundary Property of Semimartingale Reflecting BrownianMotions. Probab. Th. Rel. Fields (1), 87-97.[29] Julien Reygner (2015). Chaoticity of the Stationary Distribution of Rank-Based Interacting Diffusions. Electr. Comm. Probab. (60), 1-20.[30] Andrey Sarantsev (2015). Comparison Techniques for Competing Brownian Particles. To appear in J. Th.Probab. Available at arXiv:1305.1653.[31] Andrey Sarantsev (2016). Penalty Method for Obliquely Reflected Diffusions. Available atarXiv:1509.01777.[32] Andrey Sarantsev (2015). Triple and Simultaneous Collisions of Competing Brownian Particles. Electr. J.Probab. (29), 1-28.[33] Andrey Sarantsev (2015). Two-Sided Infinite Systems of Competing Brownian Particles. Available atarXiv:1509.01859.[34] Timo Seppalainen (1997). A Scaling Limit for Queues in Series. Ann. Appl. Probab. (4), 885-872.[35] Mykhaylo Shkolnikov (2011). Competing Particle Systems Evolving by Interacting L´evy Processes. Ann.Appl. Probab. (5), 1911-1932.[36] Mykhaylo Shkolnikov (2012). Large Systems of Diffusions Interacting Through Their Ranks. Stoch. Proc.Appl. (4), 1730-1747.[37] Daniel W. Stroock (2011). Probability Theory. An Analytic View . Second edition. Cambridge UniversityPress.[38] Alain-Sol Sznitman (1986). A Propagation of Chaos Result for Burgers’ Equation. Probab. Th. Rel. Fields (4), 581-613.[39] Ruth J. Williams (1995). Semimartingale Reflecting Brownian Motions in the Orthant. Stochastic networks ,IMA Vol. Math. Appl. , 125-137. Springer-Verlag.[40] Ruth J. Williams (1987). Reflected Brownian Motion with Skew-Symmetric Data in a Polyhedral Domain. Probab. Th. Rel. Fields (4), 459-485. Department of Statistics and Applied Probability, University of California, Santa Barbara E-mail address ::