Stationary Distributions of the Atlas Model

In this article we study the Atlas model, which constitutes of Brownian particles on $ \mathbb{R} $, independent except that the Atlas (i.e., lowest ranked) particle $ X_{(1)}(t) $ receive drift $ \gamma dt $, $ \gamma\in\mathbb{R} $. For any fixed shape parameter $ a>2\gamma_- $, we show that, up to a shift $ \frac{a}{2}t $, the entire particle system has an invariant distribution $ \nu_a $, written in terms an explicit Radon-Nikodym derivative with respect to the Poisson point process of density $ a e^{a\xi} d\xi $. We further show that $ \nu_a $ indeed has the product-of-exponential gap distribution $ \pi_a $ derived in Sarantsev and Tsai (2016). As a simple application, we establish a bound on the fluctuation of the Atlas particle $ X_{(1)}(t) $ uniformly in $ t $, with the gaps initiated from $ \pi_a $ and $ X_{(1)}(0)=0 $.


Introduction
In this article we study the (infinite) Atlas model. Such a model consists of a semi-infinite collection of particles X i (t), i = 1, 2, . . ., performing independent Brownian motions on R, except that the Atlas (i.e., lowest ranked) particle receives a drift of strength γ ∈ R. To rigorously define the model, we recall that x = (x i ) ∞ i=1 ∈ R N is rankable if there exists a ranking permutation p : N → N such that x p(i) ≤ x p(j) , for all i < j ∈ N. To ensure that such a ranking permutation is unique, we resolve ties in lexicographic order. That is, if x p(i) = x p(j) for i < j, then p(i) < p(j). We then let p x ( · ) : N → N denote the unique ranking permutation for a given, rankable x. Fix independent standard Brownian motions W 1 , W 2 , . . .. For suitable initial conditions, the infinite Atlas model X(t) = (X i (t)) ∞ i=1 is given by the unique weak solution of the following system of Stochastic Differential Equations (SDEs) dX i (t) = γ1{p X(t) (i) = 1}dt + dW i (t), i ∈ N. (1.1) To state the well-posedness results of (1.1), consider the following configuration space and note that lim i→∞ x i = ∞ necessarily implies that x is rankable. It is shown in [Sar17a, Theorem 3.2], for any fixed γ ∈ R and any given x ∈ U, the system (1.1) admits a unique weak solution X(t) starting from the initial condition x, such that P(X(t) ∈ U, ∀t ≥ 0) = 1. See also [Shk11,IKS13]. The interest of the Atlas model originates from the study of diffusions with rank-based drifts [Fer02,FK09]. In particular, the Atlas model was first introduced, in finite dimensions, as a simple special case of rank-based diffusions [Fer02]. Due to their intriguing properties, rank-based diffusions have been intensively studied in various generality. See [BFK05,BFI + 11, IKS13,Sar17b] and the references therein. The infinite-dimensional system (1.1) considered here was introduced by Pal and Pitman [PP08]. Parts of the motivation was to understand the effect of a drift exerted on a large (but finite) collection of Brownian particles [Ald02,TT15]. In particular, it was shown in [PP08] that, for γ > 0, the system (1.1) admits a stationary gap distribution of i.i.d. Exp(2γ), which indicates that the drift γdt is balanced by the push-back of a crowd of particles of density 2γ. To state the previous result more precisely, given a rankable x = (x i ) ∞ i=1 , we let (x (1) ≤ x (2) ≤ . . .) denote the corresponding ranked points, i.e., x (i) = x (px) −1 (i) , and consider the corresponding gaps is also a stationary gap distribution of the Atlas model. Unlike π, the distribution π a has exponentially growing particle density away from the Atlas particle. In this article, we go one step further and show that, in fact, up to a deterministic shift at 2 of each particle, the entire particle system {X i (t) + at 2 } ∞ i=1 has a stationary distribution. This extends the result of [ST17] on stationary gap distributions. In the following we use ∈ U denote the corresponding configuration space, and let µ a denote the Poisson point process on R with density ae aξ dξ. It is standard to show (e.g., using techniques from [Pan13, Section 2.2]) that µ a is supported on V. Let Γ(α) := ∞ 0 ξ −1−α e −ξ dξ denote the Gamma function, and let Gamma(α, β) ∼ 1 Γ(α) β α ξ −1−α e −βξ 1 {ξ>0} dξ denote the Gamma distribution. The following is the main result.
Remark 1.2. Under ν a , the Atlas particle X (1) and the gap process For the special case γ = 0, the Atlas model (1.1) reduces to independent Brownian motions. In this case, it is well known that the Poisson point process µ a is quasi-stationary [Lig78], and the shift − a 2 t can be easily calculated from the motion of independent Brownian particles. Here we show that, with a drift γdt exerted on the Atlas particle X (1) (t), a stationary distribution is obtained by taking V (x) := 2γx (1) to be the potential. Indeed, under such a choice of V (x), we have that γ1{p . This explains why we should expect the stationary distribution ν a as in (1.3). The proof of Theorem 1.1 amounts to justifying the aforementioned heuristic in the setting of infinite dimensional diffusions with discontinuous drift coefficients. We achieve this through finite-dimensional, smooth approximations, and using the explicit expressions of semigroups from Girsanov's theorem to take limits.
Due to their simplicity, product-of-exponential stationary gap distributions have been intensively searched within competing Brownian particle systems, in both finite and infinite dimensions. See [Sar17a] and the references therein. To date, derivations of product-ofexponential stationary gap distributions have been relying on the theory of Semimartingale Reflecting Brownian Motions (SRBM), e.g., [Wil95]. On the other hand, given the expression (1.3) of ν a , the gap distribution (1.4) follows straightforwardly from Rényi's representation [Rén53]. Theorem 1.1 hence provides an alternative derivation of the product-ofexponential distribution π a without going through SRBM.
Our methods should generalize to the case of competing Brownian particle systems with finitely many non-zero drift coefficients, i.e., Here we consider only the Atlas model for simplicity of notations.
A natural question, following the discovery a stationary gap distribution, is the longtime behavior of the Atlas particle X (1) (t) under such a gap distribution. For the i.i.d. Exp(2) gap distribution π, this question was raised in [PP08] and answered in [DT17]. It was shown in [DT17] that X (1) (t) fluctuates at order t 1 4 around its starting location, and scales to a 1 4 -fractional Brownian motion, as t → ∞. As a simple application of Theorem 1.1, under the stationary gap distribution π a and X (1) (0) = 0, we establish an exponential tail bound, uniformly in t, of the fluctuation Atlas particle around its expected location − at 2 . This shows that the fluctuation of X (1) (t) stays bounded under π a , in sharp contrast with the t 1 4 fluctuation obtained in [DT17].
Acknowledgements. I thank Amir Dembo, Ioannis Karatzas, Andrey Sarantsev and Ofer Zeitouni for enlightening discussions. I am grateful to Andrey Sarantsev for many useful comments on the presentation of this article. This work was partially supported by a Junior Fellow award from the Simons Foundation to Li-Cheng Tsai.
Performing the change of variable ζ ′ := e aζ , we see that For any given threshold ξ ∈ R, we let µ a,ξ denote the restriction of the Poisson point process µ a onto (−∞, ξ]. For the restricted process, we have Exp(ia).
Iterating this argument to higher order derivatives, we obtain sup s≤t,x∈R n ,|β|≤k |∂ β u(t, x)|e ξ(|x 1 |+...+|xn|) < ∞, ∀ξ, t, k < ∞. (2.12) The PDE (2.9) has stationary distribution e V ε (x) n i=1 e ax i dx i (not a probability distribution, since the total mass is infinite). More precisely, integrate u(t, x) against the aforementioned distribution to get Taking time derivative using (2.9) and (2.12), followed by integrations by parts The next step is to take the limit ε → 0 in (2.13), for fixed n. This amounts to establishing the convergence of the term E x φ s (X n,ε (t)) . To this end, we use Girsanov's theorem to write (2.14) where H(t) := (W i (t)+ at 2 +x i ) n i=1 consists of independent, drifted Brownian motions starting from x = (x i ) n i=1 , and the terms F (t) and F ε (t) are stochastic exponentials given by Taking the difference of (2.14)-(2.15), followed by using the Cauchy-Schwarz inequality, we obtain For the two terms in (2.18), we next show that: i ) the first term is bounded; and ii ) the second term vanishes as ε → 0. Hereafter, we use c(a 1 , a 2 , . . .) to denote a finite, deterministic constant, that may change from line to line, but depends only on the designated variables a 1 , a 2 , . . ..