Stationary gap distributions for infinite systems of competing Brownian particles

Abstract

Consider the infinite Atlas model: a semi-infinite collection of particles driven by independent standard Brownian motions with zero drifts, except for the bottom-ranked particle which receives unit drift. We derive a continuum one-parameter family of product-of-exponentials stationary gap distributions, with exponentially growing density at infinity. This result shows that there are infinitely many stationary gap distributions for the Atlas model, and hence resolves a conjecture of Pal and Pitman (2008) [PP08] in the negative. This result is further generalized for infinite systems of competing Brownian particles with generic rank-based drifts.2010 Mathematics Subject Classification. 60H10, 60J60, 60K35.

Two-Sided Infinite Systems of Competing Brownian Particles

“…Assume that σ n = 1 for all n ∈ Z, and sup |g n | < ∞. It was shown in [47] that we always have a one-parameter product-of-exponentials family of stationary gap distributions π a , a ∈ R. In contrast with finite systems, we do not need to impose any stability condition similar to (3). Therefore, the weak limit of Z(t) as t → ∞ depends on the initial distribution of Z(0).…”

“…Finite systems were studied in the following articles: [22,23,42,9,25] (triple and multiple collisions of particles); [36,4], [43,Section 2] (stationary distribution π for the gap process); [26,24,44] (convergence Z(t) ⇒ π as t → ∞ with an exponential rate); concentration of measure, [35,37]; see also miscellaneous papers [28,40,41,44]. One-sided infinite systems of competing Brownian particles (X n ) n≥1 were introduced in [36] and further studied in [50,23,43,47,13].…”

“…where the union in the right-hand side of (46) is taken over all (47) k, l ∈ Z; q − , q + ∈ Q, q − < q + ; m = 1, 2, . .…”

“…. Therefore, it suffices to show that (48) P (D(k, l, q − , q + , m)) = 0 for all k, l, q − , q + , m from (47).…”

“…Overview of the proof. Similarly to the proof of the main result in [47], we approximate this two-sided infinite system by finite systems of competing Brownian particles in stationary gap distributions, with suitably chosen uniformly bounded drifts. These stationary gap distributions have product-of-exponential form, which match the infinite poduct-of-exponentials distribution π a,b .…”