Brownian Particles with Rank‐Dependent Drifts: Out‐of‐Equilibrium Behavior

Abstract

We study the long‐range asymptotic behavior for an out‐of‐equilibrium, countable, one‐dimensional system of Brownian particles interacting through their rank‐dependent drifts. Focusing on the semi‐infinite case, where only the leftmost particle gets a constant drift to the right, we derive and solve the corresponding one‐sided Stefan (free‐boundary) equations. Via this solution we explicitly determine the limiting particle‐density profile as well as the asymptotic trajectory of the leftmost particle. While doing so we further establish stochastic domination and convergence to equilibrium results for the vector of relative spacings among the leading particles. © 2019 Wiley Periodicals, Inc.

A note on transportation cost inequalities for diffusions with reflections

“…Our inequalities are dimension-free: that is, the constant C is independent of the number of particles. This allows us to extend the inequality to infinite competing particle systems such as the infinite Atlas model [10,14,26,35]. This is an improvement over the dimension-dependent inequalities in papers [34,36] where applications of such inequalities can be found.…”

Equilibrium fluctuation of the Atlas model

“…In retrospect, the Neumann boundary condition represents the conservation of particles at x = 0. It is shown in [3] that at the equilibrium density we consider here, sup s∈[0,t] {ε 1/2 |X (0) (ε −1 t)|} → 0 almost surely. That is, at the scale ε −1/2 of space, the lowest rank particle stays very close to x = 0.…”

“…Our strategy of proving Theorem 1.1 is to focus on the empirical measure. While this strategy has been widely used for interacting particle systems, in the context of Atlas model, or more generally diffusions with rank-dependent drifts, analyzing empirical measure is a new approach that has only been used here and in [3]. It completely bypasses the need of local times, which is a major a challenge when analyzing diffusions with rankdependent drifts.…”

The infinite Atlas process: Convergence to equilibrium

“…From [15,Theorem 1] we learn that at critical spacing density λ = 2 the unit drift to the leftmost particle compensates the spreading of bulk particles to the left, thereby keeping the gaps at equilibrium. Such interplay between spacing density and drift is re-affirmed by [4], which shows that initial spacing law µ Theorem 1.1. Suppose the atlas ∞ process starts at Z(0) = (z j ) j≥1 such that for eventually non-decreasing θ(m) with inf m {θ(m − 1)/θ(m)} > 0 and β ∈ [1, 2),…”