Infinite systems of competing Brownian particles

Abstract

Abstract. Consider a system of infinitely many Brownian particles on the real line. At any moment, these particles can be ranked from the bottom upward. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank. The gaps between consecutive particles form the (infinite-dimensional) gap process. We find a stationary distribution for the gap process. We also show that if the initial value of the gap process is stochastically larger than this stationary distribution, this process converges back to this distribution as time goes to infinity. This continues the work by Pal and Pitman (2008). Also, this includes infinite systems with asymmetric collisions, similar to the finite ones from Karatzas, Pal and Shkolnikov (2016).

The infinite Atlas process: Convergence to equilibrium

“…3.12(ii)] that keeping the same driving Brownian motions {B j (s)} and initial configuration X(0), the spacing vector Z(t) is pointwise decreasing in γ. Further, by [19,Cor. 3.10(ii)], the first k − 1 spacings increase when all particles to the right of X k (0) are removed.…”

“…By Prohorov's theorem, it remains to verify that {ν (m,k) tm } are uniformly tight, namely, to provide a uniform in m tail-decay for k j=1 Z j (t m ) in the corresponding atlas m system. To this end, recall [19,Cor. 3.10(ii)] that under the same driving Brownian motions {B j (s)} and initial configuration, the first k spacings increase when all particles to the right of X k+1 (0) are removed.…”

“…Proof. Recall [19,Cor. 3.12(ii)] that keeping the same driving Brownian motions {B j (s)} and initial configuration X(0), the spacing vector Z(t) is pointwise decreasing in γ.…”

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

“…Lacking such connection here, in Section 2 we build on [38] to prove Proposition 1.7. Combining these comparison results in Section 3 with large deviation estimates for i.i.d.…”

Stochastic differential games with competing Brownian particles and related Isaacs' equations

“…The Brownian particle systems of this type are not only of interest in their own right, but also arise as universal scaling limits for systems of jump processes on the integer lattice with local interactions. Karatzas et al and Sarantsev studied the systems of competing Brownian particles with asymmetric collisions. In these systems, the local time of collision between two particles can split unevenly between them, and the parameters of the collisions are decided by the ranks of the particles involved in the collisions.…”