Quasi-stationarity of the Dyson Brownian motion with collisions
Arnaud Guillin, Boris Nectoux, Liming Wu
TL;DR
This work analyzes the long-time behavior of a Dyson Brownian motion-like system with collisions in a constrained domain. By combining multivalued SDE techniques with regularity properties of both killed and non-killed semigroups, the authors establish the existence and uniqueness of a quasi-stationary distribution for the collision regime ($\gamma\in(0,1/2)$) and prove exponential convergence of the killed process to this QSD, along with a principal eigenvalue and positive eigenfunction. They also extend the framework to the non-collision regime ($\gamma>1/2$), where the process remains in the interior and standard Lyapunov arguments yield analogous results. The results rely on density and strong Feller properties, topological irreducibility, and a careful treatment of the singular drift near the boundary. Overall, the paper advances quasi-stationarity theory for elliptic/hypoelliptic systems with singular interactions and boundary collisions, particularly in the Dyson Brownian motion context.
Abstract
In this work, we investigate the ergodic behavior of a system of particules, subject to collisions, before it exits a fixed subdomain of its state space. This system is composed of several one-dimensional ordered Brownian particules in interaction with electrostatic repulsions, which is usually referred as the (generalized) Dyson Brownian motion. The starting points of our analysis are the work [E. C{é}pa and D. L{é}pingle, 1997 Probab. Theory Relat. Fields] which provides existence and uniqueness of such a system subject to collisions via the theory of multivalued SDEs and a Krein-Rutman type theorem derived in [A. Guillin, B. Nectoux, L. Wu, 2020 J. Eur. Math. Soc.].