Infinite-dimensional Dyson Brownian motion(s)
Abstract
We study infinite-dimensional Dyson Brownian motions obtained as limits of finite systems without rescaling the actual stochastic dynamics. For β=2, we construct determinantal processes on an extended space of initial data and prove convergence of their finite-dimensional distributions under essentially optimal conditions. This extends the seminal results of Katori and Tanemura. The additional parameters record information at infinity and enter through an associated Laguerre-Pólya entire function. Moreover, for explicit classes of configurations, we establish convergence on path space and the Markov property. We prove rescaled long-time convergence, in finite-dimensional distributions, to the stationary extended Sine process from arbitrary symmetric initial configurations with power-law counting exponent q∈(0,2). This extends the integer lattice relaxation result of Katori and Tanemura which was the only such result for explicit deterministic initial conditions. For β≥1, we prove convergence of finite particle systems from regular initial data to the unique strong solution of an infinite-dimensional stochastic differential equation in a certain rigid-path-regularity class. This extends seminal works of Tsai and Osada. We finally derive a stochastic partial differential equation of Burgers-type for the Stieltjes transform of the dynamics.
Related
- Parametric estimation for stochastic wave equations driven by an infinite dimensional fractional Brownian motion
- Support of Dyson Brownian Motion
- Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials
- The singular Gaussian scaling limit for a tagged particle in the infinite-dimensional Dyson model at β=2
- Infinite-Particle Determinantal Dynamics from Brownian Motion on GLN(ℂ)
BibTeX
@misc{assiotis2026infinitedimensional,
title = {{Infinite-dimensional Dyson Brownian motion(s)}},
author = {Theodoros Assiotis and Fengyi Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26386},
eprint = {2609.26386},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26386},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Infinite-dimensional Dyson Brownian motion(s)
AU - Theodoros Assiotis
AU - Fengyi Li
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26386
UR - https://arxiv.org/abs/2609.26386
AB - We study infinite-dimensional Dyson Brownian motions obtained as limits of finite systems without rescaling the actual stochastic dynamics. For β=2, we construct determinantal processes on an extended space of initial data and prove convergence of their finite-dimensional distributions under essentially optimal conditions. This extends the seminal results of Katori and Tanemura. The additional parameters record information at infinity and enter through an associated Laguerre-Pólya entire function. Moreover, for explicit classes of configurations, we establish convergence on path space and the Markov property. We prove rescaled long-time convergence, in finite-dimensional distributions, to the stationary extended Sine process from arbitrary symmetric initial configurations with power-law counting exponent q∈(0,2). This extends the integer lattice relaxation result of Katori and Tanemura which was the only such result for explicit deterministic initial conditions. For β≥1, we prove convergence of finite particle systems from regular initial data to the unique strong solution of an infinite-dimensional stochastic differential equation in a certain rigid-path-regularity class. This extends seminal works of Tsai and Osada. We finally derive a stochastic partial differential equation of Burgers-type for the Stieltjes transform of the dynamics.
ER -