Probability Theory and Related Fields

, Volume 166, Issue 3–4, pp 801–850

Infinite dimensional stochastic differential equations for Dyson’s model

Article

Abstract

In this paper we show the strong existence and the pathwise uniqueness of an infinite-dimensional stochastic differential equation (SDE) corresponding to the bulk limit of Dyson’s Brownian Motion, for all \(\beta \ge 1\). Our construction applies to an explicit and general class of initial conditions, including the lattice configuration \(\{x_i\}=\mathbb {Z}\) and the sine process. We further show the convergence of the finite to infinite-dimensional SDE. This convergence concludes the determinantal formula of Katori and Tanemura (Commun Math Phys 293(2):469–497, 2010) for the solution of this SDE at \(\beta =2\).

Keywords

Dyson’s Brownian motion Dyson’s model Stochastic differential equations Infinite-dimensional Strong existence Pathwise uniqueness Correlation function 

Mathematics Subject Classification

Primary 60K35 Secondary 60J60 82C22 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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