Infinite Dimensional Stochastic Differential Equations for Dyson's Model

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 (DBM), 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 (2010) for the solution of this SDE at $\beta =2$.

40 pages; no figure. PTRF in press; updated to match published version. Mayor improvement (Theorem 1.4 and 1.5) introduced in v2

References in corpus (1)

• Zeros of Airy Function and Relaxation Process

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