A system of differential equations for the airy process

Abstract

The Airy process τ → Ar is characterized by its finite-dimensional distribution functions Pr (Ar < ξ1; …Arm ξm): For m = 1 itis known that Pr (Ar < ξ) is expressible in terms of a solution to Painlevé II. We show that eachfinite-dimensional distribution function is expressible in terms of a solution to a system of differential equations. © 2003 Applied Probability Trust.