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arXiv 1408.0632math.PRmath-phmath.MP
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源于 Airy 随机点场的无穷维随机微分方程

Infinite-dimensional stochastic differential equations arising from Airy random point fields

Hirofumi Osada, Hideki Tanemura

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94 pages, A revised version
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AI总结：

本文为 Airy$_{β}$ 随机点场构造了可逆的无标号扩散过程，推导并证明其无穷维随机微分方程（ISDEs）强解的存在性与路径唯一性，且该解为 $N$ 粒子系统软边缘极限，同时提出了一种求解 ISDEs 的新方法。

摘要：

Airy$_{β}$ 随机点场（$β=1,2,4$）是作为高斯随机矩阵特征值的软边缘标度极限而出现的随机点场。我们为每个 $β=1,2,4$ 构造了关于 Airy$_{β}$ 随机点场可逆的无标号扩散过程。我们推导了描述上述无标号扩散过程标号随机动力学的无穷维随机微分方程（ISDEs）。我们证明了这些 ISDEs 强解的存在性与路径唯一性。此外，该 ISDE 的解是软边缘极限下描述 $N$ 粒子系统动力学随机微分方程解的极限。我们由此建立了无标号动力学关于 Airy 随机点场可逆的随机动力学构造方法。当 $β=2$ 时，该解等于由 Prähofer--Spohn、Johansson、Katori--Tanemura 和 Corwin--Hammond 等人通过时空相关函数定义的随机动力学。我们开发了一种新方法，使得这些 ISDEs 具有唯一的强解。我们预期该方法适用于源自随机矩阵理论的随机动力学的其他软边缘标度极限。

英文摘要：

The Airy$_{β}$ random point fields ($ β= 1,2,4$) are random point fields emerging as the soft-edge scaling limits of eigenvalues of Gaussian random matrices. We construct the unlabeled diffusion reversible with respect to the Airy$_{β}$ random point field for each $ β= 1,2,4$. We identify the infinite-dimensional stochastic differential equations (ISDEs) describing the labeled stochastic dynamics for the unlabeled diffusion mentioned above. We prove the existence and pathwise uniqueness of strong solutions of these ISDEs. Furthermore, the solution of the ISDE is the limit of the solutions of the stochastic differential equations describing the dynamics of the $ N $-particle system in the soft-edge limit. We thus establish the construction of the stochastic dynamics whose unlabeled dynamics are reversible with respect to the Airy random point fields. When $ β=2 $, the solution equals the stochastic dynamics defined by the space-time correlation functions obtained by Prähofer--Spohn, Johansson, Katori--Tanemura, and Corwin--Hammond, among others. We develop a new method whereby these ISDEs have unique, strong solutions. We expect that our approach is valid for other soft-edge scaling limits of stochastic dynamics arising from the random matrix theory.

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