Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases
Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases
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DOI：
10.3842/sigma.2022.049
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发表时间：
2021-06
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通讯作者：
Taiki Endo;M. Katori;Noriyoshi Sakuma
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作者：
Taiki Endo;M. Katori;Noriyoshi Sakuma
我们研究了三种一维随机对数气体的流体动力学极限，称为戴森的布朗运动模型，它的手征版本，和Bru-Wishart过程中研究的动态随机矩阵理论。我们定义了测度值过程，使得它们的柯西变换解复杂的Burgers型方程。我们表明，这些偏微分方程的特征曲线的方法的应用程序提供的函数方程有关的柯西变换的措施在任意时间与那些在初始时间。我们将柯西变换的函数方程转化为测度的R-变换和S-变换的函数方程，它们在自由概率论中起着核心作用。所得到的R-变换和S-变换的函数方程比Cauchy变换的函数方程简单，并且对于包括自由累积量序列的显式计算都是有用的。一些结果是使用自由卷积的概念进行论证的。
We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve the complex Burgers-type equations. We show that applications of the method of characteristic curves to these partial differential equations provide the functional equations relating the Cauchy transforms of measures at an arbitrary time with those at the initial time. We transform the functional equations for the Cauchy transforms to those for the R-transforms and the S-transforms of the measures, which play central roles in free probability theory. The obtained functional equations for the R-transforms and the S-transforms are simpler than those for the Cauchy transforms and useful for explicit calculations including the computation of free cumulant sequences. Some of the results are argued using the notion of free convolutions.