Gibrat定律的动力学和平均场描述
Kinetic and mean field description of Gibrat's law
摘要：
我们介绍并分析了一个线性动力学模型，该模型描述了社会中企业数量的概率密度的演变，其中微观变化率服从Gibrat提出的所谓比例效应定律。尽管其明显的简单性，动力学模型的可能的平均场的限制是不同的。在某些情况下，渐近极限可以用一阶偏微分方程来描述。在其他情况下，平均场方程是具有非恒定扩散系数的线性扩散，其也模拟几何布朗运动并且可以解析地研究。在这种情况下，它示出的解决方案的大时间的行为表示，对于一个大类的初始数据，由对数正态分布与恒定的平均值和方差在时间上以精确的速率呈指数增长。动力学和扩散模型之间的关系允许引入一个易于实现的表达式来计算对数正态分布的傅立叶变换。
英文摘要：
We introduce and analyze a linear kinetic model that describes the evolution of the probability density of the number of firms in a society, in which the microscopic rate of change obeys to the so-called law of proportional effect proposed by Gibrat. Despite its apparent simplicity, the possible mean field limits of the kinetic model are varied. In some cases, the asymptotic limit can be described by a first-order partial differential equation. In other cases, the mean field equation is a linear diffusion with a non constant diffusion coefficient that models also the geometric Brownian motion and can be studied analytically. In this case, it is shown that the large-time behavior of the solution is represented, for a large class of initial data, by a lognormal distribution with constant mean value and variance increasing exponentially in time at a precise rate. The relationship between the kinetic and the diffusion models allow to introduce an easy-to- implement expression for computing the Fourier transform of the lognormal distribution.
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