论文标题
稳定的分布和伪配置与分数通风函数有关
Stable distributions and pseudo-processes related to fractional Airy functions
论文作者
论文摘要
在本文中，我们研究了与Lévy稳定下属的奇数热型方程相关的伪过程。本文的目的是双重的。我们首先表明，次级伪过程的伪密度可以表示为对具有广义伽马分布参数的阻尼振荡的期望。这种随机表示也是作为分数扩散方程的解决方案，涉及一个高阶Riesz-纤维算子，该操作员概括了奇数级热型方程。然后，我们证明，如果稳定的下属具有合适的指数，那么随时间变化的伪过程将成为真正的lévy稳定过程。该结果使我们能够获得一个功率序列表示，以用于指数$ν> 1 $的任意不对称稳定过程和偏斜的参数$β$的概率密度函数，其中$ 0 <\lvertβ\ lvert <1 $。为了进行分析，我们使用的方法是基于对高阶Riesz-Feller Operator的研究中出现的分数通风函数的研究。
In this paper we study pseudo-processes related to odd-order heat-type equations composed with Lévy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be represented as an expectation of damped oscillations with generalized gamma distributed parameters. This stochastic representation also arises as the solution to a fractional diffusion equation, involving a higher-order Riesz-Feller operator, which generalizes the odd-order heat-type equation. We then prove that, if the stable subordinator has a suitable exponent, the time-changed pseudo-process becomes a genuine Lévy stable process. This result permits us to obtain a power series representation for the probability density function of an arbitrary asymmetric stable process of exponent $ν>1$ and skewness parameter $β$, with $0<\lvertβ\lvert<1$. The methods we use in order to carry out our analysis are based on the study of a fractional Airy function which emerges in the investigation of the higher-order Riesz-Feller operator.
