具有二次扩散系数和线性力的Pearson族遍历扩散过程的大偏差
Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force
AI总结：
本文在统一框架下研究Pearson族遍历扩散过程的大偏差性质，通过Level 1和Level 2分析显式计算特定观测量的大偏差，并用于长轨迹数据的参数统计推断，得到Pearson线性力两个推断参数的显式速率函数。
摘要：
具有二次扩散系数和线性力的Pearson族遍历扩散过程，其特征是其整数矩的显式动力学以及朝向稳态的显式弛豫谱性质。除了具有高斯稳态的Ornstein-Uhlenbeck过程之外，Pearson族的其他代表性例子包括收敛到Gamma分布的Square-Root过程或Cox-Ingersoll-Ross过程、收敛到Beta分布的Jacobi过程、reciprocal-Gamma过程（对应于布朗运动的指数泛函）收敛到Inverse-Gamma分布、Fisher-Snedecor过程以及Student过程，因此最后三个稳态表现出重尾。本文的目标是在一个统一框架中分析这些不同扩散过程的大偏差性质。我们首先考虑Level 1，涉及大时间窗口$T$上的时间平均观测量：我们写出一般观测量的第一重标度累积量，并识别出其大偏差可以从相应变形生成元的主导特征值显式计算的具体观测量。然后利用关于时间平均密度的Level 2显式大偏差来分析从非常长的随机轨迹数据中对模型参数的统计推断，从而得到Pearson线性力的两个推断参数的显式速率函数。
英文摘要：
The Pearson family of ergodic diffusions with a quadratic diffusion coefficient and a linear force are characterized by explicit dynamics of their integer moments and by explicit relaxation spectral properties towards their steady state. Besides the Ornstein-Uhlenbeck process with a Gaussian steady state, the other representative examples of the Pearson family are the Square-Root or the Cox-Ingersoll-Ross process converging towards the Gamma-distribution, the Jacobi process converging towards the Beta-distribution, the reciprocal-Gamma process (corresponding to an exponential functional of the Brownian motion) that converges towards the Inverse-Gamma-distribution, the Fisher-Snedecor process, and the Student process, so that the last three steady states display heavy-tails. The goal of the present paper is to analyze the large deviations properties of these various diffusion processes in a unified framework. We first consider the Level 1 concerning time-averaged observables over a large time-window $T$ : we write the first rescaled cumulants for generic observables and we identify the specific observables whose large deviations can be explicitly computed from the dominant eigenvalue of the appropriate deformed-generator. The explicit large deviations at Level 2 concerning the time-averaged density are then used to analyze the statistical inference of model parameters from data on a very long stochastic trajectory in order to obtain the explicit rate function for the two inferred parameters of the Pearson linear force.
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