随机指数增长现象学
Phenomenology of stochastic exponential growth
摘要：
随机指数增长可以在各种背景下观察到，包括分子自催化，核裂变，人口增长，宇宙膨胀，病毒式社交媒体帖子和金融市场。然而，这些随机动力学的现象学建模的文献主要集中在一个模型，几何布朗运动（GBM），它可以被描述为线性漂移和线性乘性噪声的朗之万方程的解决方案。使用最近的实验结果，对单个细菌细胞大小的随机指数增长，我们激励需要一个更一般的类的现象学模型的随机指数增长，这是一致的观察，即平均重新缩放的分布是近似平稳在很长一段时间。我们表明，这种行为是不一致的GBM，而是与幂律乘性噪声与正分数的权力。因此，我们认为这一般类随机指数增长的唯象模型，提供解析解，并确定重要的无量纲组合的模型参数，确定形状的均值重新缩放分布。我们还提供了一个处方，从实验观察到的随机增长轨迹的鲁棒性推断模型参数。
英文摘要：
Stochastic exponential growth is observed in a variety of contexts, including molecular autocatalysis, nuclear fission, population growth, inflation of the universe, viral social media posts, and financial markets. Yet literature on modeling the phenomenology of these stochastic dynamics has predominantly focused on one model, Geometric Brownian Motion (GBM), which can be described as the solution of a Langevin equation with linear drift and linear multiplicative noise. Using recent experimental results on stochastic exponential growth of individual bacterial cell sizes, we motivate the need for a more general class of phenomenological models of stochastic exponential growth, which are consistent with the observation that the mean-rescaled distributions are approximately stationary at long times. We show that this behavior is not consistent with GBM, instead it is consistent with power law multiplicative noise with positive fractional powers. Therefore, we consider this general class of phenomenological models for stochastic exponential growth, provide analytical solutions, and identify the important dimensionless combination of model parameters which determines the shape of the mean-rescaled distribution. We also provide a prescription for robustly inferring model parameters from experimentally observed stochastic growth trajectories.
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