随机动力学:内源与外源噪声的数学模型及其应用  

Stochastic dynamics: Models for intrinsic and extrinsic noises and their applications

作　　者：马易安 钱紘 叶晓峰 

机构地区：[1]Department of Applied Mathematics, University of Washington

出　　处：《中国科学：数学》2017年第12期1693-1702,共10页Scientia Sinica：Mathematica

摘　　要：复杂系统与过程的数学建模需要用随机动力学(stochastic dynamics)的思想和方法.随机动力学的理论有着两种不同的数学表述:随机过程(stochastic processes)和随机动力系统(random dynamical systems).后者是比前者更为精细的数学模型,它不但给出对应于每一个初值的随机过程,还全面地描述不同初值的多条随机轨道如何同时随时间变化.前者恰恰表述了有内在随机性的个体的运动,而后者则反映了多个相同的确定性个体同时经历同一个随机环境.本文称这两种情形为内源噪声和外源噪声.两者都在化学和生物学中有广泛的应用.近年来兴起的以图G(V,E)为基础的概率布尔网络正是一类以{0,1}~V为状态空间的随机动力系统(RDS).本文介绍有关离散时间离散空间的RDS,同时也给出一个它在统计推断隐Markov模型的收敛速率估算中的应用.Mathematical modeling for complex systems and processes requires concepts from and techniques for stochastic dynamics. The theory of stochastic dynamics has two different mathematical representations：Stochastic processes and random dynamical systems. The latter is a more refined mathematical description of reality; it provides not only a stochastic trajectory following one initial condition, but also describes how the entire phase space, with all initial conditions, changes with time. The former represents the stochastic motion of individual systems with intrinsic noise while the latter describes many systems experiencing a common deterministic law of motion which is changing with time due to environmental fluctuations. We call these two situations with intrinsic and extrinsic noises; both have wide applications in chemistry and biology. The recently developed, graph G（V, E） based probabilistic Boolean networks is precisely a class of random dynamical systems（RDS） with discrete state space {0, 1}~V. This paper introduces discrete-time RDS with discrete state space as well as discusses its applications in estimating a rate of convergence in hidden Markov model inference.

关 键 词：MARKOV链 随机映射 随机矩阵 概率布尔网络 随机梯度下降法 数学生物学 

分 类 号：O19[理学—数学]