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

Abstract

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 $G(\mathscr{V},\mathscr{E})$ based probabilistic Boolean networks is precisely a class of random dynamical systems (RDS) with discrete state space $\{0,1\}^{\mathscr{V}}$. 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.