Stochastic Dynamics: Markov Chains, Random Transformations and Applications

Abstract

Stochastic dynamical systems, as a rapidly growing area in applied mathematics, has been a successful modeling framework for biology, chemistry and data science. Depending upon the origin of uncertainties in an application problem, the theory of stochastic dynamics has two different mathematical representations: stochastic processes and random dynamical systems (RDS). RDS is a more refined mathematical description of the reality; it provides not only the stochastic trajectory following one initial condition, but also describes how the entire phase space, with all initial conditions, simultaneously changes with time. Stochastic processes represent the stochastic movement of individual system. RDS, however, describes the motions of many systems that experience a common deterministic law that is randomly changing with time due to extrinsic noises, which represent fluctuating environment or complex external signal. The dynamics of an RDS may exhibit a quite counterintuitive phenomenon called noise-induced synchronization: the stochastic motions of noninteracting systems under a common noise synchronize; their trajectories become close to each other, while the individual one remains stochastic. In Chapter 2, I establish some elementary contradistinctions between Markov chain (MC) and RDS descriptions of stochastic dynamical systems with discrete time and discrete state space setting. In particular I study the linear representation of the RDS and show the expectation of the matrix-valued random variable is in fact the transition probability matrix of the corresponding MC induced by i.i.d. RDS. In Chapter 3, I study the metric entropy of MC and its corresponding RDS, and establish several inequalities about entropies and entropy productions. Next in Chapter 4 and Chapter 5, the theory of noise-induced synchronization is introduced together with a more intuitive version of the multiplicative ergodic theory, and then is applied to hidden Markov models for developing an efficient algorithm of parameter inference. In Chapter 6, The multi-dimensional Ornstein-Uhlenbeck process is used to study the dynamics of a free-draining polymer, in particular, the mean looping time. This work points to a future direction for stochastic model reduction.