Asymptotic Behaviors and Perturbation Analysis of Stochastic Dynamics and Applications to Complex Systems

Abstract

The concept of hierarchical structures prevails among scientific points of view on complex systems. From one level to another in a hierarchical structure, with proper scales in both space and time, entire new laws emerge in the limits. Not only in natural science, but in formal science, mathematicians have widely applied this idea of scaling to prove several celebrated limit theorems. In the first part of the present work, new limit laws in conditional probability are shown. These new laws can be regarded as the mathematical foundation of Gibbisian canonical ensemble theory. Based on the new laws, the canonical ensemble theory can be generalized to strongly coupled heterogeneous systems. Another parallel canonical ensemble theory by Boltzmann is also discussed with applications to sample frequencies of the phenotype among a population of cells. In the second part, asymptotic behaviors of nonlinear dynamical systems under random perturbations have a full analysis. In particular, an underlying kinematic basis of the landscape of the dynamics emerges in the deterministic limit. These results provide a lens for helping us look through stochastic limit-cycle oscillations from different perspectives. In addition, entropy, entropy production, free energy dissipation of complex dynamics at the mesoscopic scale and the corresponding limiting behaviors at the macroscopic scale are depicted by the language of probability theory.