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Stochastic dynamical systems arise naturally across nearly all areas ofscience and engineering. Typically, a dynamical system model is based on someprior knowledge about the underlying dynamics of interest in whichprobabilistic features are used to quantify and propagate uncertaintiesassociated with the initial conditions, external excitations, etc. From aprobabilistic modeling standing point, two broad classes of methods exist, i.e.macro-scale methods and micro-scale methods. Classically, macro-scale methodssuch as statistical moments-based strategies are usually too coarse to capturethe multi-mode shape or tails of a non-Gaussian distribution. Micro-scalemethods such as random samples-based approaches, on the other hand, becomecomputationally very challenging in dealing with high-dimensional stochasticsystems. In view of these potential limitations, a meso-scale scheme isproposed here that utilizes a meso-scale statistical structure to describe thedynamical evolution from a probabilistic perspective. The significance of thisstatistical structure is two-fold. First, it can be tailored to any arbitraryrandom space. Second, it not only maintains the probability evolution aroundsample trajectories but also requires fewer meso-scale components than themicro-scale samples. To demonstrate the efficacy of the proposed meso-scalescheme, a set of examples of increasing complexity are provided. Connections tothe benchmark stochastic models as conservative and Markov models along withpractical implementation guidelines are presented.

Probabilistic Evolution of Stochastic Dynamical Systems: A Meso-scale Perspective