Identifying dynamical structures in the physical space of stochastic processes

Abstract

Characterizing dynamical patterns in the (physical) state space of stochastic processes can be a challenging task. From two visualization techniques, the observable-representation and k-means clustering, a unified framework to identify such structures is developed. The only information required is the system transition matrix R (a quantity that can be directly accessed from experimental data). The approach is illustrated through the analysis of random searches for targets distributed in patchy environments. The protocol —for R constructed from a typical tracked long trajectory— is able to reveal the shape and locations of all the landscape patches. The method constitutes a valuable new tool to study the underlying geometry of general stochastic processes.