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We study the synchronization behavior of discrete-time Markov chains on countable state spaces. Representing a Markov chain in terms of a random dynamical system, which describes the collective dynamics of trajectories driven by the same noise, allows for the characterization of synchronization via random attractors. We establish the existence and uniqueness of a random attractor under mild conditions and show that forward and pullback attraction are equivalent in our setting. Additionally, we provide necessary and sufficient conditions for reaching the random attractor, or synchronization respectively, in a time of finite mean. By introducing insulated and synchronizing sets, we structure the state space with respect to the synchronization behavior and characterize the shape of the random attractor.

We study a prototypical non-polynomial decision-making model for which agents in a population potentially alternate between two consumption strategies, one related to the exploitation of an unlimited but considerably expensive resource and the other a comparably cheaper but restricted and slowly renewable source. In particular, we study a model following a Boltzmann-like exploration policy, enhancing the accuracy at which the exchange rates are captured with respect to classical polynomial approaches by considering sigmoidal functions to represent the cost-profit relation in both exploit strategies. Additionally, given the intrinsic timescale separation between the decision-making process and recovery rates of the renewable resource, we use geometric singular perturbation theory to analyze the model. We further use numerical analysis to determine parameter ranges for which the model undergoes bifurcations. These bifurcations, being related to critical states of the system, are relevant to the fast transitions between strategies. Hence, we design controllers to regulate such rapid transitions by taking advantage of the system's criticality., Comment: 40 pages, 20 figures

Coherent sets are time-dependent regions in the physical space of nonautonomous flows that exhibit little mixing with their neighborhoods, robustly under small random perturbations of the flow. They thus characterize the global long-term transport behavior of the system. We propose a framework to extract such time-dependent families of coherent sets for nonautonomous systems with an ergodic driving dynamics and (small) Brownian noise in physical space. Our construction involves the assembly and analysis of an operator on functions over the augmented space of the associated skew product that, for each fixed state of the driving, propagates distributions on the corresponding physical-space fibre according to the dynamics. This time-dependent operator has the structure of a semigroup (it is called the Mather semigroup), and we show that a spectral analysis of its generator allows for a trajectory-free computation of coherent families, simultaneously for all states of the driving. Additionally, for quasi-periodically driven torus flows, we propose a tailored Fourier discretization scheme for this generator and demonstrate our method by means of three examples of two-dimensional flows.

Functional central limit theorems for random processes viewed as rough paths were obtained in a wide range of models and settings in recent years. Each result of this kind has described and identified the limit Brownian rough path within the specifically suitable set-up. Taking a unified point of view, we provide simple and exact formulae, of Green-Kubo type, that characterize the limit Brownian rough path and demonstrate how they naturally apply in each case., Comment: 22 pages