Analysis of Random Transport in Chains using Modern Tools from Systems and Control Theory

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Random transport in chains is often modelled by continuous-time Markov processes on a finite, discrete state-space. In this proposal, we focus on cases where the transition rates of the process are either deterministic and vary periodically in time or are given as random variables. Both cases are well motivated by applications: periodic excitations are ubiquitous in systems biology, where biological organisms are exposed to the 24 hours solar day; in epidemiology where infection rates depend on annual rhythms; and in models of vehicular traffic where traffic lights follow a periodic operating pattern. Random rates are typically used to model uncertainty or variability in the exact values of the rates. For example, a recent paper studied the process of mRNA translation with the transition rates of the ribosomes along the mRNA molecule modeled as random variables. In these and many other application areas, stochastic models of the type we plan to investigate have become very popular. Thus, besides the mathematical analysis of these models, that forms the core of this proposal, applications of the main results to models from biology and physics will also be addressed.Central to the mathematical analysis are different classes of associated large-scale systems of ordinary differential equations. This includes the so-called Master Equation (sometimes also called the Pauli Master equation) as well as mean-field approximations thereof with varying degrees of accuracy. Recent results of the applicants have shown that the solutions of the periodic Master Equation as well as the solutions of certain periodic mean-field approximations in systems biology converge (under suitable conditions) to a unique periodic limit solution. Modern tools from systems and control theory like the theory of cooperative and contractive systems were pivotal for these findings. The applicants also recently proved the existence of a random attractor for a large class of random dynamical systems associated with irreducible, finite-state Markov processes with time-independent transition rates, and plan to extend these results to random dynamical systems with periodic coefficients. Starting from these results, this project aims at:- understanding the behavior of periodically driven stochastic systems for a wide range of model classes: Master Equations, Mean-Field Approximations and Random Dynamical Systems- understanding the relation between these different types of model classes, both qualitatively and quantitatively- designing systems with periodic trajectories with a desirable behavior, e.g., maximal throughput in traffic modelsTo reach this goal, both analytical and numerical methods will be used, with a strong emphasis on methods from mathematical systems and control theory.