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

链中的随机输运通常用有限离散状态空间上的连续时间马尔可夫过程来建模。在这个建议中，我们关注的是过程的转换速率是确定性的，并且随时间周期性变化，或者是作为随机变量给出的情况。这两种情况都有很好的应用动机：周期性激励在系统生物学中是普遍存在的，其中生物有机体暴露于24小时太阳日；在流行病学中，感染率取决于年节律；在车辆交通模型中交通灯遵循周期性的运行模式。随机速率通常用于模拟速率精确值的不确定性或可变性。例如，最近的一篇论文研究了mRNA的翻译过程，将核糖体沿mRNA分子的转移速率建模为随机变量。在这些和许多其他应用领域，我们计划研究的这种类型的随机模型已经变得非常流行。因此，除了这些模型的数学分析（这是本提案的核心）之外，还将讨论将主要结果应用于生物学和物理学模型的问题。数学分析的中心是不同类别的相关的常微分方程的大规模系统。这包括所谓的主方程（有时也称为泡利主方程）以及其精度不同程度的平均场近似。申请者最近的研究结果表明，周期主方程的解以及系统生物学中某些周期平均场近似的解（在适当条件下）收敛于一个唯一的周期极限解。来自系统和控制理论的现代工具，如合作和收缩系统理论，对这些发现至关重要。申请者最近还证明了随机吸引子的存在，这类随机动力系统与不可约的、有限状态的马尔可夫过程有关，具有时间无关的过渡速率，并计划将这些结果扩展到具有周期系数的随机动力系统。从这些结果出发，本项目旨在：-理解周期驱动随机系统的行为为广泛的模型类；主方程，平均场近似和随机动力系统-理解这些不同类型的模型类之间的关系，定性和定量-设计具有理想行为的周期性轨迹的系统，例如，交通模型中的最大吞吐量为了达到这一目标，将使用解析和数值方法，并强调数学系统和控制理论的方法。

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.