离散时间马尔可夫主体通过势相互作用
Discrete Time Markovian Agents Interacting Through a Potential
摘要：
研究了由大量相互作用的马尔可夫链组成的多智能体系统的离散时间随机模型。相互作用粒子的演化是通过一个时间不均匀的转移概率内核，取决于“梯度”的势场。粒子反过来通过它们的累积输入动态地修改势场。上述形式的相互作用马尔可夫过程已被建议作为响应于外部刺激（如化学梯度）的主动生物运输的模型。基本的数学挑战之一是发展一个一般的理论，稳定性的相互作用马尔可夫系统和相应的非线性马尔可夫过程中出现的大代理限制。这样的理论将是关键的数学理解的互动结构的形成，结果从复杂的反馈之间的代理和潜在的领域。它也将是开发长期忠实于基本模型的模拟方案的关键因素。本文的目的是研究当粒子数N和时间参数n趋于无穷大时上述随机系统的定性性质。在这方面，一个确定性的非线性动力系统，在随机模型的混沌极限的传播中出现的渐近性质，起着关键的作用。我们证明了在适当的条件下，这个动力系统有唯一的不动点。这一结果使我们能够研究潜在的随机模型的稳定性。我们表明，作为N \rightarrow \infty，随机系统很好地近似的动力系统，一致的时间。结果表明，对于任意初始化的系统，当N \rightarrow \infty和n \rightarrow \infty时，相互作用粒子的势场和经验测度收敛到动力系统的唯一不动点.在一般情况下，这种相互作用的马尔可夫系统的模拟是一个计算艰巨的任务。我们提出了一个基于粒子的近似的动态势场，它允许一个数值上易于处理的模拟方案。结果表明，这种模拟方案很好地近似真实的物理系统，在无限的时间范围内一致。
英文摘要：
A discrete time stochastic model for a multiagent system given in terms of a large collection of interacting Markov chains is studied. The evolution of the interacting particles is described through a time inhomogeneous transition probability kernel that depends on the 'gradient' of the potential field. The particles, in turn, dynamically modify the potential field through their cumulative input. Interacting Markov processes of the above form have been suggested as models for active biological transport in response to external stimulus such as a chemical gradient. One of the basic mathematical challenges is to develop a general theory of stability for such interacting Markovian systems and for the corresponding nonlinear Markov processes that arise in the large agent limit. Such a theory would be key to a mathematical understanding of the interactive structure formation that results from the complex feedback between the agents and the potential field. It will also be a crucial ingredient in developing simulation schemes that are faithful to the underlying model over long periods of time. The goal of this work is to study qualitative properties of the above stochastic system as the number of particles (N) and the time parameter (n) approach infinity. In this regard asymptotic properties of a deterministic nonlinear dynamical system, that arises in the propagation of chaos limit of the stochastic model, play a key role. We show that under suitable conditions this dynamical system has a unique fixed point. This result allows us to study stability properties of the underlying stochastic model. We show that as N \rightarrow \infty, the stochastic system is well approximated by the dynamical system, uniformly over time. As a consequence, for an arbitrarily initialized system, as N\rightarrow \infty and n \rightarrow \infty, the potential field and the empirical measure of the interacting particles are shown to converge to the unique fixed point of the dynamical system. In general, simulation of such interacting Markovian systems is a computationally daunting task. We propose a particle based approximation for the dynamic potential field which allows for a numerically tractable simulation scheme. It is shown that this simulation scheme well approximates the true physical system, uniformly over an infinite time horizon.
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