论文标题
马尔可夫链的定量粗粒
Quantitative coarse-graining of Markov chains
论文作者
论文摘要
粗粒技术在降低随机模型的复杂性方面起着核心作用，通常以映射为特征，该映射将系统的完整状态投射到较小的变量集中，该变量捕获了系统的基本特征。从连续的马尔可夫链开始，在这项工作中，我们提出和分析了有效的动力学，该动力学近似于粗粒链中的动力学信息。在不假设明确的比例分离的情况下，我们提供了足够的条件，在该条件下，该有效动力学靠近原始系统，并在近似误差上提供定量界限。我们还将有效的动力学和相应的误差界限与有关马尔可夫链的平均文献进行了比较，这些文献涉及明确的规模分离。我们在一个说明性的测试示例中演示了我们的发现。
Coarse-graining techniques play a central role in reducing the complexity of stochastic models, and are typically characterised by a mapping which projects the full state of the system onto a smaller set of variables which captures the essential features of the system. Starting with a continuous-time Markov chain, in this work we propose and analyse an effective dynamics, which approximates the dynamical information in the coarse-grained chain. Without assuming explicit scale-separation, we provide sufficient conditions under which this effective dynamics stays close to the original system and provide quantitative bounds on the approximation error. We also compare the effective dynamics and corresponding error bounds to the averaging literature on Markov chains which involve explicit scale-separation. We demonstrate our findings on an illustrative test example.
