Markov链的定量粗粒化
Quantitative coarse-graining of Markov chains
AI总结：
针对连续时间Markov链，本文提出一种无需显式尺度分离的有效动力学来近似粗粒化链，并建立近似误差的定量界。
摘要：
粗粒化技术在降低随机模型复杂性方面发挥着核心作用，其通常特征是一个映射，该映射将系统的完整状态投影到一组更小的变量上，而这些变量捕捉系统的本质特征。本文从连续时间Markov链出发，提出并分析了一种有效动力学，用以近似粗粒化链中的动力学信息。在不假设有显式尺度分离的情况下，我们给出了使该有效动力学与原系统保持接近的充分条件，并给出了近似误差的定量界。我们还将该有效动力学及相应误差界与涉及显式尺度分离的Markov链平均化文献进行了比较，并通过一个说明性测试示例展示了研究结果。
英文摘要：
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.
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