When modeling a physical system using a Markov chain, it is often instructive to compute its probability distribution at statistical equilibrium, thereby gaining insight into the stationary, or long-term, behavior of the system. Computing such a distribution directly is problematic when the state space of the system is large. Here, we look at the case of a chemical reaction system that models the dynamics of cellular processes, where it has become popular to constrain the computational burden by using a finite state projection, which aims only to capture the most likely states of the system, rather than every possible state. We propose an efficient method to further narrow these states to those that remain highly probable in the long run, after the transient behavior of the system has dissipated. Our strategy is to quickly estimate the local maxima of the stationary distribution using the reaction rate formulation, which is of considerably smaller size than the full-blown chemical master equation, and from there develop adaptive schemes to profile the distribution around the maxima. We include numerical tests that show the efficiency of our approach.

中文翻译：

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有限状态投影，用于使用反应速率方程式近似固定溶液至化学主方程式。

使用马尔可夫链对物理系统进行建模时，通常有启发性的是在统计平衡时计算其概率分布，从而深入了解系统的静态或长期行为。当系统的状态空间很大时，直接计算这样的分布是有问题的。在这里，我们看一个化学反应系统的案例，该系统模拟细胞过程的动力学，其中已经流行通过使用有限状态投影来限制计算负担的方法，该方法仅旨在捕获系统最可能的状态，而不是所有可能的状态。我们提出了一种有效的方法，可以在系统的瞬态行为消除后，将这些状态进一步缩小到长期可能保持高度状态的状态。我们的策略是使用反应速率公式快速估算平稳分布的局部最大值，该公式的大小要比成熟的化学主方程要小得多，然后从中开发自适应方案来描绘最大值附近的分布。我们包含的数值测试表明了我们方法的有效性。