It is well known that stochastically modeled reaction networks that are complex balanced admit a stationary distribution that is a product of Poisson distributions. In this paper, we consider the following related question: supposing that the initial distribution of a stochastically modeled reaction network is a product of Poissons, under what conditions will the distribution remain a product of Poissons for all time? By drawing inspiration from Crispin Gardiner's "Poisson representation" for the solution to the chemical master equation, we provide a necessary and sufficient condition for such a product-form distribution to hold for all time. Interestingly, the condition is a dynamical "complex-balancing" for only those complexes that have multiplicity greater than or equal to two (i.e. the higher order complexes that yield non-linear terms to the dynamics). We term this new condition the "dynamical and restricted complex balance" condition (DR for short).

中文翻译：

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具有高阶络合物的反应网络的时间相关产品形式泊松分布。

众所周知，复杂平衡的随机建模反应网络可以接受泊松分布产物的平稳分布。在本文中，我们考虑以下相关问题：假设随机建模的反应网络的初始分布是泊松的产物，那么在什么条件下该分布将一直保持泊松的产物？通过从克里斯平·加德纳（Crispin Gardiner）的“泊松表示”中获得化学主方程式解的灵感，我们为这种产品形式的分布始终保持提供了必要和充分的条件。有趣的是，条件是动态的“复杂平衡”，仅适用于多重度大于或等于2（即 产生动力学非线性项的高阶复合物）。我们将此新条件称为“动态且受限的复杂平衡”条件（简称DR）。