连续时间分裂-交互-聚合过程的副本-平均场极限
Replica-mean-field limit of continuous-time fragmentation-interaction-aggregation processes
AI总结：
本文引入连续分裂-交互-聚合过程，研究其副本-平均场版本，并证明在无限副本极限下泊松假设成立，为具有点状交互的网络动力学提供了可处理性。
摘要：
许多现象可以被建模为具有点状交互的网络动力学。然而，大多数相关动力学无法实现计算上的可处理性。为了克服这一困难，泊松假设（Poisson Hypothesis）将节点间的交互时间替换为独立的泊松过程，从而在诸如计算神经科学中基于强度的模型等若干情况下实现了可处理性。这一假设通常仅作为猜想提出，或通过数值验证。在本研究中，我们通过类比先前在离散时间中引入的过程，引入了一类连续时间过程，称为连续分裂-交互-聚合过程（continuous fragmentation-interaction-aggregation processes）。每个节点的状态由关联点过程的随机强度描述，它聚合来自邻居的到达，并在离开时发生分裂。我们考虑了此类过程的副本-平均场（replica-mean-field）版本，这是一个由目标网络的随机交互副本组成的物理系统。推广了在离散时间及基于兴奋性强度的神经动力学这一特定情形下证明的结果，我们证明了泊松假设在无限副本数量的极限下成立。
英文摘要：
Many phenomena can be modeled as network dynamics with punctuate interactions. However, most relevant dynamics do not allow for computational tractability. To circumvent this difficulty, the Poisson Hypothesis regime replaces interaction times between nodes by independent Poisson processes, allowing for tractability in several cases, such as intensity-based models from computational neuroscience. This hypothesis is usually only conjectured, or numerically validated. In this work, we introduce a class of processes in continuous time called continuous fragmentation-interaction-aggregation processes, by analogy with previously introduced processes in discrete time. The state of each node, described by the stochastic intensity of an associated point process, aggregates arrivals from its neighbors and is fragmented upon departure. We consider the replica-mean-field version of such a process, which is a physical system consisting of randomly interacting copies of the network of interest. Generalizing results proved in discrete time and in the particular case of excitatory intensity-based neural dynamics, we prove that the Poisson Hypothesis holds at the limit of an infinite number of replicas.
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