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arXiv 2102.01052math.PR
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具有激发和抑制的非线性Hawkes过程的平均场极限

Mean-field limits for non-linear Hawkes processes with excitation and inhibition

Peter Pfaffelhuber, Stefan Rotter, Jakob Stiefel

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摘要：

研究了具有$N$结点的完全图上的多元非线性Hawkes过程$Z^N$.每个顶点要么是兴奋性的（概率$p$），要么是抑制性的（概率$1-p$）。我们采取$Z^N$的平均场极限，导致一个多元点过程$\bar Z$。如果是$p\neq\tfrac12$，我们用$N$重新标度相互作用强度，发现极限强度过程解一个确定性卷积方程，并且$\bar Z$的所有分量是独立的。在临界情况下，$p=\tfrac12$，我们通过$N^{1/2}$重新标度并获得极限强度，该极限强度解随机卷积方程并且$\bar Z$的所有分量是条件独立的。

英文摘要：

We study a multivariate, non-linear Hawkes process $Z^N$ on the complete graph with $N$ nodes. Each vertex is either excitatory (probability $p$) or inhibitory (probability $1-p$). We take the mean-field limit of $Z^N$, leading to a multivariate point process $\bar Z$. If $p\neq\tfrac12$, we rescale the interaction intensity by $N$ and find that the limit intensity process solves a deterministic convolution equation and all components of $\bar Z$ are independent. In the critical case, $p=\tfrac12$, we rescale by $N^{1/2}$ and obtain a limit intensity, which solves a stochastic convolution equation and all components of $\bar Z$ are conditionally independent.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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