论文标题
非线性鹰派工艺的平均场限制对Erdős-rényi-graph的抑制作用
Mean-field limits for non-linear Hawkes processes with inhibition on a Erdős-Rényi-graph
论文作者
论文摘要
我们在$ q $-erdős-rényi-Graph上使用$ n $ nodes研究了多元的非线性鹰队流程$ z^n $。每个顶点是兴奋性（概率$ p $）或抑制性（概率$ 1-P $）。如果$ p \ neq \ tfrac12 $，我们采用$ z^n $的平均场限制，从而导致多元点过程$ \ bar z $。我们将交互强度缩减为$ n $，发现极限强度过程求解了确定性卷积方程，并且$ \ bar z $的所有组件都是独立的。平均场限制周围的波动会收敛到随机卷积方程的解。在关键情况下，$ p = \ tfrac12 $，我们通过$ n^{1/2} $恢复并讨论启发式上和数字上的困难。
We study a multivariate, non-linear Hawkes process $Z^N$ on a $q$-Erdős-Rényi-graph with $N$ nodes. Each vertex is either excitatory (probability $p$) or inhibitory (probability $1-p$). If $p\neq\tfrac12$, we take the mean-field limit of $Z^N$, leading to a multivariate point process $\bar Z$. 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. The fluctuations around the mean field limit converge to the solution of a stochastic convolution equation. In the critical case, $p=\tfrac12$, we rescale by $N^{1/2}$ and discuss difficulties, both heuristically and numerically.
