We study the long time behavior of the solution to some McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic external quantity (we call it the external current). For constant current, we obtain the convergence to the invariant measure. Using a perturbation method, we extend this result to more general external currents. Finally, we prove the result for the non-linear McKean-Vlasov equation.

中文翻译：

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相互作用神经元的平均场模型的长时间行为

我们研究了由泊松过程驱动的某些 McKean-Vlasov 随机微分方程 (SDE) 解的长时间行为。在神经科学中，该 SDE 模拟了大型网络中尖峰神经元的膜电位的渐近动态。我们证明，对于足够小的交互参数，任何解决方案都会收敛到唯一的（在这种情况下）不变测度。为此，我们首先获得跳跃率的全局界限，并推导出满足该跳跃率的 Volterra 型积分方程。然后我们用一个确定性的外部量（我们称之为外部电流）临时替换方程的相互作用部分。对于恒定电流，我们获得了对不变测度的收敛。使用扰动方法，我们将此结果扩展到更一般的外部电流。最后，