McKean模型的慢-快动力学：随机重置与扩散
Slow-fast dynamics of the McKean model with stochastic resetting and diffusion
- Imperial College London（帝国理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究随机重置与扩散对McKean慢-快系统的影响，通过绝热近似得到快变量NESS，推导慢变量平均方程并分析其稳定不动点，揭示重置与扩散对系统长期行为的联合作用。
AI中文摘要：
本文研究了随机重置和扩散对由分段线性McKean模型给出的慢-快动力系统的联合效应。即，快变量$v$受到有效扩散系数为$D$的高斯白噪声的影响，并在由速率为$r$的泊松过程生成的随机时间序列上重置为固定值$v_r$。假设重置发生在快时间尺度上，我们在绝热近似下冻结慢变量$w$，并将快变量的非平衡稳态（NESS）确定为修正的Fokker-Planck方程的解。我们证明NESS可以用抛物柱面函数表示，其渐近行为使我们能够在扩散较小的极限下恢复相应的无扩散NESS。该NESS用于推导慢动力学的平均方程，其解收敛到依赖于$D$、$r$和$v_r$的稳定不动点$w^*$。该不动点有效地决定了整个系统的长期行为。最后，我们分析了重置发生在与慢变量相同时间尺度上的情况。我们展示了慢变量现在如何由于重置引起的快零斜线分支之间的切换而经历噪声振荡，并在无扩散情况下推导出$w$的相应NESS。
英文摘要：
In this paper we investigate the combined effects of stochastic resetting and diffusion on a slow--fast dynamical system given by the piecewise-linear McKean model. That is, the fast variable $v$ is subject to Gaussian white noise with effective diffusivity $D$ and is reset to a fixed value $v_r$ at a random sequence of times generated from a Poisson process with rate $r$. Assuming that resetting occurs on the fast timescale, we freeze the slow variable \(w\) under an adiabatic approximation and determine the resulting non-equilibrium stationary state (NESS) of the fast variable as the solution of a modified Fokker--Planck equation. We show that the NESS can be expressed in terms of parabolic cylinder functions, whose asymptotic behavior allows us to recover the corresponding NESS without diffusion in the small-diffusion limit. The NESS is used to derive an averaged equation for the slow dynamics, whose solution converges to a stable fixed point \(w^*\) that depends on \(D\), \(r\) and \(v_r\). This fixed point effectively determines the long-time behaviour of the full system. Finally, we analyze the regime in which resetting occurs on the same timescale as the slow variable. We show how the slow variable now undergoes noisy oscillations due to resetting-induced switching between branches of the fast nullcline and derive the corresponding NESS for $w$ in the non-diffusive case.
