Slow-fast dynamics of the McKean model with stochastic resetting and diffusion

Abstract

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.