Diffusive slow-resetting switching dynamics

Develop a generalized switching-system analysis for the slow-resetting McKean model that incorporates Gaussian fluctuations, including the small-diffusion regime in which the slow dynamics remains near the fast nullcline except for rare noise-induced transitions caused by resetting or large Gaussian fluctuations.

Background

The paper analyzes the piecewise-linear McKean slow–fast model under stochastic resetting and diffusion. When resetting is fast relative to the slow variable, the authors derive a nonequilibrium stationary state for the fast variable and an averaged equation for the slow variable. When resetting is slow and diffusion is absent, resetting induces stochastic switching between the attracting outer branches of the fast nullcline, allowing the authors to derive stationary branch densities for the slow variable.

The unresolved extension concerns the slow-resetting regime with Gaussian diffusion. In that setting, the slow variable is no longer confined to the fast nullcline between resetting events, so the two-state branch-switching description used in the non-diffusive analysis does not directly apply. The authors identify the small-diffusion limit as a potentially tractable setting because the dynamics should remain close to the fast nullcline except during rare noise-induced transitions.