ON THE STOCHASTIC SYNCHRONIZATION OF THE WINFREE MODEL WITH A MULTIPLICATIVE NOISE∗

We study long-time behaviors of stochastic Winfree oscillators under the effect of multiplicative white noise. Under additive white noise, it is well known that the deterministic drift cannot make oscillators be accumulated in a small area, where the ensemble forms a smooth density profile as in the heat equation. However, with the multiplicative noise, especially when the noise strength is proportional to the drift coefficient as in the geometric Brownian motion, we get a stochastic convergence toward an equilibrium. Based on previous results from the stochastic Kuramoto model, we establish the emergence of synchronization for the stochastic Winfree model. Moreover, the proposed argument works in a more general setting, “the gradient system with multiplicative noise”. In particular, it explains and enlarges a previous condition of convergence in the stochastic Kuramoto model via the framework of a generalized Winfree model, and it can be extended to the corresponding kinetic model from the mean-field limit.