We estimate emergent dynamics of Winfree oscillators under noise as in a heat bath. In a large coupling regime, The oscillators start accumulating in a small area in a short time while they spread by heat in long-time horizon, which is called a practical synchronization. Without noise, the deterministic Winfree model exhibits the oscillator death, emerging with a convergence of the phase ensemble. The additive noise, however, is expected to destroy the stability of the equilibrium. In this talk, by tracking the running maximum of the phase processes, we estimate the escaping probability from a small space interval near the equilibrium. This result quantitatively explains the robustness of the practical synchronization, which indicates that the finite-time emergent behavior from finite oscillators is close to the deterministic dynamics, known as the large deviation theory. Our approach produces explicit bounds on probabilities, relying on comparisons with the Ornstein-Uhlenbeck processes. It is hence optimal in the sense that the linearized model gives the same order.

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