PATHWISE ROBUSTNESS IN THE INSTABILITY OF THE INCOHERENT STATE TO THE KURAMOTO-FOKKER-PLANCK EQUATION WITH RANDOM INPUTS

초록

We present the pathwise robustness of an instability phenomenon for the incoherent state to the kinetic Kuramoto-Fokker-Planck equation with random inputs in a large coupling regime. For the case in which coupling strength and initial fluctuation are deterministic, it is well known that the incoherent state is nonlinearly unstable by constructing an exponentially growing mode. In contrast, when uncertainties are present in the coupling strength and initial data, we show that the nonlinear fluctuations around the incoherent state exist globally and grow exponentially fast along the sample path as well. Thus, the instability phenomenon of the incoherent state is robust with respect to random inputs as long as the supports of natural frequency distribution are confined in a sufficiently small symmetric interval around zero.

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