Asymptotic solutions to nonlinear Hawkes processes: A systematic classification of the steady-state solutions

The linear Hawkes point process is a first-order non-Markovian stochastic model of intermittent bursty dynamics. While its nonlinear extensions, called nonlinear Hawkes processes, are expected to be more powerful in describing the coexistence of excitatory and inhibitory effects (or negative feedback) as occurs, for instance, in seismic and neural systems, such nonlinear Hawkes processes have been found hitherto to be analytically intractable due to the interplay between their non-Markovian and nonlinear characteristics, with no analytical solutions available. Here we systematically classify the solutions of the nonlinear Hawkes processes and then present their various exact/asymptotic solutions using the field master equation approach introduced previously by us. We report explicit power-law formulas for the steady-state intensity distributions Pss(λ)∝λ-1-a, where the tail exponent a is expressed analytically as a function of parameters of the nonlinear Hawkes models. We introduce the basic analytical tools for advanced Hawkes modeling, particularly for model calibration to time-series data in various complex systems.