Ubiquitous power law intensity distributions in the self-excited nonlinear Hawkes processes

Self-excited mechanisms are prevalent in various complex systems, such as physical, seismic, neural, financial, and social systems, and one of the corresponding minimal models is the nonlinear Hawkes process. While this model has been popular and used for various data analyses of complex systems, its analytical properties have not been documented yet because of its non-Markovian and nonlinear characteristics. In this talk, we present its asymptotic solutions using the field master equation approach previously introduced by the authors. We first convert the original non-Markovian dynamics onto a high-dimensional Markovian field dynamics. By deriving the corresponding field master equation, the power law intensity distributions are found for a broad class of nonlinear Hawkes processes. Our work highlights the ubiquity of power laws in complex systems.