Master equations for general non-Markovian processes: the Hawkes process and beyond

The Markovian process is one of the most important classes of stochastic processes. The Markovian process is defined as a stochastic process whose time evolution is independent of the system's entire history and has been extensively studied using the master equation and Fokker-Planck equation approaches. In contrast, non-Markovian processes -- where time evolution depends on the full history of the system -- have not been systematically explored, except for a few special cases, such as semi-Markovian processes. In this talk, we present a recent master-equation approach to general non-Markovian jump processes [1-4]. Beginning with a general non-Markovian jump process, we derive the corresponding master equation through a Markovian-embedding approach. The Markovian embedding is a scheme to add a sufficient number of auxiliary variables to convert a non-Markovian model to a high-dimensional Markovian model. For the case of our model, the one-dimensional non-Markovian model is shown to be equivalent to a Markovian stochastic field theory, and we derive the field master equation correspondingly [4]. As an application, we examine the nonlinear Hawkes process, a history-dependent and self-exciting model frequently used in studying complex systems [1-3]. Additionally, we explore the stochastic thermodynamic framework for general jump processes [5] as another example.