Spatio-Temporal Intensity Estimation for Point Processes on Complicated Domains

Spatio-temporal point processes provide a flexible framework for modeling events occurring over space and time, with applications in fields such as epidemiology, geophysics, and urban systems. In this work, we present a unified framework for spatio-temporal intensity estimation on complex domains. The proposed approach builds on a nonparametric penalized likelihood formulation, where smoothness is enforced through differential operators defined consistently with the geometry of the domain. This allows for a principled incorporation of spatial structure without relying on ad hoc distance definitions. The methodology accommodates a wide range of domain types, including planar regions, Riemannian manifolds, and linear networks, within a common modeling perspective.