Spatio-Temporal Log-Gaussian Cox-Hawkes Processes with Inhibition and Excitation for Stochastic Star Formation

We establish a connection between the stochastic self-propagating star-formation model and spatio-temporal point processes by showing that the SSPSF update law admits a conditional Poisson representation. Building on this connection, we propose a spatio-temporal log-Gaussian Cox-Hawkes process as a continuous point process model for stochastic star formation. The model represents star-formation events as point patterns driven jointly by deterministic galactic structure, latent spatio-temporal background variation, and dependence on past events. Its key feature is that the deterministic mean field, latent Gaussian random field, and history-dependent interaction field enter through a single log-intensity. This log-scale construction differs from additive Cox-Hawkes formulations and allows the history effect to be signed: past events may either increase or decrease future local intensity while the conditional intensity remains positive. The resulting framework provides an interpretable point-process model for representing latent clustering, self-excitation, local inhibition, and event-driven propagation in stochastic star formation. Beyond linking SSPSF to spatio-temporal point-process theory, it offers a continuous stochastic formulation for analysing the propagation of star formation in galaxies and for interpreting observational surveys of star-forming regions within a unified statistical model.