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

Technical Summary: Spatio-Temporal Log-Gaussian Cox-Hawkes Processes for Stochastic Star Formation

Problem Statement
The stochastic self-propagating star formation (SSPSF) model, originally introduced by Mueller and Arnett (1976) and refined by subsequent researchers, provides a rule-based framework for understanding how local interactions and feedback generate large-scale star-formation patterns in galaxies. However, the relationship between the classical SSPSF update laws and modern spatio-temporal point process theory has not been explicitly formulated. Furthermore, existing continuous models for clustered events, such as additive Cox-Hawkes processes, typically assume that past events only increase future intensity (excitation). They struggle to naturally represent inhibitory effects (where past events suppress future activity) while maintaining a positive conditional intensity. Additionally, standard Log-Gaussian Cox processes (LGCP) capture latent background clustering but fail to explicitly model event-driven triggering and inhibition.

Methodology
The paper proposes a unified framework that bridges the gap between discrete SSPSF models and continuous spatio-temporal point processes through the following steps:

1. Theoretical Connection: The authors first demonstrate that under suitable discretization, the SSPSF update law can be represented as a separable spatio-temporal Hawkes process. This establishes a formal link between the rule-based SSPSF mechanism and point-process theory.
2. Model Formulation: Building on this connection, the paper introduces a Spatio-Temporal Log-Gaussian Cox-Hawkes Process. The conditional intensity $\lambda(s, t | H_t)$ is defined on a log-scale rather than an additive scale:
   $$\lambda(s, t | H_t) = \exp\{ \mu(s, t) + Y(s, t) + D(s, t | H_t) \}$$
   Where:
   • $\mu(s, t)$ is a deterministic mean field representing large-scale galactic structure.
   • $Y(s, t)$ is a latent spatio-temporal Gaussian random field capturing unobserved environmental variations (e.g., gas density, metallicity).
   • $D(s, t | H_t)$ is a history-dependent dynamic field representing the cumulative effect of past events.
3. Signed History Dependence: The history term $D$ is defined as a sum of spatio-temporal kernels weighted by a parameter $\alpha \in \mathbb{R}$. Crucially, because the history term enters the log-intensity, $\alpha$ can be negative. This allows past events to either increase (excitation, $\alpha > 0$) or decrease (inhibition, $\alpha < 0$) the local intensity, while the exponential function ensures the conditional intensity remains strictly positive.
4. Recovery Time Extension: To address physical constraints and mathematical stability, a "recovery time" version of the model is introduced. In this variant, an event at time $t_i$ does not influence the intensity until time $t_i + \rho$. This delay prevents instantaneous self-triggering cascades.
5. Mathematical Analysis: The paper utilizes a Poisson embedding construction to prove the existence of the process. It establishes:
   • Non-explosion (finite events in finite time) for the non-excitatory case ($\alpha \leq 0$).
   • The possibility of explosion (infinite events in finite time) for the excitatory case ($\alpha > 0$) under specific conditions.
   • Guaranteed non-explosion for the recovery-time model for any $\alpha \in \mathbb{R}$.

Key Contributions

• Unified Framework: The paper provides the first continuous stochastic formulation that simultaneously integrates deterministic galactic structure, latent background variation, and signed history-dependent interactions (both excitation and inhibition) within a single log-intensity.
• Signed Interactions: Unlike additive Cox-Hawkes models where the triggering term is non-negative, this log-scale construction allows for inhibitory history effects without violating the positivity of the intensity function.
• Theoretical Rigor: The work provides rigorous proofs regarding the existence and explosion properties of the proposed process, distinguishing between excitatory and non-excitatory regimes and demonstrating how a recovery time parameter ensures stability.
• SSPF-Point Process Bridge: It explicitly derives the equivalence between the discretized SSPSF update law and a separable Hawkes process, offering a statistical interpretation of the classical SSPSF mechanism.

Results

• Existence: The process is well-defined up to an explosion time $T_\infty$ via Poisson embedding.
• Stability:
  • If $\alpha \leq 0$ (inhibition or neutral), the process is non-explosive ($T_\infty = \infty$ almost surely).
  • If $\alpha > 0$ (excitation), the process may explode in finite time if events occur frequently enough in a small spatial window to drive the log-intensity to infinity.
  • The introduction of a recovery time $\rho > 0$ ensures non-explosion for all values of $\alpha$, preventing instantaneous feedback loops.
• Special Cases: The model reduces to a standard Log-Gaussian Cox process when $\alpha = 0$ and to a spatio-temporal log-linear Hawkes process when the latent field $Y$ is absent.

Significance
The paper claims to offer a new, interpretable point-process framework for analyzing stochastic star formation. By moving beyond additive formulations, the model can represent complex physical phenomena such as latent clustering, self-excitation, local inhibition, and event-driven propagation within a unified statistical model. The recovery-time variant specifically offers a physically motivated interpretation where feedback delays (e.g., time required for supernova feedback to affect neighboring gas) prevent unrealistic instantaneous cascades.

The authors note that while the framework is established, full stability theory, parameter identifiability, and practical inference methods (which will likely require numerical approximation due to the intractable spatio-temporal integral in the likelihood) remain open questions for future work. The paper positions itself as a foundational step toward analyzing the propagation of star formation in galaxies and interpreting observational surveys within a unified statistical model.