Branching Diffusion for Point Processes in Time and Space

We propose a non-autoregressive branching diffusion model for generating spatio-temporal point processes. Starting from a geometric principle---the Wasserstein-Fisher-Rao (WFR) gradient flow of a generalized KL divergence toward a simple reference intensity---we obtain a tractable forward noising mechanism with two interpretable components: (i) a Langevin-type drift-diffusion step that perturbs event locations and times, and (ii) a birth-death branching step that changes the event count via location-dependent thinning (deaths) and Poisson offspring replication (births). We learn the reverse-time dynamics using a permutation-equivariant denoiser that predicts a drift field and a net-growth field, and we train it using an entropic-regularized unbalanced optimal transport (UOT), which naturally handles count mismatch between noisy and clean samples. The resulting generator produces complete spatio-temporal event sets without autoregressive simulation or explicit intensity normalization.