Count-Conserving Diffusion for Point Processes in Time and Space

Planning often needs synthetic event scenarios that meet exact counts in several regions. Existing generative models control such counts at best on average, and correcting their samples afterwards either leaves overlapping counts wrong or meets them by rewriting the sample. We formalize this problem and develop a theory of count conservation for point-process diffusion. Viewing branching diffusion as a Wasserstein–Fisher–Rao gradient flow, we derive the projection of this flow that keeps every expected count, and show through the quadratic variation of the counts that its births and deaths still let each sample's counts drift. Conservation on every sample path requires every jump to lie in the null space of the count map. On a table of cell counts, the target factorizes into a table law and a position law shared by all budgets, and this condition admits three kinds of count-neutral updates and a Markov-basis chain that provably converges under overlap. Count-Conserving Diffusion is built from these updates alone: every sample meets all counts at every step, and one network serves every budget. On California wildfire scenarios with 33 overlapping coverage areas, global projection of DSTPP samples rewrites a median of 68–84% of the output events. Our samples meet every count without correction and are closer to the real data distribution.