Coagulation-Fragmentation Duality of Infinitely Exchangeable Partitions from Coupled Mixed Poisson Species Sampling Models

Abstract

We generalize the celebrated coagulation-fragmentation duality of Pitman (1999), originally established for the PD$(\alpha,\theta)$ laws of Pitman and Yor (1997), resolving a two-decade open problem. Our framework extends the duality to processes driven by arbitrary non-negative L'evy subordinators and, for the first time, to multi-group settings with coupled dynamics. The solution is a novel four-component system built from the PHIBP, a framework developed for modeling complex microbiome species sampling (arXiv:2502.01919), which circumvents intractable analysis on traditional partition spaces. Crucially, this architecture embeds naturally within Bertoin's (2006) continuous-time fragmentation framework, resolving a foundational impasse he highlighted [Ch. 4, p. 213]: where time-reversal fails, we introduce simultaneous structural duality, where fragmentation and coalescence evolve in physical time while maintaining a pointwise dual relationship via coupled subordinators. This provides a new generative modeling framework enabling continuous-time representations of coupled genealogical and mutational dynamics -- opening avenues for complex ancestral processes such as recombination graphs. The architecture yields exact compound Poisson representations providing tractable paths to explicit joint EPPF laws, with exact sampling inherited from the PHIBP framework. Section 7 develops the $h$-biased L\'evy-It\^o coupled duality constructor, extending our framework to arbitrary Polish spaces by adapting the $h$-biased PRM structure of Pitman and Yor to Cox processes driven by a common PRM. This demonstrates that duality arises from point-process regrouping rather than features specific to interval partitions.