Path Measures for Stochastic Galaxy Formation on Layered Halo Graphs

Galaxy formation, viewed as an inference problem from incomplete information, is inherently stochastic. Reducing the full simulation state to a coarse-grained set of variables integrates out unresolved degrees of freedom, motivating an effective stochastic description of galaxy formation in reduced variables. Existing approaches have achieved substantial predictive success, but generally lack a unified statistical framework for trajectory-level galaxy assembly and history-conditioned fluctuations. We introduce a Graph Path Likelihood Model that formulates galaxy assembly histories as stochastic dynamical trajectories on hierarchical halo merger graphs, where temporal edges encode causal transport and coeval host edges encode environmental conditioning. Within this formulation, galaxy evolution is described by graph-conditioned path measures and effective actions, from which observables, likelihoods, and response diagnostics emerge from a common probabilistic description. As a first realization, we train a graph neural likelihood model for stellar and gas mass assembly histories on layered halo graphs extracted from hydrodynamic simulations. We show that it reproduces the main statistics of these histories while capturing environmentally conditioned correlated fluctuations. The path measure formulation also provides a natural setting for example fixed-graph applications, which we illustrate with the fraction of dark-matter-deficient galaxies, controlled gas-response deformations, and nonequilibrium diagnostics of environmentally dependent evolution. In particular, the present construction also admits extensions in which merger-history statistics and baryonic evolution are treated within a unified probabilistic description, potentially enabling studies of how graph structure, assembly histories, and galaxy observables respond jointly to variations in the underlying theory.