Size-structured populations with growth fluctuations: Feynman--Kac formula and decoupling

We study a size--structured population model of proliferating cells in which biomass accumulation and binary division occur at rates modulated by fluctuating internal phenotypes. We quantify how fluctuations in internal variables that influence both growth and division shape the distribution of population phenotypes. We derive conditions under which the distributions of size and internal state decouple. Under this decoupling, population--level expectations are obtained from lineage-level expectations by an exponential tilting given by the Feynman--Kac formula. We further characterize weaker (ensemble-specific) versions of decoupling that hold in the lineage or the population ensemble but not both. Finally, we provide a more general interpretation of the tilted expectations in terms of the mass-weighted phenotype distribution.