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

We study a size-structured population model in which individual cells grow at a rate determined by a fluctuating internal variable (e.g., gene expression levels). Many previous models of phenotypically heterogeneous populations can be viewed as special cases of this model, and it has previously been observed that the internal variable decouples from cell size under certain conditions. In this work, we generalize these results and connect them to the Feynman-Kac formula, which yields relationships between the lineage dynamics and population distribution in branching processes. To this end, we derive conditions for decoupling, both in the lineage and population ensemble. When decoupling occurs in both ensembles, the size dynamics can be transformed, via a random time change, into a growth-homogeneous process, and expectations can be evaluated through an exponential tilting procedure that follows from the Feynman-Kac formula. We further characterize weaker, ensemble-specific forms of decoupling that hold in either the lineage or the population ensemble, but not both. We provide a more general interpretation of tilted expectations in terms of the mass-weighted phenotype distribution