  Stochastic dynamics govern many important processes in cellular biology, and
an underlying theoretical approach describing these dynamics is desirable to
address a wealth of questions in biology and medicine. Mathematical tools exist
for treating several important examples of these stochastic processes, most
notably gene expression, and random partitioning at single cell divisions or
after a steady state has been reached. Comparatively little work exists
exploring different and specific ways that repeated cell divisions can lead to
stochastic inheritance of unequilibrated cellular populations. Here we
introduce a mathematical formalism to describe cellular agents that are subject
to random creation, replication, and/or degradation, and are inherited
according to a range of random dynamics at cell divisions. We obtain
closed-form generating functions describing systems at any time after any
number of cell divisions for binomial partitioning and divisions provoking a
deterministic or random, subtractive or additive change in copy number, and
show that these solutions agree exactly with stochastic simulation. We apply
this general formalism to several example problems involving the dynamics of
mitochondrial DNA (mtDNA) during development and organismal lifetimes.
