In this paper a stochastic model for the simultaneous growth and division
of a cell-population cohort structured by size is formulated. This probabilistic approach
gives straightforward proof of the existence of the steady-size distribution and a simple
derivation of the functional-differential equation for it. The latter one is the celebrated
pantograph equation (of advanced type). This firmly establishes the existence of the
steady-size distribution and gives a form for it in terms of a sequence of probability distribution functions. Also it shows that the pantograph equation is a key equation for other situations where there is a distinct stochastic framework.
Derfel, G., Van Brunt, B. & Wake G. (2012). A cell growth model revisited. Functional Differential Equations 19 (1/2) 71-81