Branching Feller diffusion for cell division with parasite infection

We now deal with a continuous time model for dividing cells which are infected by parasites. We assume that parasites proliferate in the cells and that their lifetimes are much shorter than the cell lifetimes. The quantity of parasites (X t : t ≥ 0) in a cell is modeled by a Feller diffusion (see Chapter 3 and Definition 4.1). The cells divide in continuous time at rate τ(x) which may depend on the quantity of parasites x that they contain. When a cell divides, a random fraction F of the parasites goes in the first daughter cell and a fraction (1 − F) in the second one. More generally, splitting Feller diffusion may model the quantity of some biological content which grows (without resource limitation) in the cells and is shared randomly when the cells divide (for example, proteins, nutriments, energy or extrachromosomal rDNA circles in yeast).