A linear cell-size dependent branching process

We consider a cell-size dependent branching process in which each cell grows at a linear rate and divides into a pair of daughter cells, preserving total size, at a rate proportional to its size. Such processes expand exponentially fast. If, on division, each possible combination of daughter sizes occurs with equal probability, then conventional analysis provides explicit values for the limiting distribution of the size of a typical cell, together with the distributions of its size just after its birth and just before its division.