Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat

We study the existence and uniqueness of solution a non-linear coupled system constituted degenerate diffusion-growth-fragmentation equation differential equation, resulting from modeling bacterial growth in chemostat. This is derived, large population approximation, stochastic individual-based model where each individual characterized by non-negative real valued trait described diffusion. Two results are highlighted. They differ their hypotheses related to influence resource on dynamics, main difficulty being non-linearity due this dependence degeneracy diffusion coefficient. Further we show that semi-group dynamics admits density probabilistic arguments, allows measure diffusiongrowth-fragmentation be function with certain Besov regularity.