Stochastic inflation is an elegant method to study inflation and cosmological perturbations. It is an approach for the dynamics of the long super-horizon perturbations which are affected by smalls-scale quantum perturbations. Our goal is to extend the analysis of stochastic inflation to the picture there is an additional source of stochasticity from the boundaries in the field space. With this motivation in mind, we study various physical and mathematical questions. We next review multiple fields inflation in diffusion dominated regime using stochastic δN formalism. This setup can be realized towards the final stages of the ultra-slow-roll setup where the classical drifts fall off exponentially and the perturbations are driven by quantum kicks. By considering both symmetric and asymmetric boundaries and studying the first crossing probabilities as well as the power spectrum, we finally study the primordial black holes (PBHs) formation in this setup and calculate the mass fraction and the contribution of PBHs in dark matter energy density for various higher dimensional field spaces.
