Persistence of Small Noise and Random initial conditions in the Wright-Fisher model

The effect of small noise in a smooth dynamical system is negligible on any finite time interval. Here we study situations when it persists on intervals increasing to infinity. Such phenomenon occurs when the system starts from initial condition, sufficiently close to an unstable fixed point. In this case, under appropriate scaling, the trajectory converges to solution of the unperturbed system, started from a certain {\em random} initial condition. In this paper we consider the Wright-Fisher diffusion from population dynamics. We show that when the selection parameter is positive and the diffusion coefficient is small, the solution, which starts near zero, follows the logistic equation with a random initial condition that comes from an approximating Feller branching diffusion. We also consider the case of discrete time dynamical systems for which random initial condition involves the limit of iterates of the deterministic system and weighted random perturbations.