Persistence of Small Noise and Random initial conditions
J. Baker, P. Chigansky, K. Hamza, F. C. Klebaner
TL;DR
The paper addresses how small stochastic perturbations persist and shape long-time behavior for one-dimensional diffusions near an unstable fixed point. It develops a new fluid-type limit on expanding time horizons by coupling the nonlinear dynamics to its linearization and then extrapolating along the deterministic flow, producing a random initial condition for the limiting ODE. The main contributions include a precise limit theorem with an explicit random initial condition $x_0=H(W)$, an explicit characterization of the limiting random variable $W$ (compound Poisson with Exp jumps) and the function $H$ (via $G$ and Schröder-type relations), and concrete examples such as Wright–Fisher diffusion with and without selection. The results provide a rigorous foundation for random-initial-condition fluid limits in population dynamics and related stochastic systems, enabling accurate long-time predictions and connections to biological models like PCR and allele-frequency dynamics.
Abstract
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 asymptotic regime 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 case of one dimensional diffusions on the positive half line, which often arise as scaling limits in population dynamics.