Persistence of Small Noise and Random initial conditions

TL;DR

This paper investigates how small noise effects can persist over long time intervals in dynamical systems, especially near unstable fixed points, leading to convergence to solutions with random initial conditions in population models.

Contribution

It provides a detailed analysis of the asymptotic behavior of one-dimensional diffusions near unstable fixed points, revealing conditions under which small noise effects persist over increasing time scales.

Findings

Small noise effects can persist indefinitely near unstable fixed points.

Trajectories converge to solutions with random initial conditions under appropriate scaling.

Results apply to population dynamics models with one-dimensional diffusions.

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.