Persistence of small noise and random initial conditions. (1st February 2019)

• Record Type: Journal Article
• Authors: Baker, J.; Chigansky, P.; Hamza, K.; Klebaner, F. C.
• Editors: Asmussen, S.; Klebaner, F.; Nerman, O.; Vatutin, V.
• Is Part Of: Advances in applied probability. Volume 50:Number A(2018)
• Page Start: 67
• Page End: 81
• Publication Date: 2019-02-01
• Subjects: Fluid approximation, -- small noise, -- dynamical system; Primary 60J60, -- Secondary 60J70, -- 60F17; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2018.71
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: The effect of small noise in a smooth dynamical system is negligible on any finite time interval; in this paper we study situations where the effect persists on intervals increasing to ∞. Such an asymptotic regime occurs when the system starts from an initial condition that is sufficiently close to an unstable fixed point. In this case, under appropriate scaling, the trajectory converges to a solution of the unperturbed system started from a certain random initial condition. In this paper we consider the case of one-dimensional diffusions on the positive half-line; this case often arises as a scaling limit in population dynamics.