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Persistence of small noise and random initial conditions

Part of:Markov processesLimit theorems

Published online by Cambridge University Press:  01 February 2019

J. Baker,
P. Chigansky,
K. Hamza and
F. C. Klebaner
J. Baker*
Affiliation:
Monash University
P. Chigansky*
Affiliation:
The Hebrew University of Jerusalem
K. Hamza*
Affiliation:
Monash University
F. C. Klebaner*
Affiliation:
Monash University
*
School of Mathematical Sciences, Monash University, Monash, VIC 3800, Australia. Email address: jeremy.baker@monash.edu
Department of Statistics, The Hebrew University, Mount Scopus, Jerusalem 91905, Israel. Email address: pavel.chigansky@mail.huji.ac.il
School of Mathematical Sciences, Monash University, Monash, VIC 3800, Australia. Email address: kais.hamza@monash.edu
School of Mathematical Sciences, Monash University, Monash, VIC 3800, Australia. Email address: fima.klebaner@monash.edu
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Abstract

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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.

Keywords

Fluid approximationsmall noisedynamical system

MSC classification

Primary:60J60: Diffusion processes
Secondary:60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)60F17: Functional limit theorems; invariance principles

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018, pp. 67 - 81
Copyright
Copyright © Applied Probability Trust 2018 

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