Hostname: page-component-7f64f4797f-rcs6z Total loading time: 0 Render date: 2025-11-04T01:42:38.656Z Has data issue: false hasContentIssue false
  • English
  • Français

Persistence of small noise and random initial conditions

Part of: Limit theorems Markov processes

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

[1]Barbour, A. D.,Chigansky, P. and Klebaner, F. C. (2016).On the emergence of random initial conditions in fluid limits.J. Appl. Prob. 53,1193–1205.Google Scholar
[2]Barbour, A. D.,Hamza, K.,Kaspi, H. and Klebaner, F. C. (2015).Escape from the boundary in Markov population processes.Adv. Appl. Prob. 47,1190–1211.Google Scholar
[3]Chigansky, P.,Jagers, P. and Klebaner, F. C. (2018).What can be observed in real time PCR and when does it show?J. Math. Biol. 76,679–695.Google Scholar
[4]Freidlin, M. I. and Wentzell, A. D. (2012).Random Perturbations of Dynamical Systems (Fundamental Principles Math. Sci. 260),3rd edn.Springer,Heidelberg.Google Scholar
[5]Gyöngy, I. and Rásonyi, M. (2011).A note on Euler approximations for SDEs with H ölder continuous diffusion coefficients.Stoch. Process. Appl. 121,2189–2200.Google Scholar
[6]Kendall, D. G. (1956).Deterministic and stochastic epidemics in closed populations. In Proc. 3rd Berkeley Symp. Math. Statist. Prob., 1954‒1955, Vol. IV,University of California Press,Berkeley, CA, pp. 149–165.Google Scholar
[7]Klebaner, F. C. (2012).Introduction to Stochastic Calculus With Applications,3rd edn.Imperial College Press,London.Google Scholar
[8]Klebaner, F. C. et al. (2011).Stochasticity in the adaptive dynamics of evolution: the bare bones.J. Biol. Dynam. 5,147–162.Google Scholar
[9]Kurtz, T. G. (1970).Solutions of ordinary differential equations as limits of pure jump Markov processes.J. Appl. Prob. 7,49–58.Google Scholar
[10]Martin, G. and Lambert, A. (2015).A simple, semi-deterministic approximation to the distribution of selective sweeps in large populations.Theoret. Pop. Biol. 101,40–46.Google Scholar
[11]Pardoux, É. (2016).Probabilistic Models of Population Evolution (Math. Biosci. Inst. Lecture Ser. 1.6).Springer,Cham.Google Scholar
[12]Thorisson, H. (2000).Coupling, Stationarity, and Regeneration.Springer,New York.Google Scholar
[13]Whittle, P. (1955).The outcome of a stochastic epidemic–a note on Bailey's paper.Biometrika 42,116–122.Google Scholar