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Persistence of small noise and random initial conditions

  • J. Baker (a1), P. Chigansky (a2), K. Hamza (a1) and F. C. Klebaner (a1)

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.

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Copyright

Corresponding author

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

References

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[1]Barbour, A. D.,Chigansky, P. and Klebaner, F. C. (2016).On the emergence of random initial conditions in fluid limits.J. Appl. Prob. 53,11931205.
[2]Barbour, A. D.,Hamza, K.,Kaspi, H. and Klebaner, F. C. (2015).Escape from the boundary in Markov population processes.Adv. Appl. Prob. 47,11901211.
[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. 76,679695.
[4]Freidlin, M. I. and Wentzell, A. D. (2012).Random Perturbations of Dynamical Systems (Fundamental Principles Math. Sci. 260),3rd edn.Springer,Heidelberg.
[5]Gyöngy, I. and Rásonyi, M. (2011).A note on Euler approximations for SDEs with H ölder continuous diffusion coefficients.Stoch. Process. Appl. 121,21892200.
[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. 149165.
[7]Klebaner, F. C. (2012).Introduction to Stochastic Calculus With Applications,3rd edn.Imperial College Press,London.
[8]Klebaner, F. C. et al. (2011).Stochasticity in the adaptive dynamics of evolution: the bare bones.J. Biol. Dynam. 5,147162.
[9]Kurtz, T. G. (1970).Solutions of ordinary differential equations as limits of pure jump Markov processes.J. Appl. Prob. 7,4958.
[10]Martin, G. and Lambert, A. (2015).A simple, semi-deterministic approximation to the distribution of selective sweeps in large populations.Theoret. Pop. Biol. 101,4046.
[11]Pardoux, É. (2016).Probabilistic Models of Population Evolution (Math. Biosci. Inst. Lecture Ser. 1.6).Springer,Cham.
[12]Thorisson, H. (2000).Coupling, Stationarity, and Regeneration.Springer,New York.
[13]Whittle, P. (1955).The outcome of a stochastic epidemic–a note on Bailey's paper.Biometrika 42,116122.

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