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Diffusion and Deterministic Systems

Published online by Cambridge University Press:  07 February 2014

M. Tyran-Kamińska
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Abstract

We show that simple diffusion processes are weak limits of piecewise continuous processes constructed within a totally deterministic framework. The proofs are based on the continuous mapping theorem and the functional central limit theorem.

Keywords

functional limit theorempiecewise monotonic mapsOrnstein-Uhlenbeck processLangevin equationdiffusion
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 1: Issue dedicated to Michael Mackey , 2014 , pp. 139 - 150
Copyright
© EDP Sciences, 2014

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References

Beck, C.. Ergodic properties of a kicked damped particle. Commun. Math. Phys., 130 (1990), 51–60. CrossRefGoogle Scholar
Beck, C., Roepstorff, G.. From dynamical systems to the Langevin equation. Phys. A, 145 (1987), 1–14. CrossRefGoogle Scholar
P. Billingsley. Convergence of probability measures, 2nd edition. John Wiley & Sons Inc., New York, 1999.
Braumann, C. A.. Itô versus Stratonovich calculus in random population growth. Math. Biosci., 206 (2007), 81–107. CrossRefGoogle ScholarPubMed
Donsker, M. D.. An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc., 1951 (1951), 12. Google Scholar
Erdös, P., Kac, M.. On certain limit theorems of the theory of probability. Bull. Amer. Math. Soc., 52 (1946), 292–302. CrossRefGoogle Scholar
S. N. Ethier, T. G. Kurtz. Markov processes. Characterization and convergence. John Wiley & Sons Inc., New York, 1986.
Feller, W.. The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math., (2) 55 (1952), 468–519. CrossRefGoogle Scholar
Givon, D., Kupferman, R.. White noise limits for discrete dynamical systems driven by fast deterministic dynamics. Phys. A 335 (2004), 385–412. CrossRefGoogle Scholar
Gottwald, G. A., Melbourne, I.. Homogenization for deterministic maps and multiplicative noise. Proc. R. Soc. A 469 (2013), 20130201. CrossRefGoogle Scholar
N. Ikeda, S. Watanabe. Stochastic differential equations and diffusion processes, 2nd edition. North-Holland Publishing Co., Amsterdam, 1989.
J. Jacod, A. N. Shiryaev. Limit theorems for stochastic processes, 2nd edition. Springer-Verlag, Berlin, 2003.
I. Karatzas, S. E. Shreve. Brownian motion and stochastic calculus, 2nd edition. Springer-Verlag, New York, 1991.
Mackey, M. C., Tyran-Kamińska, M.. Deterministic Brownian motion: The effects of perturbing a dynamical system by a chaotic semi-dynamical system. Phys. Rep., 422 (2006), 167–222. CrossRefGoogle Scholar
Melbourne, I., Stuart, A. M.. A note on diffusion limits of chaotic skew-product flows. Nonlinearity 24 (2011), 1361–1367. CrossRefGoogle Scholar
Merlevède, F., Peligrad, M., Utev, S.. Recent advances in invariance principles for stationary sequences. Probab. Surv., 3 (2006), 1–36. CrossRefGoogle Scholar
Pang, G., Talreja, R., Whitt, W.. Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv. 4 (2007), 193–267. CrossRefGoogle Scholar
Skorohod, A. V.. Limit theorems for stochastic processes. Teor. Veroyatnost. i Primenen., 1 (1956), 289–319. Google Scholar
D. W. Stroock, S. R. S. Varadhan. Multidimensional diffusion processes. Springer-Verlag, Berlin, 1979.
Tyran-Kamińska, M.. An invariance principle for maps with polynomial decay of correlations. Comm. Math. Phys., 260 (2005), 1–15. CrossRefGoogle Scholar
Tyran-Kamińska, M.. Convergence to Lévy stable processes under some weak dependence conditions. Stochastic Process. Appl., 120 (2010), 1629–1650. CrossRefGoogle Scholar
Tyran-Kamińska, M.. Weak convergence to Lévy stable processes in dynamical systems. Stoch. Dyn., 10 (2010), 263–289. CrossRefGoogle Scholar
Whitt, W.. Some useful functions for functional limit theorems. Math. Oper. Res., 5 (1980), 67–85. CrossRefGoogle Scholar
W. Whitt. Stochastic-process limits. Springer-Verlag, New York, 2002.
Wiener, N.. The differential space. J. Math. Phys., 2 (1923), 121–174. CrossRefGoogle Scholar
Wong, E., Zakai, M.. On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist., 36 (1965), 1560–1564. CrossRefGoogle Scholar