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Stochastic averaging of dynamical systems with multiple time scales forced with \(\alpha\)-stable noise. (English) Zbl 1333.34099

MSC:
34F05 Ordinary differential equations and systems with randomness
34E13 Multiple scale methods for ordinary differential equations
65C30 Numerical solutions to stochastic differential and integral equations
60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34C29 Averaging method for ordinary differential equations
Software:
Matlab; STABLE
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References:
[1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, UK, 2004.
[2] L. Arnold, P. Imkeller, and Y. Wu, Reduction of deterministic coupled atmosphere-ocean models to stochastic ocean models: A numerical case study of the Lorenz-Maas system, Dyn. Syst., 18 (2003), pp. 295–350. · Zbl 1071.76045
[3] S. Asmussen and P. W. Glynn, Stochastic Simulation: Algorithms and Analysis, Springer, New York, 2007. · Zbl 1126.65001
[4] A. N. Borodin, A limit theorem for solution of differential equations with random right-hand side, Theory Probab. Appl., 22 (1977), pp. 482–497. · Zbl 0412.60067
[5] A. S. Chaves, A fractional diffusion equation to describe Lévy flights, Phys. Lett. A, 239 (1998), pp. 13–16. · Zbl 1026.82524
[6] A. V. Chechkin and I. Pavlyukevich, Marcus versus Stratonovich for systems with jump noise, J. Phys. A, 47 (2014), 342001. · Zbl 1326.60083
[7] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Taylor & Francis, New York, Oxford, 2004. · Zbl 1052.91043
[8] P. Ditlevsen, Observation of alpha stable noise induced millennial climate changes from an ice-core record, Geophys. Res. Lett., 26 (1999), pp. 1441–1444.
[9] B. Dybiec, E. Gudowska-Nowak, and P. Hänggi, Lévy-Brownian motion on finite intervals: Mean first passage time analysis, Phys. Rev. E, 73 (2006), 046104.
[10] W. Feller, An Introduction to Probability Theory and Its Applications: Vol. \textupII, 2nd ed., John Wiley & Sons, New York, 1966. · Zbl 0138.10207
[11] M. Freidlin and A. Wentzell, Random Pertubations of Dynamical Systems, Springer-Verlag, Berlin, 1984. · Zbl 0522.60055
[12] D. Givon, R. Kupferman, and A. Stuart, Extracting macroscopic dynamics: Model problems and algorithms, Nonlinearity, 17 (2004), pp. 55–127. · Zbl 1073.82038
[13] M. Grigoriu, Numerical solution of stochastic differential equations with Poisson and Lévy white noise, Phys. Rev. E, 80 (2009), 026704.
[14] K. Hasselmann, Stochastic climate models: Part I. Theory, Tellus, 28 (1976), pp. 473–485.
[15] C. Hein, P. Imkeller, and I. Pavlyukevich, Limit theorems for \(p\)-variations of solutions of SDEs driven by additive stable Lévy noise and model selection for paleo-climatic data, in Recent Development in Stochastic Dynamics and Stochastic Analysis, Interdiscip. Math. Sci. 8, J. Duan, S. Luo, and C. Wang, eds., World Scientific, River Edge, NJ, 2009, pp. 137–150. · Zbl 1218.60027
[16] E. J. Hinch, Perturbation Methods, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0746.34001
[17] M. Huber, J. C. McWilliams, and M. Ghil, A climatology of turbulent dispersion in the troposphere, J. Atmos. Sci., 58 (2001), pp. 2377–2394.
[18] W. Just, K. Gelfert, N. Baba, A. Riegert, and H. Kantz, Elimination of fast chaotic degrees of freedom: On the accuracy of the Born approximation, J. Statist. Phys., 112 (2003), pp. 277–292. · Zbl 1031.34051
[19] W. Just, H. Kantz, C. Rodenbeck, and M. Helm, Stochastic modelling: Replacing fast degrees of freedom by noise, J. Phys. A, 34 (2001), pp. 3199–3123. · Zbl 1057.82009
[20] R. Z. Khas’minskii, A limit theorem for the solutions of differential equations with random right-hand sides, Theory Probab. Appl., 11 (1966), pp. 390–406.
[21] R. Z. Khas’minskii, On stochastic processes defined by differential equations with a small parameter, Theory Probab. Appl., 11 (1966), pp. 211–228. · Zbl 0168.16002
[22] Y. Kifer, L\(^2\) diffusion approximation for slow motion in averaging, Stoch. Dyn., 3 (2003), pp. 213–246. · Zbl 1055.34087
[23] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. · Zbl 0752.60043
[24] T. Li, B. Min, and Z. Wang, Marcus canonical integral for non-Gaussian processes and its computation: Pathwise simulation and tau-leaping algorithm, J. Chem. Phys., 138 (2013), 104118.
[25] T. Li, B. Min, and Z. Wang, Adiabatic elimination for systems with inertia driven by compound Poisson colored noise, Phys. Rev. E, 89 (2014), 022144.
[26] T. Li, B. Min, and Z. Wang, Erratum: “Marcus canonical integral for non-Gaussian processes and its computation: Pathwise simulation and tau-leaping algorithm” [J. Chem. Phys., 138 (2013), 104118], J. Chem. Phys., 140 (2014), 099902.
[27] A. Majda, I. Timofeyev, and E. Vanden-Eijnden, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math., 54 (2001), pp. 891–974. · Zbl 1017.86001
[28] A. Majda, I. Timofeyev, and E. Vanden-Eijnden, A priori tests of a stochastic mode reduction strategy, Phys. D, 170 (2002), pp. 206–252. · Zbl 1007.60064
[29] A. Majda, I. Timofeyev, and E. Vanden-Eijnden, Systematic strategies for stochastic mode reduction in climate, J. Atmos. Sci., 60 (2003), pp. 1705–1722.
[30] S. Marcus, Modeling and analysis of stochastic differential equations driven by point processes, IEEE Trans. Inform. Theory, IT-24 (1978), pp. 164–172. · Zbl 0372.60084
[31] R. G. Miller, Simultaneous Statistical Inference, Springer-Verlag, New York, Berlin, 1981. · Zbl 0463.62002
[32] A. H. Monahan and J. Culina, Stochastic averaging of idealized climate models, J. Clim., 28 (2011), pp. 3068–3088.
[33] J. P. Nolan, Numerical calculation of stable densities and distribution functions, Comm. Statist. Stoch. Models, 13 (1997), pp. 759–774. · Zbl 0899.60012
[34] B. Øksendal, Stochastic Differential Equations, Springer-Verlag, New York, Berlin, 2003.
[35] T. N. Palmer, F. J. Doblas-Reyes, A. Weisheimer, and M. J. Rodwell, Toward seamless prediction: Calibration of climate change projections using seasonal forecasts, Bull. Amer. Meteor. Soc., 89 (2008), pp. 459–470.
[36] G. C. Papanicolaou and W. Kohler, Asymptotic theory of mixing stochastic ordinary differential equations, Comm. Pure Appl. Math., 27 (1974), pp. 641–668. · Zbl 0288.60056
[37] G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Springer, New York, 2007. · Zbl 1160.35006
[38] P. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, New York, Berlin, 1990. · Zbl 0694.60047
[39] D. Rosadi and M. Deistler, Estimating the codifference function of linear time series models with infinite variance, Metrika, 73 (2009), pp. 395–429. · Zbl 1213.62143
[40] B. Saltzman, Dynamical Paleoclimatology, Academic Press, New York, 2002.
[41] K.-H. Seo and K. P. Bowman, Lévy flights and anomalous diffusion in the stratosphere, J. Geophys. Res., 105 (2000), pp. 295–302.
[42] T. H. Solomon, E. R. Weeks, and H. L. Swinney, Chaotic advection in a two-dimensional flow: Lévy flights and anomalous diffusion, Phys. D, 76 (1994), pp. 70–84. · Zbl 1194.37160
[43] T. Srokowski, Correlated Lévy noise in linear dynamical systems, Acta Phys. Polon. B, 42 (2011), pp. 3–19. · Zbl 1371.60104
[44] X. Sun, J. Duan, and X. Li, An alternative expression for stochastic dynamical systems with parametric Poisson white noise, Probab. Eng. Mech., 32 (2013), pp. 1–4.
[45] M. S. Taqqu and G. Samarodnitsky, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, CRC Press, Boca Raton, FL, 1994.
[46] M. Veillette, Stbl: Alpha Stable Distributions for MATLAB, 2012, http://www.mathworks.com/matlabcentral/fileexchange/37514-stbl–alpha-stable-distributions-for-matlab.
[47] Y. Xu, J. Duan, and W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Phys. D, 240 (2011), pp. 1395–1401. · Zbl 1236.60060
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