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Reduced \(\alpha\)-stable dynamics for multiple time scale systems forced with correlated additive and multiplicative Gaussian white noise. (English) Zbl 1390.34185
Summary: Stochastic averaging problems with Gaussian forcing have been the subject of numerous studies, but far less attention has been paid to problems with infinite-variance stochastic forcing, such as an \(\alpha\)-stable noise process. It has been shown that simple linear systems driven by correlated additive and multiplicative (CAM) Gaussian noise, which emerge in the context of reduced atmosphere and ocean dynamics, have infinite variance in certain parameter regimes. In this study, we consider the stochastic averaging of systems where a linear CAM noise process in the infinite variance parameter regime drives a comparatively slow process. We use (semi)-analytical approximations combined with numerical illustrations to compare the averaged process to one that is forced by a white \(\alpha\)-stable process, demonstrating consistent properties in the case of large time-scale separation. We identify the conditions required for the fast linear CAM process to have such an influence in driving a slower process and then derive an (effectively) equivalent fast, infinite-variance process for which an existing stochastic averaging approximation is readily applied. The results are illustrated using numerical simulations of a set of example systems.
©2017 American Institute of Physics

MSC:
34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34D20 Stability of solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
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References:
[1] Gardiner, C. W., Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, (1985), Springer-Verlag
[2] Cont, R.; Tankov, P., Financial Modelling With Jump Processes, (2004), Taylor & Francis · Zbl 1052.91043
[3] Bressloff, P. C., (2014), Springer International Publishing
[4] Hasselmann, K., Tellus, 28, 473, (1976)
[5] Majda, A.; Timofeyev, I.; Vanden-Eijnden, E., Commun. Pure Appl. Math., 54, 891, (2001) · Zbl 1017.86001
[6] Arnold, L.; Imkeller, P.; Wu, Y., Dyn. Syst., 18, 295, (2003) · Zbl 1071.76045
[7] Monahan, A. H.; Culina, J., J. Clim., 24, 3068, (2011)
[8] Mitchell, L.; Gottwald, G. A., J. Atmos. Sci., 69, 1359, (2012)
[9] Sura, P.; Newman, M.; Penland, C.; Sardeshmukh, P., J. Atmos. Sci., 62, 1391, (2005)
[10] Penland, C.; Ewald, B. D., Philos. Trans. R. Soc., 366, 2455, (2008) · Zbl 1153.60383
[11] Sura, P.; Sardeshmukh, P., J. Phys. Oceanogr., 38, 639, (2008)
[12] Sardeshmukh, P. D.; Sura, P., J. Clim., 22, 1193, (2009)
[13] Sura, P., Atmos. Res., 101, 1, (2011)
[14] Feller, W., An Introduction to Probability Theory and Its Applications, II, (1966), John Wiley & Sons · Zbl 0138.10207
[15] Penland, C.; Sardeshmukh, P. D., Chaos, 22, 023119, (2012)
[16] Khas’minskii, R. Z., Theory Probab. Appl., 11, 211, (1966) · Zbl 0168.16002
[17] Khas’minskii, R. Z., Theory Probab. Appl., 11, 390, (1966)
[18] Papanicolaou, G. C.; Kohler, W., Commun. Pure Appl. Math., 27, 641, (1974) · Zbl 0288.60056
[19] Borodin, N. N., Theor. Probab. Appl., 22, 482, (1977) · Zbl 0412.60067
[20] Freidlin, M.; Wentzell, A., Random Perturbations of Dynamical Systems, (1984), Springer-Verlag · Zbl 0522.60055
[21] Givon, D.; Kupferman, R.; Stuart, A., Nonlinearity, 17, R55, (2004) · Zbl 1073.82038
[22] Pavliotis, G. A.; Stuart, A. M., Multiscale Methods: Averaging and Homogenization, (2007), Springer
[23] Kantz, H.; Just, W.; Baba, N.; Gelfert, K.; Riegert, A., Physica D, 187, 200, (2004) · Zbl 1050.60092
[24] Srokowski, T., Acta Phys. Pol. B, 42, 3, (2011) · Zbl 1371.60104
[25] Xu, Y.; Duan, J.; Xu, W., Physica D, 240, 1395, (2011) · Zbl 1236.60060
[26] Gottwald, G. A.; Melbourne, I., Proc. R. Soc. London, Ser. A, 469, 20130201, (2013) · Zbl 1371.34084
[27] Thompson, W. F.; Kuske, R. A.; Monahan, A. H., Multiscale Model. Simul., 13, 1194, (2015) · Zbl 1333.34099
[28] Saltzman, B., Dynamical Paleoclimatology, (2002), Academic Press
[29] Sardeshmukh, P.; Penland, C., Chaos, 25, 036410, (2015) · Zbl 1374.86060
[30] Ditlevsen, P., Geophys. Res. Lett., 26, 1441, (1999)
[31] Huber, M.; McWilliams, J. C.; Ghil, M., J. Atmos. Sci., 58, 2377, (2001)
[32] Seo, K.-H.; Bowman, K. P., J. Geophys. Res., 105, 12295, (2000)
[33] Del-Castillo-Negrete, D., Phys. Fluids, 10, 576, (1998) · Zbl 1185.76850
[34] Taqqu, M. S.; Samarodnitsky, G., Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance, (1994), CRC Press
[35] Wyłomańska, A.; Chechkin, A.; Gajda, J.; Sokolov, I. M., Physica A, 421, 412, (2015) · Zbl 1395.62286
[36] Keller, J.; Kuske, R., SIAM J. Appl. Math., 61, 1308, (2001) · Zbl 0985.60013
[37] Chaves, A. S., Phys. Lett. A, 239, 13, (1998) · Zbl 1026.82524
[38] Gross, A., Stoch. Proc. Appl., 51, 277, (1994) · Zbl 0813.60039
[39] Marcus, S., IEEE Trans. Inform. Theory, IT-24, 164, (1978) · Zbl 0372.60084
[40] Chechkin, A. V.; Pavlyukevich, I., J. Phys. A, 47, 342001, (2014) · Zbl 1326.60083
[41] Sato, K.-I., Lévy Processes and Infinitely Divisible Distributions, (1999), Cambridge University Press
[42] Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations, (1992), Springer-Verlag · Zbl 0925.65261
[43] Koutrouvelis, I. A., J. Am. Stat. Assoc., 75, 918, (1980)
[44] Besbeas, P.; Morgan, B. J. T., Stat. Comput., 18, 219, (2008)
[45] Veillette, M., STBL: Alpha stable distributions for MATLAB, (2012)
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