Lévy过程驱动的动力系统的随机渐近现象
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摘要
在本文中研究两个方面的内容.一是研究Levy过程驱动系统的同步化现象.二是讨论Levy过程驱动系统的随机稳定和不稳定流形.
首先我们研究了加性及乘性Levy过程驱动的动力系统的同步化现象.具体的,我们考虑欧氏空间Rd上Levy过程驱动的耦合动力系统和其中a,b, c, d为实数或向量,Lti,i= 1,2,3,4是独立的Levy过程,连续函数f,g满足单边耗散Lipschitz条件,u*是两个Langevin方程随机稳态解之和.
在讨论了两个系统的稳态解和随机吸引子的基础上,证明了同步化结果.这揭示了耦合系统的一种动力系统渐近现象.这两个结果是Caraballo和Kloeden05年和08年的结果的进一步发展.
在随机动力系统研究中不变流形是个重要的几何研究对象,对不变流形研究有助于对随机影响下复杂动力系统性态的认识.我们考虑Marcus系统
假定矩阵S,U和非线性项f,g满足适当的条件,我们考虑了上述系统生成随机动力系统的稳定和不稳定流形.目前已有的结果都是考虑连续噪声的情形.进一步,我们考虑了随机稳定流形和相应的确定性稳定流形之间的渐近关系.
最后,我们讨论了下述系统的随机慢流形,当ε↓0+时得到了一个简单的渐近结果.
In this thesis, we study synchronization phenomena and invariant stable and un-stable manifolds for dynamical systems driven by non-Gaussian Levy noises.
First we investigate synchronization of systems under additive as well as multi-plicative Levy noises. In particular, we consider coupled SDE systems in Rd, driven by Levy motion and where a, b, c, d are constants, Lti,i= 1,2,3,4 are independent two-sided Levy mo-tions, continuous functions f, g satisfy the one-sided dissipative Lipschitz conditions, and u* is the sum of stationary solutions of two Langevin equations.
After discussing the stationary orbits and random attractors, a synchronization phenomenon is shown to occur. The synchronization result implies that coupled dy-namical systems share a dynamical feature in some asymptotic sense. This further develops the results of Caraballo and Kloeden.
Random invariant manifolds are geometric objects useful for understanding com-plex dynamics under stochastic influences. We consider a Marcus system
Under suitable assumptions on matrixes S, U and nonlinear terms f, g, invari-ant stable and unstable manifolds for random dynamical systems generated by above system are considered. The existing work in this area is for stochastic differential equations driven by noises that are continuous in time. Moreover, when the noise in- tensity is small, the random invariant manifold is represented as a perturbation of the deterministic invariant manifold.
Finally, the slow manifold of the following system is considered and an asymptotic result forε↓0+ is shown.
首先我们研究了加性及乘性Levy过程驱动的动力系统的同步化现象.具体的,我们考虑欧氏空间Rd上Levy过程驱动的耦合动力系统和其中a,b, c, d为实数或向量,Lti,i= 1,2,3,4是独立的Levy过程,连续函数f,g满足单边耗散Lipschitz条件,u*是两个Langevin方程随机稳态解之和.
在讨论了两个系统的稳态解和随机吸引子的基础上,证明了同步化结果.这揭示了耦合系统的一种动力系统渐近现象.这两个结果是Caraballo和Kloeden05年和08年的结果的进一步发展.
在随机动力系统研究中不变流形是个重要的几何研究对象,对不变流形研究有助于对随机影响下复杂动力系统性态的认识.我们考虑Marcus系统
假定矩阵S,U和非线性项f,g满足适当的条件,我们考虑了上述系统生成随机动力系统的稳定和不稳定流形.目前已有的结果都是考虑连续噪声的情形.进一步,我们考虑了随机稳定流形和相应的确定性稳定流形之间的渐近关系.
最后,我们讨论了下述系统的随机慢流形,当ε↓0+时得到了一个简单的渐近结果.
In this thesis, we study synchronization phenomena and invariant stable and un-stable manifolds for dynamical systems driven by non-Gaussian Levy noises.
First we investigate synchronization of systems under additive as well as multi-plicative Levy noises. In particular, we consider coupled SDE systems in Rd, driven by Levy motion and where a, b, c, d are constants, Lti,i= 1,2,3,4 are independent two-sided Levy mo-tions, continuous functions f, g satisfy the one-sided dissipative Lipschitz conditions, and u* is the sum of stationary solutions of two Langevin equations.
After discussing the stationary orbits and random attractors, a synchronization phenomenon is shown to occur. The synchronization result implies that coupled dy-namical systems share a dynamical feature in some asymptotic sense. This further develops the results of Caraballo and Kloeden.
Random invariant manifolds are geometric objects useful for understanding com-plex dynamics under stochastic influences. We consider a Marcus system
Under suitable assumptions on matrixes S, U and nonlinear terms f, g, invari-ant stable and unstable manifolds for random dynamical systems generated by above system are considered. The existing work in this area is for stochastic differential equations driven by noises that are continuous in time. Moreover, when the noise in- tensity is small, the random invariant manifold is represented as a perturbation of the deterministic invariant manifold.
Finally, the slow manifold of the following system is considered and an asymptotic result forε↓0+ is shown.
引文
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[52]Imkeller P., Pavlyukevich I., First exit time of SDEs driven by stable L6vy processes. Stoch. Proc. Appl.,2006,116:611-642.
[53]Imkeller P., Pavlyukevich I., Wetzel T., First exit times for Levy-driven diffusions with exponen-tially light jumps. Ann. of Prob.,2009,37:530-564.
[54]Ditlevsen P. D., Observation of α—stable noise induced millennial climate changes from an ice record. Geophys. Res. Lett.,1999,26:1441-1444.
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[57]Kloeden P.E., Pavani R., Dissipative synchronization of nonautonomous and random systems. Gamm-Mitt.,2009,32(1):80-92.
[58]Kloeden P.E., Nonautonomous attractors of switching systems. Dyna.Syst.,2006,21:209-230.
[59]Liu X., Duan J., Liu J. and Kloeden P. E., Synchronization of dissipative dynamical systems driven by non-Gausssian Levy noises. International J. Stoch. Anal.,2010 (2010), Article ID 502803,13 pages.
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[63]Liu X., Duan J., Liu J. and Kloeden P. E., Synchronization of systems of Marcus canonical equations driven by a-stable noises. Nonlinear Anal:Real World Appl.,Accepted,2009.
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[2]Da Prato G., Zabczyk J., Stochastic equations in infinite dimension. Cambridge:Cambridge University Press,1992.
[3]Ruelle D., Ergodic theory of differentiable dynamical systems. Publ.Math.Inst.Hautes Etud.Sci., 1979,50:275-306.
[4]Arnold L., Random dynamical systems. New York:Springer,1998.
[5]Woyczynski W. A., Levy processes in the physical sciences. In:Levy Processes:Theory and Applications, (Eds. Barndorff-Nielsen O. E. et al.),Boston:Birkhauser,2001,241-266.
[6]Metzler R., Krafter J.,The random walk's guide to anomalous diffusion:a fractional dynamic approach. Phys.Rep.,2000,339(1):1-72.
[7]Oksendal B., Applied stochastic control of jump diffusions. New York:Springer-Verlag,2005.
[8]Herrchen M.P., Stochastic modeling of dispersive diffusion by non-Gaussian noise. Ph.D. the-sis,Zurich:Swiss Federal Inst. of Tech.,2001.
[9]Ikeda N., Watanabe S., Stochastic differential equations and diffusion processes. Amsterdam: North-Holland Publishing Company,1989.
[10]Yang Z., Duan J., An intermediate regime for exit phenomena driven by non-Gaussian Levy noises. Stoch. Dyna.,2008,8(3):583-591.
[11]Applebaum D., Levy processes and stochastic calculus.Second edition. Cambridge:Cambridge University Press,2009.
[12]Peszat S., Zabczyk J., Stochastic partial differential equations with Levy processes. Cambridge: Cambridge University Press,2007.
[13]Fujiwara T., Kunita H., Stochastic differenntial equations of jump type and Levy flows in diffeo-morphisms group. J.Math.Kyoto Univ.,1985,25:71-106.
[14]Kunita H., Stochastic differenntial equations with jumps and stochastic flows of diffeomor-phisms. In:Ito's stochastic calculus and probability theory. Tokyo:Springer,1996,197-211.
[15]Fujiwara T., Kunita H., Canonical SDE's based on semimartingales with spatial parameters, Part 1 Stochastic flows of diffeomorphisms. Kyushu J.Math.,1999,53:265-300.
[16]Kunita H., Stochastic differential equations based on Levy processes and stochastic flows of diffeomorphisms. Real and stochastic analysis. Eds. M. M. Rao, Birkhiiser, Boston,2004,305-373.
[17]Albeverrio S., Rudiger B. and Wu J., Invariant measures and symmetry property of Levy type operators. Potent. Anal.,2000,13:147-168.
[18]Schertzer D., Larcheveque M., Duan J. et al, Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Levy stable noises. J. Math. Phys.,2000, 42:200-212.
[19]Yanovsky V., Chechin A.V., Schertzer D. et al, Levy anomalous diffusion and fractional Fokker-Planck equation. Phys. A.,2000,282:13-41.
[20]Yonezawa F., Introduction to focused session on anomalous relaxation. J. Non-Cryst. Solids, 1996,198:503-506.
[21]Billingsley P., Convergence of probability measure. Second edition. New York:Wiley,1999.
[22]Caraballo T., Kloeden P.E., The persistence of synchronization under environmental noise. Proc. R. Soc. Lond.,2005,461:2257-2267.
[23]Caraballo T., Kloeden P.E., Neuenkirch A., Synchronization of systems with multiplicative noise. Stoch.Dyna.,2008,8(1):139-154.
[24]Boxler P., Stochastische Zentrumsmannigfaltigkeiten. Ph.D. thesis, Universitat Bremen:Institut fiir Dynamische Systeme,1988.
[25]黄志远,随机分析学基础.第二版.北京：科学出版社,2001.
[26]Revuz D., Yor M., Continuous Martingale and Brownian motion. Tirth edtion. Berlin:Springle-Verlag,1999.
[27]Robinson J.C., Infinite-dimensional dynamical systems. Cambridge:Cambridge University Press,2001.
[28]Babin A.B., Vishik M.I., Attractors of evolution equations. London-New York-Tokyo:North-Holland-Amsterdam,1992.
[29]Crauel H., Flandoli F., Attractors for random dynamical systems. Probab. Theory Relat. Fields, 1994,100:365-393.
[30]Schmalfuss B., Attractors for nonautonumous and random dynamical systems perturbed by im-pulses. Discrete and Continuous Dyna. Syst.,2003,9:727-744.
[31]Schmalfuss B., The random attractor of the stochastic Lorenz system. Z. Angew. Math. Phys., 1997,48(6):951-975.
[32]Duan J., Lu K., Schmalfuss B., Smooth stable and unstable manifolds for stochastic evolutionary equations. J. Dyna. Diff. Eqns.,2004,16:949-972.
[33]Sato K.I., Levy processes and infinitely divisible distributions. Cambridge:Cambridge Univer-sity Press,1999.
[34]Situ R., Theory of stochastic differential equations with jumps and applications. New York: Springer,2005.
[35]Marcus S.I., Modelling and analysis of Stochastic differenntial equations driven by point pro-cesses. IEEE Trans. Inform. Theory.1978,24:164-172.
[36]Marcus S. I., Modelling and approximation of Stochastic differenntial equations driven by semi-maringales. stochastics,1981,4:223-245.
[37]Kurtz T. G., Pardoux E., Protter P., Stratonovich stochastic differential equations driven by gen-eral semimartingales. Ann. Inst. H. Poincar'e Probab. Statist.,1995,23:351-377.
[38]Errami M., Russo F., Vallois P., Ito formula for C1,λ functions of a cadlag process and related calculus. Probab. Theory Relat.Fields,2002,122:191-221.
[39]Nourdin I., Simon T., On the absolute continuity of levy processes with drift. Ann. of Prob., 2006,34(3):1035-1051.
[40]Bouchaud J. P., Georges A., Anomalous diffusion in disordered media:Statistic mechanics, models and physical applications. Phys. Repts.,1990,195:127-293.
[41]Scher H.,Shlesinger M.F., Bendler J.T., Time-scale invariance in transport and relaxation. Phys. Today.,1991,26-34.
[42]Shlesinger M. F., Zaslavsky G. M., Frisch.U. Levy flights and related topics in physics. (LNP 450). Berlin:Springer-Verlag,1995.
[43]Jacod J., Shiryaev A.N., Limit Theorems for Stochastic Processes. New York:Springer,1987.
[44]Bertoin J., Levy Processes. Cambridge:Cambridge University Press,1998.
[45]Janicki A., Weron A., Simulation and chaotic behavior of α—stable stochastic processes. Marcel Dekker, Inc.,1994.
[46]Castaing C., Valadier M., Convex analysis and measurable multifunctions. (LNM 580). Berlin-Heidelberg-New York:Springer-Verlag,1977.
[47]Scheutzow M., On the perfection of crude cocycles. Random & Computational Dyna.,1996,4: 235-255.
[48]Kager G., Scheutzow M., Generation of one-sided random dynamical systems by stochastic dif-ferential equations. Electronic J. Prob.,1997,2(8):1-17.
[49]Samorodnitsky G., Taqqu M. S., Stable Non-Gaussian Random Processes. New York:Chapman & Hall,1994.
[50]Arnold L., Scheutzow M., Perfect cocycles through stochastic differential equations. Probab. Theory Relat. Fields,1995,101:65-88.
[51]Kunita H., Stochastic flows and stochastic differential equations. Cambridge:Cambridge Uni-versity Press,1990.
[52]Imkeller P., Pavlyukevich I., First exit time of SDEs driven by stable L6vy processes. Stoch. Proc. Appl.,2006,116:611-642.
[53]Imkeller P., Pavlyukevich I., Wetzel T., First exit times for Levy-driven diffusions with exponen-tially light jumps. Ann. of Prob.,2009,37:530-564.
[54]Ditlevsen P. D., Observation of α—stable noise induced millennial climate changes from an ice record. Geophys. Res. Lett.,1999,26:1441-1444.
[55]Strogatz S., Sync:The Emerging Science of Spontaneous Order. New York:Huperion Press, 2003.
[56]Caraballo T., Chueshov I.D., Kloeden P.E., Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain. SIAM J. Math. Anal.,2007,38:1489-1507.
[57]Kloeden P.E., Pavani R., Dissipative synchronization of nonautonomous and random systems. Gamm-Mitt.,2009,32(1):80-92.
[58]Kloeden P.E., Nonautonomous attractors of switching systems. Dyna.Syst.,2006,21:209-230.
[59]Liu X., Duan J., Liu J. and Kloeden P. E., Synchronization of dissipative dynamical systems driven by non-Gausssian Levy noises. International J. Stoch. Anal.,2010 (2010), Article ID 502803,13 pages.
[60]Dudley R. M., Real Analysis and Probability. Cambridge:Cambridge University Press,2002.
[61]Royden H. L., Real Analysis. Third Edition. Prentice:Prentice-Hall,1988.
[62]Barndorff-Nielsen O.E.,Jensen J.L., S(?)rensen M., Some stationary processes in discrete and con-tinuous time. Adv. Appl. Prob.,1998,30:989-1007.
[63]Liu X., Duan J., Liu J. and Kloeden P. E., Synchronization of systems of Marcus canonical equations driven by a-stable noises. Nonlinear Anal:Real World Appl.,Accepted,2009.
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