Strong averaging along foliated Lévy diffusions with heavy tails on compact leaves. (English) Zbl 1380.60056
Authors’ abstract: This article shows a strong averaging principle for diffusions driven by discontinuous heavy-tailed Lévy noise, which are invariant on the compact horizontal leaves of a foliated manifold subject to small transversal random perturbations. We extend a result for such diffusions with exponential moments and bounded, deterministic perturbations to diffusions with polynomial moments of order \(p\geqslant 2\), perturbed by deterministic and stochastic integrals with unbounded coefficients and polynomial moments. The main argument relies on a result of the dynamical system for each individual jump increments of the corresponding canonical Marcus equation. The example of Lévy rotations on the unit circle subject to perturbations by a planar Lévy-Ornstein-Uhlenbeck process is carried out in detail.
Reviewer: Jacques Franchi (Strasbourg)
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60J60 | Diffusion processes |
| 60G51 | Processes with independent increments; Lévy processes |
| 58J65 | Diffusion processes and stochastic analysis on manifolds |
| 58J37 | Perturbations of PDEs on manifolds; asymptotics |
Keywords:
Markov processes on manifolds; stochastic differential equations; Lévy noise; compact leavesReferences:
| [1] | Anosov, D. B.: Averaging in systems of ordinary differential equations with fast oscillating solutions (Russian). Izv. Acad. nauk SSSR Ser. Mat. 24, 731-742 (1960) |
| [2] | Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2009) · Zbl 1200.60001 · doi:10.1017/CBO9780511809781 |
| [3] | Arnold, L.: Hasselmann’s program revisited: the analysis of stochasticity in deterministic climate models. In: Imkeller, P. et al. (eds.) Stochastic Climate Models, vol. 49, pp 141-157. Basel Birkhäuser Prog. Probab. 49, 141-157 (2001) · Zbl 1015.34037 |
| [4] | Arnold, V.: Mathematical Methods of Classical Mechanics. Springer, Berlin (1989) · Zbl 0692.70003 · doi:10.1007/978-1-4757-2063-1 |
| [5] | Arak, T. V., Zaitsev, A.: On the rate of convergence in Kolmogorov’s second uniform limit theorem. Theory Probab. Appl. 28(2), 351-374 (1983) · Zbl 0534.60023 |
| [6] | Bainov, D. D., Stoyanov, I. M.: The averaging method for a class of stochastic differential equations. Ukr. Math. J. 26(2), 186-194 (1974) · Zbl 0294.60051 |
| [7] | Bakry, D., Cattiaux, P., Guillin, A.: Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254(3), 727-759 (2008) · Zbl 1146.60058 · doi:10.1016/j.jfa.2007.11.002 |
| [8] | Bell, D. R.: The Malliavin Calculus. Longman Scientific & Techincal (1987) · Zbl 0678.60042 |
| [9] | Bismut, J. M.: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander’s conditions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 56(4), 469-505 (1981) · Zbl 0445.60049 · doi:10.1007/BF00531428 |
| [10] | Boothby, W. M.: An Introduction to Differential Manifold and Riemannian Geometry. Academic Press, INC (1986) · Zbl 0596.53001 |
| [11] | Borodin, A., Freidlin, M.: Fast oscillating random perturbations of dynamical systems with conservation laws. Ann. Inst. H. Poincaré,. Prob. Statist. 31, 485-525 (1995) · Zbl 0831.60066 |
| [12] | Bogoliubov, N. N., Krylov, N. M.: La theorie générale de la mésure dans son application a létude de systèmes dynamiques de la mécanique non-linéaire (French). Ann. Math. II 38, Zbl. 16.86 (1937) · Zbl 1191.60076 |
| [13] | Bogoliubov, N. N., Mitropolskii, Y: Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Breach, New York (1961) · Zbl 0151.12201 |
| [14] | Bramanti, M.: An Invitation to Hypoelliptic Operators and Hörmander’s Vector Fields. Springer Briefs in Mathematics (2014) · Zbl 1298.47001 |
| [15] | Brin, M., Freidlin, M. I.: On stochastic behavior of perturbed Hamiltonian systems. Ergod. Th. Dynam. Sys. 20, 55-76 (2000) · Zbl 0997.37020 · doi:10.1017/S0143385700000043 |
| [16] | Cannas, A.: Lectures on symplectic geometry. Lecture Notes in Mathematics 1764 (2001) · Zbl 1016.53001 |
| [17] | Cerrai, S.: A Khasminskii type averaging principle for stochastic reaction-diffusion equations. Ann. Probab. 19(3), 899-948 (2009) · Zbl 1191.60076 · doi:10.1214/08-AAP560 |
| [18] | Douca, R., Fortb, G., Guillinc, A.: Subgeometric rates of convergence of f-ergodic strong Markov processes. Stoch. Process. Appl. 119, 897-923 (2009) · Zbl 1163.60034 · doi:10.1016/j.spa.2008.03.007 |
| [19] | Xu, Y., Duan, J., Xu, W.: An averaging principle for stochastic dynamical systems with Levý noise. Physica D 240, 1395-1401 (2011) · Zbl 1236.60060 · doi:10.1016/j.physd.2011.06.001 |
| [20] | Down, D., Meyn, S. P., Tweedie, R. L.: Exponential and uniform ergodicity of Markov process. Ann. Probab. 23(4), 1671-1691 (1995) · Zbl 0852.60075 · doi:10.1214/aop/1176987798 |
| [21] | Freidlin, M. I., Wentzell, A. D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1984) · Zbl 0522.60055 · doi:10.1007/978-1-4684-0176-9 |
| [22] | Freidlin, M. I., Wentzell, A. D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1991) · Zbl 0522.60055 |
| [23] | Freidlin, M. I.: The averaging principle and theorems on large deviations. Russ. Math. Surv. 33(5), 107-160 (1978) · Zbl 0416.60029 · doi:10.1070/RM1978v033n05ABEH002516 |
| [24] | Garnett, L.: Foliation, the ergodic theorem and Brownian motion. J. Funct. Anal. 51, 285-311 (1983) · Zbl 0524.58026 · doi:10.1016/0022-1236(83)90015-0 |
| [25] | Gargate, I. I. G., Ruffino, P. R.: An averaging principle for diffusions in foliated spaces. Ann. Probab. 44(1), 567-588 (2016) · Zbl 1462.60074 · doi:10.1214/14-AOP982 |
| [26] | Gihman, I. I., Skorohod, A. V.: Stochastic Differential Equations. Springer, Berlin (1972) · Zbl 0242.60003 · doi:10.1007/978-3-642-88264-7 |
| [27] | Hairer, M.: On Malliavins proof of Hörmander’s theorem. Bull. Sci. math. 135, 650-666 (2011) · Zbl 1242.60085 · doi:10.1016/j.bulsci.2011.07.007 |
| [28] | Högele, M. A., Ruffino, P. R.: Averaging along foliated Lévy diffusions. Nonlinear Anal. Theory Methods Appl. 112, 1-14 (2015) · Zbl 1301.60067 · doi:10.1016/j.na.2014.09.006 |
| [29] | Kabanov, Y., Pergamenshchikov, S.: Two-Scale Stochastic Systems: Asymptotic Analysis and Control. Springer, Berlin (2003) · Zbl 1033.60001 · doi:10.1007/978-3-662-13242-5 |
| [30] | Kakutani, S., Petersen, K.: The speed of convergence in the ergodic theorem. Monat. Mathematik 91, 11-18 (1981) · Zbl 0446.28015 · doi:10.1007/BF01306954 |
| [31] | Khasminskii, R. Z.: A limit theorem for the solution of differential equations with random right-hand sides. Theory Probab. Appl. 11, 390-405 (1963) · doi:10.1137/1111038 |
| [32] | Khasminskii, R. Z.: Principle of averaging of parabolic and elliptic differential equations for Markov process with small diffusion. Theory Probab. Appl. 8, 1-21 (1963) · Zbl 0211.20701 · doi:10.1137/1108001 |
| [33] | Khasminskii, R. Z.: On stochastic processes defined by differential equations with a small parameter. Theor. Probab. Appl. 11, 211-228 (1966) · Zbl 0168.16002 · doi:10.1137/1111018 |
| [34] | Khasminskii, R. Z.: On the averaging principle for Itô stochastic differential equations (Russian). Kibernetika 4, 260-279 (1968) · Zbl 0231.60045 |
| [35] | Khasminski, R. Z., Krylov, N. M.: On the averaging principle for diffusion processes with null-recurrent fast component. Stoch. Proc. Appl. 93(2), 229-240 (2001) · Zbl 1053.60060 · doi:10.1016/S0304-4149(00)00097-1 |
| [36] | Kolomiets, V. G., Melnikov, A. I.: Averaging of stochastic systems of integral-differential equations with Poisson noise. Ukr. Math. J. 43(2), 242-246 (1991) · Zbl 0735.60060 · doi:10.1007/BF01060515 |
| [37] | Krasnoselskii, M. A., Krein, S. G.: On the averaging principle in nonlinear mechanics (Russian). Uspekhi Mat. Nauk. 10(3), 147-152 (1955) · Zbl 0064.33901 |
| [38] | Krengel, U.: On the speed of convergence of the ergodic theorem. Monat. Mathematik 86, 3-6 (1978) · Zbl 0352.28008 · doi:10.1007/BF01300052 |
| [39] | Kolmogorov, A. N.: Two uniform limit theorems for sums of independent random variables. Theory Prob. Appl. 1, 384-394 (1956) · Zbl 0079.34502 · doi:10.1137/1101030 |
| [40] | Kulik, A.: Exponential ergodicity of the solutions to SDEs with a jump noise. Stoch. Process. Appl. 119, 602-632 (2009) · Zbl 1169.60012 · doi:10.1016/j.spa.2008.02.006 |
| [41] | Kurtz, T. G., Pardoux, E., Protter, P: Stratonovich stochastic differential equations driven by general semimartingales. Annales de l’H.I.P. B 31(2), 351-377 (1995) · Zbl 0823.60046 |
| [42] | Kunita, H.: Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms. In: Rao, M. M. (ed.) Real and Stochastic Analysis. Birkhauser, 305-373 (2004) · Zbl 1082.60052 |
| [43] | Li, X. -M.: An averaging principle for a completely integrable stochastic Hamiltonian systems. Nonlinearity 21, 803-822 (2008) · Zbl 1140.60033 · doi:10.1088/0951-7715/21/4/008 |
| [44] | Marcus, S. I.: Modeling and analysis of stochastic differential equations driven by Poisson point processes. IEEE Trans. Inf. Theory 24(3), 164-172 (1978) · Zbl 0372.60084 · doi:10.1109/TIT.1978.1055857 |
| [45] | Marcus, S. I.: Modeling and analysis of stochastic differential equations driven by semimartingales. Stochastics 4(3), 223-245 (1981) · Zbl 0456.60064 · doi:10.1080/17442508108833165 |
| [46] | Pardoux, E., Veretennikov, A. Y: On the Poisson equation and diffusion approximation. Ann. Probab. 29(3), 1061-1085 (2000) · Zbl 1029.60053 |
| [47] | Pascal, M.: Rates of convergence in the central limit theorem for empirical processes. Annales de l’I.H.P. Probabilités et statistiques 22(4), 381-423 (1986) · Zbl 0615.60032 |
| [48] | Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach. Cambridge University Press (2007) · Zbl 1205.60122 |
| [49] | Protter, Ph.: Stochastic Integration and Differential Equations. Springer, Berlin (2004) · Zbl 1041.60005 |
| [50] | Wang, W., Roberts, A. R.: Average and deviation for slow-fast stochastic partial differential equations. J. Differ. Equ. 253(1), 1265-1286 (2012) · Zbl 1251.35201 · doi:10.1016/j.jde.2012.05.011 |
| [51] | Sato, K. I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999) · Zbl 0973.60001 |
| [52] | Saint Loubert Bié, E.: ÉTude d’une EDPS conduite par un bruit poissonnien. Probab. Theory Relat. Fields 111, 287-321 (1998) · Zbl 0939.60064 · doi:10.1007/s004400050169 |
| [53] | Sanders, J. A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems. Springer (2007) · Zbl 1128.34001 |
| [54] | Skorokhod, A. V.: Asymptotic Methods of the Theory of Stochastic Differential Equations (Russian). Naukova Dumka, Kiev (1987). English transl., Amer. Math. Soc., Providence, R.I (1989) · Zbl 0445.60049 |
| [55] | Stratonovich, R. L.: Topics in the theory of random noise. Gordon and Breach, New York Vol. 1 (1963), Vol. 2 (1967) · Zbl 0831.60066 |
| [56] | Stratonovich, R. L.: Conditional Markov Processes and their Application to the Theory of Optimal Control. American Elsevier (1967) · Zbl 0159.46804 |
| [57] | Tondeur, P.: Foliations on Riemannian Manifolds. Universitext. Springer, Berlin (1988) · Zbl 0643.53024 · doi:10.1007/978-1-4613-8780-0 |
| [58] | Villani, C.: Hypocoercivity. Memoirs of the American Mathematical Society, AMS Vol. 202 (2009) · Zbl 1197.35004 |
| [59] | Volsov, V. M.: Some types of calculation connected with averaging in the theory of non-linear vibrations. USSR Comput. Math. Math. Phys. 3(1), 1-64 (1962) · doi:10.1016/0041-5553(63)90122-8 |
| [60] | Veretennikov, A.: On the averaging principle for systems of stochastic differential equations. Mathe. USSR Sbornik 69, 271-284 (1991) · Zbl 0724.60069 · doi:10.1070/SM1991v069n01ABEH001237 |
| [61] | Veretennikov, A.: On large deviations in the averaging principle for SDEs with “full dependence”. Ann. Probab. 27, 284-296 (1999) · Zbl 0939.60012 · doi:10.1214/aop/1022677263 |
| [62] | Walcak, P.: Dynamics of Foliations, Groups and Pseudogroups. Birkhäuser Verlag (2004) · Zbl 1084.37022 |
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