Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations
ESAIM: Probability and Statistics, Tome 9 (2005), pp. 165-184.

We consider a differential equation with a random rapidly varying coefficient. The random coefficient is a gaussian process with slowly decaying correlations and compete with a periodic component. In the asymptotic framework corresponding to the separation of scales present in the problem, we prove that the solution of the differential equation converges in distribution to the solution of a stochastic differential equation driven by a classical brownian motion in some cases, by a fractional brownian motion in other cases. The proofs of these results are based on the Lyons theory of rough paths. Finally we discuss applications in two physical situations.

DOI : https://doi.org/10.1051/ps:2005009
Classification : 34F05,  60F05,  60G15
Mots clés : limit theorems, stationary processes, rough paths
@article{PS_2005__9__165_0,
author = {Marty, Renaud},
title = {Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations},
journal = {ESAIM: Probability and Statistics},
pages = {165--184},
publisher = {EDP-Sciences},
volume = {9},
year = {2005},
doi = {10.1051/ps:2005009},
zbl = {1136.60317},
mrnumber = {2148965},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2005009/}
}
TY  - JOUR
AU  - Marty, Renaud
TI  - Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations
JO  - ESAIM: Probability and Statistics
PY  - 2005
DA  - 2005///
SP  - 165
EP  - 184
VL  - 9
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2005009/
UR  - https://zbmath.org/?q=an%3A1136.60317
UR  - https://www.ams.org/mathscinet-getitem?mr=2148965
UR  - https://doi.org/10.1051/ps:2005009
DO  - 10.1051/ps:2005009
LA  - en
ID  - PS_2005__9__165_0
ER  - 
Marty, Renaud. Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 165-184. doi : 10.1051/ps:2005009. http://www.numdam.org/articles/10.1051/ps:2005009/

[1] P. Billingsley, Convergence of Probability Measures. Wiley (1968). | MR 233396 | Zbl 0172.21201

[2] A. Bunimovich, H.R. Jauslin, J.L. Lebowitz, A. Pellegrinotti and P. Nielaba, Diffusive energy growth in classical and quantum driven oscillators. J. Stat. Phys. 62 (1991) 793-817. | Zbl 0741.70017

[3] L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions. Prob. Th. Related Fields 122 (2002) 108-140. | Zbl 1047.60029

[4] H. Doss, Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. H. Poincaré 13 (1977) 99-125. | Numdam | Zbl 0359.60087

[5] S.N. Ethier and T.G. Kurtz, Markov processes, characterization and convergence. Wiley, New York (1986). | MR 838085 | Zbl 0592.60049

[6] J.P. Fouque, G. Papanicolaou and R. Sircar, Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press (2000). | MR 1768877 | Zbl 0954.91025

[7] J. Garnier, A multi-scaled diffusion-approximation theorem. Applications to wave propagation in random media. ESAIM: PS 1 (1997) 183-206. | Numdam | Zbl 0930.60061

[8] J. Garnier, Asymptotic behavior of the quantum harmonic oscillator driven by a random time-dependent electric field. J. Stat. Phys. 93 (1998) 211-241. | Zbl 0952.60063

[9] J. Garnier, Scattering, spreading, and localization of an acoustic pulse by a random medium, in Three Courses on Partial Differential Equations, E. Sonnendrücker Ed. Walter de Gruyter, Berlin (2003) 71-123. | Zbl 1055.74021

[10] J. Garnier and R. Marty, Effective pulse dynamics in optical fibers with polarization mode dispersion. Preprint, submitted to Wave Motion. | MR 2252753

[11] R.Z. Khasminskii, A limit theorem for solutions of differential equations with random right hand side. Theory Probab. Appl. 11 (1966) 390-406. | Zbl 0202.48601

[12] H.J. Kushner, Approximation and weak convergence methods for random processes. MIT Press, Cambridge (1994). | Zbl 0551.60056

[13] M. Ledoux, T. Lyons and Z. Qian, Lévy area of Wiener processes in Banach spaces. Ann. Probab. 30 (2002) 546-578. | Zbl 1016.60071

[14] M. Ledoux, Z. Qian and T. Zhang, Large deviations and support theorem for diffusion processes via rough paths. Stoch. Proc. Appl. 102 (2002) 265-283. | Zbl 1075.60510

[15] A. Lejay, An introduction to rough paths, in Séminaire de Probabilités XXXVII. Lect. Notes Math. Springer-Verlag (2003). | MR 2053040 | Zbl 1041.60051

[16] T. Lyons, Differential equations driven by rough signals. Rev. Mat. Iberoamer. 14 (1998) 215-310. | Zbl 0923.34056

[17] T. Lyons, Differential equations driven by rough signals (I): an extension of an inequality of L.C. Young. Math. Res. Lett. 1 (1994) 451-464. | Zbl 0835.34004

[18] T. Lyons and Z. Qian, Flow equations on spaces of rough paths. J. Funct. Anal. 149 (1997) 135-159. | Zbl 0890.58090

[19] T. Lyons and Z. Qian, System control and rough paths. Oxford Mathematical Monographs. Oxford University Press (2002). | MR 2036784 | Zbl 1029.93001

[20] R. Marty, Théorème limite pour une équation différentielle à coefficient aléatoire à mémoire longue. C. R. Acad. Sci. Paris, Ser. I 338 (2004). | MR 2038288 | Zbl 1038.60033

[21] A. Messiah, Quantum Mechanics. North Holland, Amsterdam (1962). | Zbl 0102.42602

[22] D. Middleton, Introduction to statistical communication theory. Mc Graw Hill Book Co., New York (1960). | MR 118561 | Zbl 0111.32501

[23] G. Papanicolaou, Waves in one dimensional random media, in École d'été de Probabilités de Saint-Flour, P.L. Hennequin Ed. Springer. Lect. Notes Math. (1988) 205-275. | Zbl 0667.73004

[24] G. Papanicolaou and J.B. Keller, Stochastic differential equations with two applications to random harmonic oscillators and waves in random media. SIAM J. Appl. Math. 21 (1971) 287-305. | Zbl 0205.56003

[25] G. Papanicolaou and W. Kohler, Asymptotic theory of mixing stochastic ordinary differential equations. Comm. Pure Appl. Math. 27 (1974) 641-668. | Zbl 0288.60056

[26] G. Papanicolaou, D.W. Stroock and S.R.S. Varadhan, Martingale approach to some limit theorem, in Statistical Mechanics and Dynamical systems, D. Ruelle Ed. Duke Turbulence Conf., Duke Univ. Math. Series III, Part IV (1976) 1-120. | Zbl 0387.60067

[27] G. Samorodnitsky and M.S. Taqqu, Stable non-Gaussian random processes. Chapman and Hall (1994). | MR 1280932 | Zbl 0925.60027

[28] L.I. Schiff, Quantum Mechanics. Mac Graw Hill, New York (1968). | Zbl 0068.40202

[29] K. Sølna, Acoustic Pulse Spreading in a Random Fractal. SIAM J. Appl. Math. 63 (2003) 1764-1788. | Zbl 1055.34105

[30] H.J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations. Ann. Prob. 6 (1978) 19-41. | Zbl 0391.60056

Cité par Sources :