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.
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},
mrnumber = {2148965},
zbl = {1136.60317},
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 SP - 165 EP - 184 VL - 9 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2005009/ DO - 10.1051/ps:2005009 LA - en ID - PS_2005__9__165_0 ER -
%0 Journal Article %A Marty, Renaud %T Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations %J ESAIM: Probability and Statistics %D 2005 %P 165-184 %V 9 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2005009/ %R 10.1051/ps:2005009 %G en %F PS_2005__9__165_0
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/
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