Dérivées asymptotiques associées à un système dynamique aléatoire

Abstract : We study the asymptotic behavior of the intrinsic derivatives of a curve under the effect of a smooth random dynamical system. A curve $c$ through a point $m$ on a Riemannian manifold can be lifted, by the exponential map, to the tangent space at $m$. By "intrinsic derivatives of the curve $c$ at $m$", we mean the derivatives at the origin of the lifted curve parametrized by arc length. For any given order $k$, we provide a sufficient condition on the largest two Lyapounov exponents of the system, $\lambda_1$ and $\lambda_2$ ($\lambda_1$ of multiplicity one), such that the intrinsic derivatives up to order $k$ converge. This condition is $\lambda_2-k\lambda_1<0$, so it can be satisfied even by stable random dynamical systems. The proof is based on Oseledets' Multiplicative Ergodic Theorem. It uses an expansion of the intrinsic derivatives with the help of diagrams and gives an iterative process to compute an expression of the limits. If the first Lyapounov exponent is positive, then the limits of the intrinsic derivatives are related to the unstable manifolds associated to this exponent. In order to establish the "optimality'' of the condition $\lambda_2-2\lambda_1<0$ insuring the convergence of the curvature, we study a particular class of random dynamical systems where the condition $\lambda_2-2\lambda_1<0$ is not always satisfied : isotropic Brownian flows on the unit sphere of $R^d$. Considering the images of a curve by such a flow, we prove that the square norm of its curvature vector is a diffusion process. The description of its asymptotic behavior, according to the values of the first two Lyapounov exponents, shows that this diffusion is positive recurrent if and only if $\lambda_2-2\lambda_1<0$, except in the almost surely constant case.
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Sophie Lemaire. Dérivées asymptotiques associées à un système dynamique aléatoire. Probabilités [math.PR]. Université Paris Sud - Paris XI, 1999. Français. ⟨NNT : 5600⟩. ⟨tel-01058455⟩

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