Rotation Numbers For Linear Stochastic Differential Equations

Arnold, Ludwig ; Imkeller, Peter

Ann. Probab., Tome 27 (1999) no. 1, p. 130-149 / Harvested from Project Euclid

Résumé

Let $dx=\sum_{i=0}^{infty}$ be a linear SDE in $\mathbb{R}^d$, generating the flow $\Phi_t$ of linear isomorphisms. The multiplicative ergodic theorem asserts that every vector $v\in\mathbb{R}^d\{0}$ possesses a Lyapunov exponent (exponential growth rate) $/lambda(v)$ under $\Phi_t$, which is a random variable taking its values from a finite list of canonical exponents $\lambda_i$ realized in the invariant Oseledets spaces $E_i$. We prove that, in the case of simple Lyapunov spectrum, every 2-plane $p$ in $\mathbb{R}^d$ possesses a rotation number $\rho(p)$ under $\Phi_t$ which is defined as the linear growth rate of the cumulative inffinitesimal rotations of a vector $v_t$ inside $\Phi_t(p)$. Again, $\rho(p)$ is a random variable taking its values from a finite list of canonical rotation numbers $\rho_{ij}$ realized in span $(E_i, E_j)$. We give rather explicit Furstenberg-Khasminski-type formulas for the $\rho_{i,j}$. This carries over results of Arnold and San Martin from random to stochastic differential equations, which is made possible by utilizing anticipative calculus.

Détail

Publié le : 1999-01-14
Classification:  Stochastic differential equation,  random dynamical system,  cocycle,  stochastic flow,  anticipative calculus,  multiplicative ergodic theorem,  rotation number,  Furstenberg-Khasminskii formula,  60H10,  93E03,  34D08,  34F05