32 research outputs found
A Sampling Theorem for Rotation Numbers of Linear Processes in
We prove an ergodic theorem for the rotation number of the composition of a
sequence os stationary random homeomorphisms in . In particular, the
concept of rotation number of a matrix can be generalized
to a product of a sequence of stationary random matrices in .
In this particular case this result provides a counter-part of the Osseledec's
multiplicative ergodic theorem which guarantees the existence of Lyapunov
exponents. A random sampling theorem is then proved to show that the concept we
propose is consistent by discretization in time with the rotation number of
continuous linear processes on ${\R}^{2}.
An averaging principle for diffusions in foliated spaces
Consider an SDE on a foliated manifold whose trajectories lay on compact
leaves. We investigate the effective behavior of a small transversal
perturbation of order . An average principle is shown to hold such
that the component transversal to the leaves converges to the solution of a
deterministic ODE, according to the average of the perturbing vector field with
respect to invariant measures on the leaves, as goes to zero. An
estimate of the rate of convergence is given. These results generalize the
geometrical scope of previous approaches, including completely integrable
stochastic Hamiltonian system.Comment: Published at http://dx.doi.org/10.1214/14-AOP982 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org