70 research outputs found
Averaging of Hamiltonian flows with an ergodic component
We consider a process on , which consists of fast motion along
the stream lines of an incompressible periodic vector field perturbed by white
noise. It gives rise to a process on the graph naturally associated to the
structure of the stream lines of the unperturbed flow. It has been shown by
Freidlin and Wentzell [Random Perturbations of Dynamical Systems, 2nd ed.
Springer, New York (1998)] and [Mem. Amer. Math. Soc. 109 (1994)] that if the
stream function of the flow is periodic, then the corresponding process on the
graph weakly converges to a Markov process. We consider the situation where the
stream function is not periodic, and the flow (when considered on the torus)
has an ergodic component of positive measure. We show that if the rotation
number is Diophantine, then the process on the graph still converges to a
Markov process, which spends a positive proportion of time in the vertex
corresponding to the ergodic component of the flow.Comment: Published in at http://dx.doi.org/10.1214/07-AOP372 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Deviations of ergodic sums for toral translations II. Boxes
We study the Kronecker sequence on the torus
when is uniformly distributed on We
show that the discrepancy of the number of visits of this sequence to a random
box, normalized by , converges as to a Cauchy
distribution. The key ingredient of the proof is a Poisson limit theorem for
the Cartan action on the space of dimensional lattices.Comment: 56 pages. This is a revised and expanded version of the prior
submission
Scaling limits of recurrent excited random walks on integers
We describe scaling limits of recurrent excited random walks (ERWs) on
integers in i.i.d. cookie environments with a bounded number of cookies per
site. We allow both positive and negative excitations. It is known that ERW is
recurrent if and only if the expected total drift per site, delta, belongs to
the interval [-1,1]. We show that if |delta|<1 then the diffusively scaled ERW
under the averaged measure converges to a (delta,-delta)-perturbed Brownian
motion. In the boundary case, |delta|=1, the space scaling has to be adjusted
by an extra logarithmic term, and the weak limit of ERW happens to be a
constant multiple of the running maximum of the standard Brownian motion, a
transient process.Comment: 12 page
- …