Jaksic V., Pillet C.-A.
Ergodic Properties of Classical Dissipative Systems I
(294K, postscript)
ABSTRACT. We consider a class of models in which a Hamiltonian system
A, with a finite number of degrees of freedom, is brought into contact
with an infinite heat reservoir B. We develop the formalism required
to describe these models near thermal equilibrium. Using a combination of
abstract spectral techniques and harmonic analysis we investigate the
singular spectrum of the Liouvillean L of the coupled system A+B.
We provide a natural set of conditions which ensure that the spectrum of
L is purely absolutely continuous except for a simple eigenvalue at zero.
It then follows from the spectral theory of dynamical systems (Koopmanism)
that the system A+B is strongly mixing.
>From a probabilistic point of view, we study a new class of random processes
on finite dimensional manifolds: non-Markovian Ornstein-Uhlenbeck processes.
The paths of such a process are solutions of a random integro-differential
equation with Gaussian noise which is a natural generalization of the well
known Langevin equation. In this context we establish that, under appropriate
conditions, the OU process is strongly mixing even when the Langevin equation
has memory and is driven by non-white noise.