From discrete- to continuous-time ergodic theorems (original) (raw)

An ergodic theorem for repeated and continuous measurement

2000

We prove an ergodic theorem for repeated measurement, indicating its significance for quantum trajectories in discrete time. We roughly sketch the extension to continuous time, and some connections to the algebraic theory of quantum Markov processes.

A Multiplicative Ergodic Theorem for Discontinuous Random Dynamical Systems and Applications

Cornell University - arXiv, 2012

Motivated by studying stochastic systems with non-Gaussian Lévy noise, spectral properties for a type of linear cocycles are considered. These linear cocycles have countable jump discontinuities in time. A multiplicative ergodic theorem is proved for such linear cocycles. Then, the result is illustrated for two linear stochastic systems with general Lévy motions.

Ergodic theorems for lower probabilities

Proceedings of the American Mathematical Society, 2016

We establish an Ergodic Theorem for lower probabilities, a generalization of standard probabilities widely used in applications. As a byproduct, we provide a version for lower probabilities of the Strong Law of Large Numbers.

On rates in mean ergodic theorems

Mathematical Research Letters, 2011

We create a general framework for the study of rates of decay in mean ergodic theorems. As a result, we unify and generalize results due to Assani, Cohen, Cuny, Derriennic, and Lin dealing with rates in mean ergodic theorems in a number of cases. In particular, we prove that the Cesàro means of a power-bounded operator applied to elements from the domain of its abstract one-sided ergodic Hilbert transform decay logarithmically, and this decay is best possible under natural spectral assumptions.