Abstract It is shown that each locally compact second countable non-(T) group G admits non-strongly ergodic weakly mixing IDPFT Poisson actions of any possible Krieger type. These are amenable if and only amenable. If has the Haagerup property, then (and then) these can be chosen 0-type. amenable, Bernoulli arbitrary

The present paper is concerned with some self-interacting diffusions (Xt, t ≥ 0) living on R. These diffusions are solutions to stochastic differential equations: dXt = dBt − g(t)∇V (Xt − μt)dt where μt is the empirical mean of the process X, V is an asymptotically strictly convex potential and g is a given function. The authors have still studied the ergodic behavior of X and proved that it is...

where Ψ and g are smooth functions, ξε t is a “fast” ergodic diffusion whileXε t is a “slow” diffusion type process, κ ∈ (0, 1/2). Under the assumption that g has zero barycenter with respect to the invariant distribution of the fast diffusion, we derive the main result from the moderate deviation principle for the family (ε−κ ∫ t 0 g(ξ ε s)ds)t≥0, ε ↘ 0 which has an independent interest as wel...

Let G be a countably infinite group, and let μ be a generating probability measure on G. We study the space of μ-stationary Borel probability measures on a topological G space, and in particular on ZG, where Z is any perfect Polish space. We also study the space of μ-stationary, measurable G-actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak...

The connection between ergodic theory and the theory of von Neumann algebras goes back to the very beginning of the theory of “rings of operators”. Maximal inequalities in ergodic theory provide an important tool in classical analysis. In this paper we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic theorem, thereby connecting these different aspects of ergod...

Let (Xi)1i=0 be a V -uniformly ergodic Markov chain on a general state space, and let be its stationary distribution. For g : X! R, de ne Wk(g) := k 1=2 k 1 X i=0 g(Xi) (g) : It is shown that if jgj V 1=n for a positive integer n, then ExWk(g) n converges to the n-th moment of a normal random variable with expectation 0 and variance 2 g := (g ) (g) + 1 X j=1 Z g(x)Exg(Xj) (g) 2 : This extends t...

The study of ergodic theorems from the viewpoint of computable analysis is a rich field of investigation. Interactions between algorithmic randomness, computability theory and ergodic theory have recently been examined by several authors. It has been observed that ergodic measures have better computability properties than non-ergodic ones. In a previous paper we studied the extent to which non-...

Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H , and = {Tt : t ∈ G} a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L(x)= {z ∈H : infs∈G supt∈G ‖Ttsx−z‖ = inf t∈G ‖Ttx−z‖} for each x ∈ C and L( )= ⋂x∈C L(x). In this paper, we prove that ⋂s∈G conv{Ttsx : t ∈ G}⋂L( ) is nonempty for each x ∈ C if and only if there exist...

We present a construction showing that a class of sets C that is Glivenko-Cantelli for an i.i.d. process need not be Glivenko-Cantelli for every stationary ergodic process with the same one dimensional marginal distribution. This result provides a counterpoint to recent work extending uniform strong laws to ergodic processes, and a recent characterization of universal Glivenko Cantelli classes.