ERGODIC THEOREM FOR INFINITE ITERATED FUNCTION SYSTEMS

A set of contraction maps of a metric space is called an iterated function systems. Iterated function systems with condensation, can be considered infinite iterated function systems. Infinite iterated function systems on compact metric spaces were studied. Using the properties of Banach limit and uniform contractiveness, it was proved that the random iterating algorithms for infinite iterated function systems on compact metric spaces-satisfy ergodicity. So the random iterating algorithms for iterated function systems with condensation satisfy ergodicity, too.

Stochastic quantization and ergodic theorem for density of diffusions

For a given probability density function p（x） on R^d, we construct a （non-stationary） diffusion process xt, starting at any point x in R^d, such that 1/T ∫_o^T δ（xt-x）dt converges to p（x） almost surely. The rate of this convergence is also investigated. To find this rate, we mainly use the Clark-Ocone formula from Malliavin calculus and the Girsanov transformation technique.

Semi-uniform Sub-additive Ergodic Theorems for Skew-product Quasi-flows

The aim of this paper is to extend the semi-uniform ergodic theorem and semi-uniform sub-additive ergodic theorem to skew-product quasi-flows. Furthermore, more strict inequalities about these two theorems are established. By making use of these results, it is feasible to get uniform estimation of the Lyapunov exponent of some special systems even under non-uniform hypotheses

ERGODIC THEOREMS FOR NON-LIPSCHITZIAN SEMIGROUPS WITHOUT CONVEXITY

Let G be a semitopological semigroup. Let C be a nonempty subset of a Hilbert space and J ={T t:t∈G} be a representation of G as asymptotically nonexpansive type mappings of C into itself such that the common fixed point set F(J) of J in C is nonempty. It is proved that ∩s∈G co {T ts x:t∈G}∩F(J) is nonempty for each x ∈ C if and only if there exists a nonexpansive retraction P of C onto F(J) such that PT s=T sP=P for all s∈G and P(x) is in the closed convex hull of {T sx:s∈G}, x∈C . This result shows that many key conditions in [1-4, 9, 12-15 ] are not necessary.

Central limit theorems for ergodic continuous-time Markov chains with applications to single birth processes

We obtain sufficient criteria for central limit theorems （CLTs） for ergodic continuous-time Markov chains （CTMCs）. We apply the results to establish CLTs for continuous-time single birth processes. Moreover, we present an explicit expression of the time average variance constant for a single birth process whenever a CLT exists. Several examples are given to illustrate these results.

A Generalization of a Theorem of Liao

Let X be a metrizable space and let φ：R× X → X be a continuous flow on X. For any given {φt}-invariant Borel probability measure, this paper presents a {φt}-invariant Borel subset of X satisfying the requirements of the classical ergodic theorem for the contiImous flow （X, {φt}）. The set is more restrictive than the ones in the literature, but it might be more useful and convenient, particularly for non-uniformly hyperbolic systems and skew-product flows.

Functional ergodic limits for occupation time processes of site-dependent branching Brownian motions in R

We consider a kind of site-dependent branching Brownian motions whose branching laws depend on the site-branching factor σ（·）. We focus on the functional ergodic limits for the occupation time processes of the models in IR. It is proved that the limiting process has the form of λζ（·）, where A is the Lebesgue measure on R and ζ（·） is a real-valued process which is non-degenerate if and only if cr is integrable. When ζ（·） is non-degenerate, it is strictly positive for t 〉 0. Moreover, ζ converges to 0 in finite-dimensional distributions if the integral of a tends to infinity.

Integral expressions of Lyapunov exponents for autonomous ordinary differential systems

In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space ? d , not necessarily compact, by Liaowise spectral theorems that give integral expressions of Lyapunov exponents. In the context of smooth linear skew-product flows with Polish driving systems, the results are still valid. This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.

ON THE RANGE OF RANDOM WALKS IN RANDOM ENVIRONMENT

The range of random walk on Z￣d in symmetric random environment is investigated. As results, it is proved that the strong law of large numbers for the range of random walk on Zdin some random environments holds if d≥ 3, and a weak law of large numbers holds for d= 1.