On existence of invariant measures

Let G be a Lie group, H ≤ G a closed subgroup and M ≈ G/H. In [14] André Weil gave a necessary and sufficient condition for the existence of invariant measures on homogeneous spaces of arbitrary locally compact groups. For Lie groups using the structure theory we give a neater necessary and sufficient condition for the existence of a G-invariant measure on M, cf. Theorems (2.1) and (3.2) in the introduction.     

Some weak and strong laws of large numbers for D[0,1]-valued random variables

Pointwise Weak Law of Large Numbers and Weak Law of Large Numbers in the norm topology of D[0,l] are shown to be equivalent under uniform convex tightness and uniform integrability conditions for weighted sums of a sequence of random elements in D[0,1]. Uniform convex tightness and uniform integrability conditions are jointly characterized. Marcinkiewicz–Zygmund–Kolmogorov's and Brunk– Chung's Strong Laws of Large Numbers are derived in the setting of D[0,l]-space under uniform convex tightness and uniform integrability conditions. Equivalence of pointwise convergence, convergence in the Skorokhod topology and convergence in the norm topology f o r sequences in D[0,l] is studied     

Absolutely continuous invariant measures forC 2 unimodal maps satisfying the Collet-Eckmann conditions

Summary In this paper we show that unimodal mappingsf[0, 1][0, 1] have absolutely continuous measures of positive entropy if these maps areC ² and satisfy the so-called Collet-Eckmann conditions. No conditions on the Schwarzian derivative off are assumed.     

On the existence of invariant probability measures

Let (X,A) be a measureable space andT:X →X a measurable mapping. Consider a family ℳ of probability measures onA which satisfies certain closure conditions. IfA ₀⊂A is a convergence class for ℳ such that, for everyA ∈A ₀, the sequence ((1/n) Σ _i ^=0/n−1 1 _A ∘T ⁱ) converges in distribution (with respect to some probability measurev ∈ ℳ), then there exists aT-invariant element in ℳ. In particular, for the special case of a topological spaceX and a continuous mappingT, sufficient conditions for the existence ofT-invariant Borel probability measures with additional regularity properties are obtained.     

On the existence of equivalent finite invariant measures

Summary Necessary and sufficient conditions are given for the existence of a finite measure which is equivalent to a given measure and invariant with respect to each transformation in a given commutative semigroup of measurable null-invariant point transformations. This result was already known for denumerably generated semigroups. A complementary result is proved which states that if one such equivalent measure exists, then there exists a unique equivalent measure which agrees with the original measure on the invariant sets.Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Grant No. AFOSR-68-1394.     

The existence of invariant probability measures for a group of transformations

SupposeG is a group of measurable transformations of aσ-finite measure space (X,A, m). A setA ∈A is weakly wandering underG if there are elementsg _n ∈G such that the setsg _nA, n=0, 1,…, are pairwise disjoint. We prove that the non-existence of any set of positive measure which is weakly wandering underG is a necessary and sufficient condition for the existence of aG-invariant, probability measure defined onA and dominating the measurem in the sense of absolute continuity. This paper was written while the author was visiting the Technische Universitat Berlin as a research fellow of the Alexander von Humboldt Foundation.     

Stein’s method for invariant measures of diffusions via Malliavin calculus

Given a random variable F$F$ regular enough in the sense of the Malliavin calculus, we are able to measure the distance between its law and any probability measure with a density function which is continuous, bounded, strictly positive on an interval in the real line and admits finite variance. The bounds are given in terms of the Malliavin derivative of F$F$. Our approach is based on the theory of Itô diffusions and the stochastic calculus of variations. Several examples are considered in order to illustrate our general results.     

L~2[0,1]上的正交小波基

从尺度因子 M =4的正交小波基出发 ,利用折叠方法得到了 L 2 [0 ,1 ]空间的正交小波基 .这种小波不同于折叠前的小波基 ,它是完全限制在有限区间 [0 ,1 ]上 ,且保持小波基的正交性 ,并在使用过程中拥有更大的灵活性 .也可用类似方法对一般尺度小波进行折叠     

Infinite-dimensional diffusion processes as gibbs measures onC[0,1]^{Z^d }

Summary An infinite lattice system of interacting diffusion processes is characterized as a Gibbs distribution on with continuous local conditional probabilities. Using estimates for the Vasserstein metric onC[0, 1], Dobrushin's contraction technique is applied in order to obtain information about macroscopic properties of the entire diffusion process.     

Transformations on [0, 1] with infinite invariant measures

Under certain regularity conditions a real transformation with indifferent fixed points has an infinite invariant measure equivalent to Lebesgue measure. In this paper several ergodic properties of such transformations are established.     

The semiring of continous [0, 1]-valued functions

In this paper, we study the properties of prime ideals in semirings of continuous functions with values in the unit interval [0, 1] on topological spaces. We describe maximal and pure ideals of such semirings. We study homomorphisms of semirings of continuous [0, 1]-valued functions. In terms of semirings of functions we characterize some properties of compacta. We show that the theory of ideals in these semirings differs from the case of rings of continuous functions.     

Non-uniform hyperbolicity and existence of absolutely continuous invariant measures

We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicityalongtheleavesimpliesexistenceofafinitenumberofergodicabsolutely continuous invariant probability measures which describe the asymptotics of almost every point. The main technical tool is an extension for sequences of maps of a result of de Melo and van Strien relating hyperbolicity to recurrence properties of orbits. As a consequence of our main result, we also obtain a partial extension of Keller’s theorem guaranteeing the existence of absolutely continuous invariant measures for non-uniformly hyperbolic one dimensional maps.     

BKW-operators on the interval [0,1]

Uniqueness sets inC([0, 1]) are studied for test functions {1,t,…,t ⁿ}. As an application, we investigateBKW-operators on the interval [0,1].     

The existence of invariantσ-finite measures for a group of transformations

SupposeG is a group of measurable transformations of aσ-finite measure space (X, A, m). The main result of this paper gives necessary and sufficient conditions for the existence of aG-invariant,σ-finite measure defined onA and dominating the measurem in the sense of absolute continuity. An example is also given of aσ-finite nonatomic measure space (X, A, m) together with a countable groupG of its measurable transformations such that noG-invariant,σ-finite nonatomic measure exists onA. Whether the Lebesgue space ([0, 1],L, λ) provides such an example, depends on set-theoretic assumptions. This paper was written while the author was visiting the Technische Universitat Berlin as a research fellow of the Alexander von Humboldt Foundation.     

[0,1]-拓扑空间中的Starplus-连通性

本文在I-拓扑空间中引入Starplus-连通性的概念,并且研究它的性质.借助于水平拓扑,得到了下面结果:Starplus-连通性是一般拓扑学中连通性的好推广而且被连续映射保持.一个Starplus-连通的模糊集一定是O-连通的.另外,模糊单位区间和模糊实直线是Starplus-连通的.     

On the existence ofσ-finite invariant measures for operators

Several necessary and sufficient conditions are given for the existence of aσ-finite invariant measure for a positive operator onL _∞. They are ofσ-type: the entire space is an increasing union of setsX _k each of which is well-behaved. To the Memory of Shlomo Horowitz Research in part supported by the National Science Foundation (U.S.A.).     

The law of the iterated logarithm in C[0,1]

The law of the iterated logarithm is proved for C[0,1] valued random variables under conditions related to those used to establish the central limit theorem.Supported in part by NSF Grant GP 18759.