统计自相似集与随机不变测度

设犡是一完备可分度量空间，犓（ω）为Ｇｒａｆ随机模型下的随机递归集．该文构造了一列随机不变测度μ狀（狀≥１），它们是Ｈｕｔｃｈｉｎｓｏｎ确定模型下不变测度的推广；证明了存在一随机概率测度μ ，使得Ｓｕｐｐμ ＝犓（ω）且μ狀→μ （狀→∞）（弱收敛）；得到了μ狀的一些局部性质．     

随机模糊集与随机集

本文研究了三个方面的工作：一是定义了一种模糊集上的可测结构，从而定义了随机模糊集，这些定义都与论域Ｘ上的拓扑结构无关。将通常意义下的集合看成特殊模糊集得到的通常集合上的超可测结构与文（３）中的定义一致；二是给出了随机模糊集、随机集的一些等价条件；三是研究了随机模糊集、随机集的分布与其有限维落影族的关系。     

随机集的独立性与同分布性的刻划

本文绘出了随机集的独立性与同分布性的刻化，所给出的准则比Hess[4]，[5]更易于验证．     

Monotone Nonincreasing Random Fields on Partially Ordered Sets. I

For an arbitrary poset H and measure ρ on H × _R (where _R is the real axis), we construct a monotone decreasing stochastic field η_ρ and compute its finite-dimensional distributions. In the case where H is a Λ-semilattice and the measure ρ satisfies additional conditions, we compute various characteristics of the field η_ρ such as the expectation of the field value at a point, variance of the field value at a point, and correlation function of the field. The described construction of random fields gives a new method for constructing positive definite functions on posets. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 301, 2003, pp. 92–143.     

Absolute Curvature Measures

For arbitrary sets of positive reach in euclidean space a new kind of absolute curvature measures is introduced. These measures possess similar section and projection properties as their signed counterparts—the Lipschitz-Killing curvature measures. In the present paper interpretations as invariant measures of the sets of colliding planes and as mean projection measures are given.     

Scalar Curvature and Q-Curvature of Random Metrics

We define a family of probability measures on the set of Riemannian metrics lying in a fixed conformal class, induced by Gaussian probability measures on the (logarithms of) conformal factors. We control the smoothness of the resulting metric by adjusting the decay rate of the variance of the random Fourier coefficients of the conformal factor. On a compact surface, we evaluate the probability of the set of metrics with non-vanishing Gauss curvature, lying in a fixed conformal class. On higher-dimensional manifolds, we estimate the probability of the set of metrics with non-vanishing scalar curvature (or Q-curvature), lying in a fixed conformal class.     

Invariance Conditions for Random Curvature Models

A class of probability models is introduced with the objective of representing certain properties of the geometric optics of the human eye. Astigmatic probability laws are those in which the extreme curvature values in the anterior corneal surface, measured at circularly arranged and equally spaced locations, are displaced by an approximate 90 deg angular separation. The relationship between the symmetry invariance of these probability laws for curvature data and probability laws for the ranking permutations associated with the ordering of these data is obtained. A distinction is made between the condition in which the components of the curvature ensemble are represented as real numbers from that in which these curvatures are color-coded and take value on a finite totally ordered set. A constructive principle for astigmatic laws is outlined based on algebraic arguments for the analysis of structured data.     

Matrix Measures, Random Moments, and Gaussian Ensembles

We consider the moment space M_n\mathcal{M}_{n} corresponding to p×p real or complex matrix measures defined on the interval [0,1]. The asymptotic properties of the first k components of a uniformly distributed vector (S_1,n, ... , S_n,n)^* ~ U (M_n)(S_{1,n}, \dots , S_{n,n})^{*} \sim\mathcal{U} (\mathcal{M}_{n}) are studied as n→∞. In particular, it is shown that an appropriately centered and standardized version of the vector (S _1,n ,…,S _k,n )^∗ converges weakly to a vector of k independent p×p Gaussian ensembles. For the proof of our results, we use some new relations between ordinary moments and canonical moments of matrix measures which are of their own interest. In particular, it is shown that the first k canonical moments corresponding to the uniform distribution on the real or complex moment space M_n\mathcal{M}_{n} are independent multivariate Beta-distributed random variables and that each of these random variables converges in distribution (as the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively.     

Random Measures and Applications

Abstract

A mapping Z(·) from a δ-ring ?₀(?) into the vector space of random variables L ^p (P) is a vector-valued measure if it is σ-additive in the metric of its range. It is a vector measure if the range is a Banach space and a random measure if also its values are independent on disjoint sets. An important reason for this study is to construct integrals relative to such Zs, which typically do not have finite variation. For this, it is essential to find a controlling (σ-finite) measure for Z that is not available if 0 <p < 1, and here the random measure is taken to be p-stable and utilize properties of infinitely divisible distributions. In the case of p = 2, Z(·) induces a bimeasure, and if p > 2 is an integer it induces a polymeasure, either of which need not be (signed) measures on product spaces. Important applications lead to all these possibilities. In all those cases, a detailed analysis of vector-valued set functions is presented, with special focus for the cases of 0 <p < 1 and p = 2 where probability and Bochner's L ^{2, 2} boundedness plays a key role. Specialization if Z is stationary, harmonizable, and/or isotropic are discussed using the group structure of ? ⁿ , n ≥ 1, extending it for an lca group G. If Z is Banach valued or a quasi-martingale measure, methods of obtaining integrals are outlined in the last section, and open problems motivated by applications are pointed out at various places.     

Random Diophantine equations,I

We consider additive Diophantine equations of degree k in s   variables and establish that whenever s?3k+2$s?3k+2$ then almost all such equations satisfy the Hasse principle. The equations that are soluble form a set of positive density, and among the soluble ones almost all equations admit a small solution. Our bound for the smallest solution is nearly best possible.     

Invariant Measures and Lower Ricci Curvature Bounds

Potential Analysis - Given a metric measure space $(X,d,mathfrak {m})$ that satisfies the Riemannian Curvature Dimension condition, RCD?(K,N), and a compact subgroup of isometries G ≤...     

On a Class of Random Variational Inequalities on Random Sets

We study a class of random variational inequalities on random sets and give measurability, existence, and uniqueness results in a Hilbert space setting. In the special case where the random and the deterministic variables are separated, we present a discretization technique based on averaging and truncation, prove a Mosco convergence result for the feasible random set, and establish norm convergence of the approximation procedure.     

Super-Tree Random Measures

We use supercritical branching processes with random walk steps of geometrically decreasing size to construct random measures. Special cases of our construction give close relatives of the super-(spherically symmetric stable) processes. However, other cases can produce measures with very smooth densities in any dimension.     

一类推广的中曲率水平集的运动

研究了根据一个光滑曲面的主曲率函数的经典运动短时存在及唯一性,推广了Evans和Spruck的结果。     

Conjugate Points on a Geodesic with Random Curvature

We study conjugate points on a renewable geodesic on which the curvature is a random process. We construct the upper bound for the mean distance between neighboring conjugate points.     

Jacobi Fields along a Geodesic with Random Curvature

A notion of a renewable geodesic on which the curvature is a random process is introduced. It is shown that the modulus of the Jacobi field along such a geodesic grows exponentially. At the same time, the existence with probability 1 of infinitely many conjugate points is demonstrated.     

Slicing Sets and Measures, and the Dimension of Exceptional Parameters

We consider the problem of slicing a compact metric space Ω with sets of the form $\pi_{\lambda}^{-1}\{t\}$ , where the mappings π _λ :Ω→?, λ∈?, are generalized projections, introduced by Yuval Peres and Wilhelm Schlag in 2000. The basic question is: Assuming that Ω has Hausdorff dimension strictly greater than one, what is the dimension of the “typical” slice $\pi_{\lambda}^{-1}\{t\}$ , as the parameters λ and t vary. In the special case of the mappings π _λ being orthogonal projections restricted to a compact set Ω??², the problem dates back to a 1954 paper by Marstrand; he proved that for almost every λ there exist positively many t∈? such that $\dim\pi_{\lambda }^{-1}\{t\} = \dim\varOmega- 1$ . For generalized projections, the same result was obtained 50 years later by Järvenpää, Järvenpää and Niemelä. In this paper, we improve the previously existing estimates by replacing the phrase “almost all λ” with a sharp bound for the dimension of the exceptional parameters.     

Envelopes of Sets of Measures, Tightness, and Markov Control Processes

We introduce upper and lower envelopes for sets of measures on an arbitrary topological space, which are then used to give a tightness criterion. These concepts are applied to show the existence of optimal policies for a class of Markov control processes. Accepted 22 May 1998