Strong Markov Properties for Markov Random Fields

Markov properties and strong Markov properties for random fields are defined and discussed. Special attention is given to those defined by I. V. Evstigneev. The strong Markov nature of Markov random fields with respect to random domains such as [0, L], where L is a multidimensional extension of a stopping time, is explored. A special case of this extension is shown to generalize a result of Merzbach and Nualart for point processes. As an additional example, Evstigneev's Markov and strong Markov properties are considered for independent increment jump processes.     

Asymptotic Expansions for the Distribution of the Maximum of Gaussian Random Fields

Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ^N . They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian field above an high level inside S and on the border S. Depending on the geometry of the border we give up to N+1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown.     

Hausdorff-type Measures of the Sample Path of Gaussian Random Fields

Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.     

On High-Level Exceedances of Gaussian Fields and the Spectrum of Random Hamiltonians

We study the almost sure asymptotic structure of high-level exceedances by Gaussian random field (x), xV with correlated values, where {V} is a family of -dimensional cubes increasing to Z . The results are applied to the study of the asymptotic behaviour of extreme eigenvalues of random Schrödinger operator in V.     

Anisotropic Fractional Brownian Random Fields as White Noise Functionals

This investigation aims at a new construction of anisotropic fractional Brownian random fields by the white noise approach. Moreover, we investigate its distribution and sample properties （stationariness of increments, self-similarity, sample continuity） which will furnish some useful views to future applications.     

Geometry of \mathbb{Z}^d and the Central Limit Theorem for Weakly Dependent Random Fields

We study the asymptotic distribution of where A is a subset of , A _N = A[–N, N] ^d , v(A) = lim _N card(A _N) (2N+1) ^–d (0, 1) and X is a stationary weakly dependent random field. We show that the geometry of A has a relevant influence on the problem. More specifically, S _N(A, X) is asymptotically normal for each X that satisfies certain mixting hypotheses if and only if has a limit F(n; A) as N for each . We also study the class of sets A that satisfy this condition.     

Models for Stationary Max-Stable Random Fields

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.     

Fractional Generalized Random Fields of Variable Order

Abstract

We study the class of random fields having their reproducing kernel Hilbert space isomorphic to a fractional Sobolev space of variable order on ? ⁿ . Prototypes of this class include multifractional Brownian motion, multifractional free Markov fields, and multifractional Riesz–Bessel motion. The study is carried out using the theory of generalized random fields defined on fractional Sobolev spaces of variable order. Specifically, we consider the class of generalized random fields satisfying a pseudoduality condition of variable order. The factorization of the covariance operator of the pseudodual allows the definition of a white-noise linear filter representation of variable order. In the ordinary case, the Hölder continuity, in the mean-square sense, of the class of random fields introduced is proved, and its mean-square Hölder spectrum is defined in terms of the variable regularity order of the functions in the associated reproducing kernel Hilbert space. The pseudodifferential representation of variable order of the resulting class of multifractal random fields is also defined. Some examples of pseudodifferential models of variable order are then given.     

次线性奇异积分算子在局部域的Herz空间上的有界性

令 1 p ∞ ,0 0 ,K为局部域 .本文将着重讨论一类线性分数次积分算子在 Herz空间K (α,p,l;K )上的有界性     

Almost Sure Asymptotics for Extremes of Non-stationary Gaussian Random Fields

In this paper, the authors prove an almost sure limit theorem for the maxima of non-stationary Caussian random fields under some mild conditions related to the covariance functions of the Gaussian fields. As the by-products, the authors also obtain several weak convergence results which extended the existing results.     

On the Essential Self-Adjointness of Wick Powers of Relativistic Fields and of Fields Unitary Equivalent to Random Fields

The essential self-adjointness on a natural domain of the sharp-time Wick powers of the relativistic free field in two space-time dimension is proven. Other results on Wick powers are reviewed and discussed.     

Linear Least Squares Estimation of Regression Models for Two-Dimensional Random Fields

We consider the problem of estimating regression models of two-dimensional random fields. Asymptotic properties of the least squares estimator of the linear regression coefficients are studied for the case where the disturbance is a homogeneous random field with an absolutely continuous spectral distribution and a positive and piecewise continuous spectral density. We obtain necessary and sufficient conditions on the regression sequences such that a linear estimator of the regression coefficients is asymptotically unbiased and mean square consistent. For such regression sequences the asymptotic covariance matrix of the linear least squares estimator of the regression coefficients is derived.     

Macroscaling Limit Theorems for Filtered Spatiotemporal Random Fields

This article addresses the problem of defining a general scaling setting in which Gaussian and non-Gaussian limit distributions of linear random fields can be obtained. The linear random fields considered are defined by the convolution of a Green kernel, satisfying suitable scaling conditions, with a non-linear transformation of a Gaussian centered homogeneous random field. The results derived cover the weak-dependence and strong-dependence cases for such Gaussian random fields. Extension to more general random initial conditions defined, for example, in terms of non-linear transformations of χ²-random fields, is also discussed. For an example, we consider the random fractional diffusion equation. The vectorial version of the limit theorems derived is also formulated, including the limit distribution of the parabolically rescaled solution to the Burgers equation in the cases of weakly and strongly dependent initial potentials.     

Characterizing Efficient Empirical Estimators for Local Interaction Gibbs Fields

The expectation of a local function on a stationary random field can be estimated from observations in a large window by the empirical estimator, that is, the average of the function over all shifts within the window. Under appropriate conditions, the estimator is consistent and asymptotically normal. Suppose that the field is a Gibbs field with known finite range of interactions but otherwise unknown potential. We show that the empirical estimator is efficient if and only if the function is (equivalent to) a sum of functions each of which depends only on the values of the field on a clique of sites. This revised version was published online in June 2006 with corrections to the Cover Date.     

Non-Gaussian Complex Random Fields, their Skeletons and Path Measures

This work investigates complex random fields Z, which have a rotation invariant path measure. Fields of this type are constructed and analyzed in terms of (pathwise convergent) L²-expansions, and quasi invariance properties of their path measures are studied. The results are used to investigate ℋL²(Z), the space of holomorphic L²-functionals of Z. Conditions are given such that every F∈ℋL²(Z) admits an L²-power series expansion, and a general skeleton theorem is proved, which justifies the notion ‘holomorphic’. Mathematics Subject Classifications (2000) 60G07, 60G30, 60G60. T. Deck: Financial support from FCP, Portugal, is gratefully acknowledged.     

Multi-Label Markov Random Fields as an Efficient and Effective Tool for Image Segmentation,Total Variations and Regularization

One of the classical optimization models for image segmentation is the well known Markov Random Fields (MRF) model. This model is a discrete optimization problem, which is shown here to formulate many continuous models used in image segmentation. In spite of the presence of MRF in the literature, the dominant perception has been that the model is not effective for image segmentation. We show here that the reason for the non-effectiveness is due to the lack of access to the optimal solution. Instead of solving optimally, heuristics have been engaged. Those heuristic methods cannot guarantee the quality of the solution nor the running time of the algorithm. Worse still, heuristics do not link directly the input functions and parameters to the output thus obscuring what would be ideal choices of parameters and functions which are to be selected by users in each particular application context.We describe here how MRF can model and solve efficiently several known continuous models for image segmentation and describe briefly a very efficient polynomial time algorithm, which is provably fastest possible, to solve optimally the MRF problem. The MRF algorithm is enhanced here compared to the algorithm in Hochbaum (2001) by allowing the set of assigned labels to be any discrete set. Other enhancements include dynamic features that permit adjustments to the input parameters and solves optimally for these changes with minimal computation time. Several new theoretical results on the properties of the algorithm are proved here and are demonstrated for images in the context of medical and biological imaging. An interactive implementation tool for MRF is described, and its performance and flexibility in practice are demonstrated via computational experiments.We conclude that many continuous models common in image segmentation have discrete analogs to various special cases of MRF and as such are solved optimally and efficiently, rather than with the use of continuous techniques, such as PDE methods, which restrict the type of functions used and furthermore, can only guarantee convergence to a local minimum.     

相应于随机自相似分形的记忆函数和分数次积分

For a physics system which exhibits memory, if memory is preserved only at points of random self-similar fractals, we define random memory functions and give the connection between the expectation of flux and the fractional integral. In particular, when memory sets degenerate to Cantor type fractals or non-random self-similar fractals our results coincide with that of Nigmatullin and Ren et al. .     

Fractional Generalized Random Fields on Bounded Domains

Using the theory of generalized random fields on fractional Sobolev spaces on bounded domains, and the concept of dual generalized random field, this paper introduces a class of random fields with fractional-order pure point spectra. The covariance factorization of an α-generalized random field having a dual is established, leading to a white-noise linear-filter representation, which reduces to the usual Markov representation in the ordinary case when α∈N and the covariance operator of the dual random field is local. Fractional-order differential models commonly arising from anomalous diffusion in disordered media can be studied within this framework.     

Restriction of Tangent Bundle and Semistability

Let G be a simple linear algebraic group defined over ? and P ? G a maximal proper parabolic subgroup such that m: = dim _? G/P ≥ 5. Let ι: Z ₁ ∩ Z ₂?G/P be a smooth complete intersection such that degree(Z _i ) ≥ (m ? 1)·index(G/P)/m, i = 1, 2. Then the vector bundle ι*T(G/P) → Z ₁ ∩ Z ₂ is semistable.     

随机结构空间的数学期望及应用

设(E，S，Ω，f)是随机结构空间，当(E，S，Ω，f)是随机度量空间，随机赋范空间，随机内积空间时，其向量的随机度量,随机范数，随机内积是随机变量．证明了它们的数学期望分别是拟度量，拟范数，内积．应用关于数学期望的结果，进而得到了随机Hilbert空间中线性连续泛函的Riesz表示定理．