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.     

Random Fields with Multifractional Regularity Order on Heterogenous Fractal Domains

In this article, we study the effect of the geometry of a domain with variable local dimension on the regularity/singularity of the restriction of a multifractional random field on such a domain. The theories of reproducing kernel Hilbert spaces (RKHS) and generalized random fields are applied. Fractional Sobolev spaces of variable order are considered as RKHSs of random fields satisfying certain elliptic multifractional pseudodifferential equations. The multifractal spectra of these random fields are trivial due to the regularity assumptions on the variable order of the fractional derivatives. In this article, we introduce a family of RKHSs defined by isomorphic identification with the trace on a compact heterogeneous fractal domain of a fractional Sobolev space of variable order. The local regularity/singularity order of functions in these spaces, which depends on the variable order of the fractional Sobolev space considered and on the local dimension of the domain, is derived. We also study the spectral properties of the family of models introduced in the mean-square sense. In the Gaussian case, random fields with sample paths having multifractional local Hölder exponent are covered in this framework.     

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.     

Tangent Fields and the Local Structure of Random Fields

A tangent field of a random field X on ^N at a point z is defined to be the limit of a sequence of scaled enlargements of X about z. This paper develops general properties of tangent fields, emphasising their rich structure and strong invariance properties which place considerable constraints on their form. The theory is illustrated by a variety of examples, both of a smooth and fractal nature.     

Time-Localization of Random Distributions on Wiener Space II: Convergence,Fractional Brownian Density Processes

For a random element X of a nuclear space of distributions on Wiener space C([0,1],R ^d ), the localization problem consists in projecting X at each time t[0,1] in order to define an S(R ^d )-valued process X={X(t),t[0,1]}, called the time-localization of X. The convergence problem consists in deriving weak convergence of time-localization processes (in C([0,1],S(R ^d )) in this paper) from weak convergence of the corresponding random distributions on C([0,1],R ^d ). Partial steps towards the solution of this problem were carried out in previous papers, the tightness having remained unsolved. In this paper we complete the solution of the convergence problem via an extension of the time-localization procedure. As an example, a fluctuation limit of a system of fractional Brownian motions yields a new class of S(R ^d )-valued Gaussian processes, the fractional Brownian density processes.     

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     

随机环境中广义随机游动的灭绝概率

随机环境中广义随机游动(GRWRE)是随机环境中随机游动(RWRE)的推广．该文构造了非负整数集上的GRWRE,证明了这种模型的存在性,并计算了灭绝概率．     

Path Collapse for an Inhomogeneous Random Walk

On an open interval we follow the paths of a Brownian motion which returns to a fixed point as soon as it reaches the boundary and restarts afresh indefinitely. We determine that two paths starting at different points either cannot collapse or they do so almost surely. The problem can be modelled as a spatially inhomogeneous random walk on a group and contrasts sharply with the higher dimensional case in that if two paths may collapse they do so almost surely.     

Some Processes Associated with Fractional Bessel Processes

Let be a d-dimensional fractional Brownian motion with Hurst parameter H and let be the fractional Bessel process. Itôs formula for the fractional Brownian motion leads to the equation . In the Brownian motion case is a Brownian motion. In this paper it is shown that X_t is not an -fractional Brownian motion if H 1/2. We will study some other properties of this stochastic process as well.     

Large Deviations of a Storage Process with Fractional Brownian Motion as Input

We study probabilities of large extremes of the storage process Y(t) = sup _t (X() - X(t) - c( - t)), where X(t) is the fractional Brownian motion. We derive asymptotic behavior of the maximum tail distribution for the process on fixed or slowly increased intervals by a reduction the problem to a large extremes problem for a Gaussian field.     

A Central Limit Theorem for the Generalized Quadratic Variation of the Step Fractional Brownian Motion

This paper gives a central limit theorem for the generalized quadratic variation of the step fractional Brownian motion. We first recall the definition of this process and the statistical results on the estimation of its parameters.     

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）.     

Fractional Brownian Density Process and Its Self-Intersection Local Time of Order k

The fractional Brownian density process is a continuous centered Gaussian ( ^d )-valued process which arises as a high-density fluctuation limit of a Poisson system of independent d-dimensional fractional Brownian motions with Hurst parameter H. ( ( ^d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure of the (initial) Poisson random measure on ^d is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k 2 if and only if Hd < k/(k – 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand¹². This result extends to a non-Markovian case the relationship known for (Markovian) symmetric -stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case H 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove.     

广义分数布朗运动的若干性质

本文, 我们定义了一类新的分数布朗运动, 研究了它的局部非决定性和局部时的联合连续性, 最后给出了它的水平集的Hausdorff维数的上、下界.     

离散型随机变量的次序统计量的分布

本文论述了离散型随机变量的次序统计量的分布律及其有关推论 .     

Random Dynamical Systems and Stationary Solutions of Differential Equations Driven by the Fractional Brownian Motion

Abstract

Linear and semilinear stochastic evolution equations with additive noise, where the forcing term is an infinite dimensional fractional Brownian motion are studied. Under usual dissipativity conditions the equations are shown to define random dynamical systems which have unique, exponentially attracting fixed points. The results are applied to stochastic parabolic PDE's. They are also applicable to standard finite-dimensional dissipative stochastic equation driven by fractional Brownian motion.     

Fractional normal inverse Gaussian diffusion

A fractional normal inverse Gaussian (FNIG) process is a fractional Brownian motion subordinated to an inverse Gaussian process. This paper shows how the FNIG process emerges naturally as the limit of a random walk with correlated jumps separated by i.i.d. waiting times. Similarly, we show that the NIG process, a Brownian motion subordinated to an inverse Gaussian process, is the limit of a random walk with uncorrelated jumps separated by i.i.d. waiting times. The FNIG process is also derived as the limit of a fractional ARIMA processes. Finally, the NIG densities are shown to solve the relativistic diffusion equation from statistical physics.     

Instrumental Variable Estimation for Linear Stochastic Differential Equations Driven by Fractional Brownian Motion

Abstract

We investigate the asymptotic properties of instrumental variable estimators of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by fractional Brownian motion.