Research On The Sample Orbit Properties Of Anisotropic Random Fields

Anisotropic random fields can describe the real world more truthfully than isotropic random fields,so it becomes a hot spot of research.There are two main aspects of research content of anisotropic random field:one is the modelling of anisotropic ran?dom fields,the other is the studies on the properties of the sample path of anisotropic random fields.In this thesis,we study mainly the sample path properties related to the anisotropic random fields,which mainly concern about the hitting probabilities,the dimension and Hausdorff measure results related to the random sets of the anisotropic random fields,the spectral conditions of local non-determinism and properties of local time for the random fields and so on.Details are as follows:In Chapter 1,we introduce some models related to anisotropic random fields,and summarize the research background and status of the sample path properties of the anisotropic random fields.Preliminaries are also given in this chapter.In Chapter 2,we study the hitting probabilities of a class of Gaussian random field-s.A class of Gaussian random fields which are anisotropic in time variable(isotropic in space variable)are provided according to the needs of research,which have more flexible covariance structure.By using the potential theory and fractal theory,the upper and lower bounds for hitting probabilities of the random fields are derived.In order to illustrate the random fields,a several interesting examples are constructed.In Chapter 3 we discuss the intersection of two independent Gaussian random field-s considered in Chapter 2,and obtain the sufficient conditions under which they inter-sect with each other.Due to the flexibility of the covariance,the sufficient conditions can be determined not only by the parameters of anisotropic metric in time variable but also by the function used to make the covariance more flexible.In Chapter 4,we consider the dimension results of space-anisotropic Gaussian random fields(which are isotropic in time variable).In order to adapt the anisotropy of random fields in space-variable,we introduce a new metric ? in space-variable.Then by using the potential theory and packing dimension profile,we obtain the ?-Hausdorff and packing dimension of the image and and the uniform Hausdorff dimension results.In Chapter 5,we study the hitting probability and dimension results of a class of Gaussian random fields which are anisotropic both in time-variable and in space-variable.As in Chapter 4,we use ? to adapt the anisotropy of random fields in space-variable,and p to the anisotropy of random fields in time-variable.By combining prop-erly the two methods used in dealing with time-anisotropic random fields and space-anisotropic random fields,we get the the hitting probabilities and dimension results of space-time anisotropic Gaussian random fields.In Chapter 6,the first objective is to obtain the spectral conditions of strong local nondeterminism with respect to some function ? for real valued stationary increment Gaussian random fields.By means of change-of-variable in polar ordinate with re-spect to some positive definite matrix,we get the spectral conditions for strong local?-nondeterminism with ? under more common conditions,which means that ? can be a mapping not only as identity function.If let ? be some power function,we also obtain the Hausdorff measure of the range of space-time anisotropic Gaussian random fields with stationary increments which are the second objective of this chapter.In Chapter 7,we study a special class of non-Gaussian random fields-harmonizable operator-scaling stable random fields.This kind of random fields are shown to satisfies the stable-type local nondeterminism.Then by making use of the local nondeterminism,we provide the sufficient conditions for the existence and jointly continuity of harmonizable operator-scaling stable random fields.In the last chapter,the research contents of this thesis are summarized,and the research work and main innovation points are given.The shortcomings and future research contents are also pointed out.