Sample Path Properties Of Several Time-Space Anisotropic Random Fields

The theory of random fields is identified naturally in probability theory and study-ing a system of stochastic differential equations.Since many data sets from various areas have different geometric and probabilistic characteristics,the random fields are applied in a wide range of scientific areas including hydrology,geostatistics,image processing and spatial statistics.The modelling of anisotropic random fields and the s-tudying properties of the sample path of anisotropic random fields are two main themes for the research content of anisotropic random fields.In this thesis,we study mainly the sample path properties related to several anisotropic random fields,which mainly con-cern about the hitting probabilities,the intersections for two independents anisotropic random fields,the dimension results related to the random sets of the anisotropic ran-dom fields,Hausdorff measure functions and continuity in the index of local times for the stable random fields and so on.Details of eight chapters are as follows:In Chapter 1,we introduce the research background?research status and some models for the sample path properties of the anisotropic random fields.Preliminaries are also given in this chapter.In Chapter 2,we study the Hausdorff dimension of intersections for two indepen-dent time-space anisotropic Gaussian random fields.In order to get our aims,we set up Gaussian random field models which are anisotropic both in time variable and in space variable.We give the second moments of every components of the Gaussian random fields governed by time metrics,conditional variance and strong local nondeterminis-m conditions in this chapter.By using the potential theory and covering theory,we first obtain the hitting probabilities of the random fields.After that,by using the re-sults of the hitting probabilities,we study and get the upper and lower bounds for the intersection probabilities.At last,we obtain the Hausdorff dimensions with positive probability of the intersection set for two independent time-space anisotropic Gaussian random fields.In Chapter 3,we study the packing dimensions of time-space anisotropic Gaussian random fields.By introducing a non-decreasing,right continuous function?,defining the lower index and the upper index for?at 0,and assuming that the second moments of the increments for every components of the Gaussian random fields satisfy generally conditions,we study the packing dimensions of time-space anisotropic Gaussian ran-dom fields on[0,1]^Nand on any bounded Borel set with the help packing dimension profile.The results show that the packing dimensions of the random fields not only are related to the time-space anisotropic indexes,but also the lower and upper indexes of function?.In Chapter 4,we study the Hausdorff measures of the range and the graph for a class of space anisotropic Gaussian processes with non-stationary increments.In con-trast to the extensive studies of sample paths'fractal properties of random fields with stationary increments,it is complexity for the dependence structure of non-stationary increments Gaussian processes.Under the conditions of self-similarity,bounded sec-ond moments and strong local nondeterminism for the space anisotropic Gaussian pro-cesses,we use Lamperti theorem to make non-stationary increments processes to be stationary processes.Then we get that the Hausdorff measure functions of the range and the graph are related to the space anisotropic indexes.In Chapter 5,we study the hitting probabilities of a time-space anisotropic random fields and study the intersections of two independent time-space anisotropic random fields.The time-space anisotropic random fields considered in this chapter may not possess the Gaussianity,they may be non-Gaussian random fields.The conditions here are given by several density functions.Under some conditions,for the intersections of two independent time-space anisotropic random fields,and the intersections with space set,the upper and lower bounds of intersection probabilities are governed by Haus-dorff measure and capacity respectively.Inspired by the study of nonlinear stochastic heat equations,we give an example on non-Gaussian random fields which satisfies the conditions in this chapter.In Chapter 6,we discuss Hausdorff dimension of the inverse image and Hausdorff dimension?packing dimension of the level set of time-space anisotropic random fields considered in Chapter 5.For the level set,the results under the anisotropic metrics?,?are more beautiful than that under the Euclidean metric,and the two different results can't obtained from each other.This means that it is nice to use anisotropic metrics?,?to study the properties of the time-space anisotropic random fields.In Chapter 7,we study the existence and the joint continuity of local time for a class of stable random fields.The conditions are related with index H for the stable random fields.The results show that the law of the local times of X^H(t)converges weakly to that of the local time of X^H0when H tends to H₀.In Chapter 8,we summarize the research contents of this thesis and give the re-search work and main innovation points.We also point out the shortcomings and future research contents.