Sample Path Properties Of Some Gaussian Random Fields

Random fields,as a more general form of stochastic process,are widely used in various scientific fields because they can better depict the uncertainty phenomenon.Meanwhile,the research of random fields is also a hot topic in modern probability theory,and the property of sample path of random fields is an important part of the theoretical research of random fields and give rise to wide attention and study.However,the research is not perfect on some random fields with good properties and structures,and some critical problems have not been solved,which needs further research.So it is necessary to continue to make deep research and develop theory of random fields.This thesis,mainly studies the relevant sample path properties of some Gaussian random fields,contents as follows:1.Based on the different condition of existence for renormalization and nonrenor-malization of self-intersection local time,the renormalized self-intersection local time of bifractional Brownian motion is discussed.Firstly,the existence of renormalized self-intersection local time of bifractional Brownian motion is obtained by the mean square convergence;Secondly,the smoothness of renormalized self-intersection local time of bifractional Brownian motion is obtained by Maliavin calculus.2.Polar function of anisotropic random fields is studied with the time variable taken value in Euclidean space.This part firstly gives the hypothesis of the generalized probability density function satisfying more cases.On the basis of this hypothesis,the probability of the polar function being a continuous function satisfying certain conditions is obtained by the methods of small ball probability and capacity,and this part also researches the upper and lower bounds of Hausdorff measure,Hausdorff dimension and packing dimension of the polar function.3.The existence and smoothness of local time of spherical Gaussian random fields are researched.Firstly,the existence of local time of spherical Gaussian random fields is studied in the sense of L~2;Secondly,the smoothness of local time of spherical Gaussian random fields is studied in the sense of Meyer-Watanabe by the chaos expansion of Malliavin calculus;Finally,the existence and smoothness of self-intersection local time of spherical Gaussian random fields are researched.4.The joint continuity of local time of spherical Gaussian random fields is dis-cussed.Firstly,the existence of intersection local time of two mutually independent spherical Gaussian random fields is studied by the method of Fourier analysis;Second-ly,the joint continuity of intersection local time of spherical Gaussian random fields is researched by standard method;Finally,the H?lder condition of the intersection local time is established.5.The Hausdorff measures of image set and graph set of spherical Gaussian random fields are discussed.Firstly,band-limited random field is used to solve the independence of spherical Gaussian random fields,and the upper bound of Hausdorff measure of image set is researched by small ball probability;Secondly,the lower bound of Hausdorff measure of image set is researched by the sojourn time and the relationship between-density of measure function on the sphere and Hausdorff measure;Finally,the upper and lower bounds of Hausdorff measure of graph set of spherical Gaussian random fields are discussed.