The Hitting Probability And Local Time Of A Class Of Gaussian Stochastic Processes

Gaussian processes are an ubiquitous and important stochastic process with extensive applications in the field of natural science research.In this paper,properties of the sample path of a class of Gaussian stochastic processes are studied,mainly studying hitting probability,the existence and joint continuity and H(?)lder conditions of local time.This thesis consists of four chapters,which are arranged as follows:In Chapter 1,we introduce the research background,research history and current status,research contents and main methods for sample path properties of stochastic process,and give the preliminary knowledge.In Chapter 2,we study the hitting probability of a class of Gaussian stochastic processes.In order to study the covariance function of Gaussian processes more generally,the upper and lower bounds of the incremental second moment of the Gaussian stochastic process are related to a function ,where the function is a non-descending right continuous function,and the has an upper and lower index at 0.By using the potential theory and fractal theory,the upper and lower bound of the hitting probability of this class of Gaussian processes are derived.The Hausdorff measure of the upper bound and the capacity of lower bound are determined by the upper index and lower index of the function ,respectively.In Chapter 3,we extend the study of local time of Gaussian processes to Gaussian random fields.We consider the existence,joint continuity and H(?)lder conditions of local time of a class of time-anisotropic(spatial isotropy)Gaussian random fields.Let be introduced as above,which will make the choice of covariance structure of Gaussian random fields more flexible,that is,the upper and lower bounds of the covariance structure will not only depend on the anisotropic distance,but also depend on the function .The Fourier analysis method is used to obtain the sufficient conditions for existence of local time of Gaussian random fields.In the discussion of joint continuity and H(?)lder conditions of local time,considering the function as a regular varying function,the joint continuity of the local times of Gaussian random field is obtained by using the higher older moments estimation of local time and the multi-parameter version of Kolmogorov’s continuity theorem.In addition,the H(?)lder condition for this class of Gaussian random fields is obtained.Finally,several different Gaussian random fields are given to illustrate the main results.In Chapter 4,we summarize and analyze the contents of the previous research,point out the shortcomings in the research and the contents of further research in the future.