Derivatives Of Local Times For Some Gaussian Fields

In this thesis,we mainly study the derivatives of local time for some class of Gaus-sian random fields and related problems,including existence condition,sample path properties and convergence rate for approximation of local time's derivatives.The full text is divided into five chapters.In Chapter 1,we first introduce the basic concepts of(N,d)-Gaussian random fields,its local time and local time's derivatives and then present research method like local non-determinism,moment method and so on.Moreover,we analyze the research status of local time and its derivatives for Gaussian random fields in recent decades.In Chapter 2,we research the sufficient condition for existence of local time's derivatives,using(2,d)-Gaussian random fields as an example.We get a sufficient condition for existence of local time's derivatives for Gaussian random fields composed of Gaussian processes with local non-determinism,and research its H(?)lder's Continuity.Using the same method,we obtain the convergence rate for approximation of local time's derivatives in L~pSpace.Moreover,for the Gaussian random fields composed of Gaussian processes with strong local non-determinism,we verify the exponential integrability for its local time's derivatives.In Chapter 3,we study the necessary condition for existence of local time's deriva-tives for(2,d)-Gaussian random fields in L~2.First,using the assumption for local non-determinism in Chapter 2,we prove that when the local time exists,the existence condi-tion in L~p(p?1)for local time's derivatives in Chapter 2 is also the necessary condition for existence of local time's derivatives at x ?0.Then for general cases,we give and prove the divergence condition in L~2for derivatives'approximation at x ?0 and x ?0seperately and then obtain necessary condition for the existence of local time's deriva-tives at any point in R~din L~2.In Chapter 4,we study the convergence rate for approximation of local time's derivatives.We prove seperately that for one Gaussian process and(2,d)-Gaussian random fields,the m-th(m?1)moment of difference between local time's derivatives and its approximation,multiplied by a normalizing factor,converges to the m-th mo-ment of a random variable,which is composed of the root of local time multiplied by an independent standard normal random variable.Then we use moment method to deduce a convergence in law for random variables from the moment convergence.Moreover,for one Gaussian process,we give a limit theorem in law for random process in C[0,?).In Chapter 5,we make a summary of this thesis,and point out the innovations of this thesis and where can be improved in the future.