The Multifractal Analysis For The Sample Paths Of Certain Types Of Gaussian Processes

The multifractal analysis for stochastic processes is one of the most active direction in the investigation of random fractal and stochastic processes in recent years. Additive Levy processes, which possess very abundant and interesting structure, and stem from the study of intersection and self-intersection of Levy processes. More and more scholars,such as D.Khoshnevisan,Z. Shi and Y.M.Xiao etc.investigated these processes recently. However, many properties of Additive Levy processes and even of Additive Brownian motion are not clear. The aim of this paper is to discuss the multifractal nature of the sample paths for some Gauss processes , including the Additive Brownian Motion. The main results are as follows:(I) The Hausdorff dimension of fast points of the increment for Additive Brownian Motion.Suppose T > 0,we consider the Hausdorff dimension ofwhere and Bt(a) denote fast points determimed by the local increment and by the increment in the direction of coordinate of the Additive Brownian motion X = {X(t) : t RN+} respectively. We obtain the result as follows: for 0 < a < ,(II) The packing dimension of fast points of the increment for Additive Brownian Motion .Suppose T > 0, AT(a) and Bt(a) mentioned above. In this paper. We obtain the result as follows: for 0 < a < 1,Dim(Ar(a)) = Dim(BT(a)) = N a.s.(III) The packing dimension of fast points of the increment for Wiener sheet. Suppose T > 0, we consider the Packing dimension ofthey denote fast points determined by the increment in the direction of coordinate, by the local increments and by the rectangle increments of the Wiener sheet W = respectively, we obtain the result as follows : for 0 < a < 1,and: for 0 < a < 1,Dim(GT(a)) = N a.s.