The Multifractal Analysis For The Sample Paths Of Multi-parameter Wiener Process

The multifractal analysis of random process has been paid close attention in recent years. Many scholars have started the observation about it. Based on the results of the iterated logarithm law and the uniform modulus of continuity for Brownian motion, Orey and Taylor(1972) discussed the multifractal decomposition of one dimension white noise. They obtained the Hausdorff dimension of the so-called fast points that lead to the failure of the iterated logarithm law for Brownian motion. However, the multifractal decomposition of high dimension white noise, that is the multifractal analytic properties of the sample paths for multi-parameter Wiener process, hasn't been discussed.In this paper, we discuss the multifractal analysis properties for the sample paths of multi-parameter Wiener process. Because of the partially-ordered nature in the multi-parameter index space, the discussion about the multifractal analytic properties of high dimension white noise is much more complex than that of one dimension, and there exist several forms of the multifractal decomposition of the multi-parameter sets. In this paper, we investigate the multifractal decomposition of the sample paths of Brownian sheet in three forms determinded by the increments as follow :(I) The Hausdorff dimension of fast points determinded by the increments in the direction of coordinate.Suppose that T > 0, LetBy means of constructing a Cantor-like set K included in ET-(α), the lower bound of the dimension of ET(OL) is estimated. Finally we gain the result as follow: 0 < α <,(II) The Hausdorff dimension of fast points determinded by the local increments.Let?,the method differing from (I) lead to the dimension upper bound estimate of FT (α). We gain the result as follow: VO < α < dim(III) The Hausdorff dimension of fast points determinded by the rectangle increments.distinct difference between the pobability properties of the rectangle increments and that of the local increments of Brownian sheet leads to the failure of normal methods used in the past, we attempt to overcome the difficulties mentioned above with a new method, and manage to get the result as follow: 0 < α < 1,...