A Class Of Operator Stable Processes Sample Path Of Nature

Random fractal, which involves probability, classical analysis and geometry,is a new mathematics branch. The fractal properties of the sample path ofstochastic processes, which is an important component of the random fractaltheory, have become one of the most important and active research fields ofrandom fractal. With many favorable properties and structures emerging inthe strictly stable processes, it is necessary to generalize them and discuss theirrelated fractal properties. The purpose of this thesis is to investigate the exactHausdorff measure function and Packing measure of the range set and graphset of the dilation-stable process. The result partly solves an open problemsuggested by Xiao. lim sup type laws are also established of the iterated logarithmfor its sojourn time and first passage time. On the other hand, we find theexact Hausdorff measure function for the product sets of the independent stableprocesses. In the following we summarize them.Dilation-stable processes is a class of operator stable processes, and theirexponents are diagonal matrixes. This kind of processes is the natural generalityon the index of the strictly stable processes. The background of it is clear.Therefore, there are theoretic and applied values in studying the sample pathproperties of dilation-stable processes. To be precise, we first conjecture andshow- the exact Hausdorff measure functions for the range of the dilation-stableprocesses. To obtain the lower bound, we mainly make use of the density theorem,which is a useful tool. In the proof of the upper bound, we need to constructan economic covering for the range. We classify the cubes in the state spaceR~d into "good" cubes and "bad" cubes, according to the property of sojourntime of the process. The proof is more complicated. For the dilation-stableprocess is different from the process with independent stable components, itscomponents need not to be independent. We must overcome the problems comefrom the nonindependent. But we obtain some interesting results in the proof.For example, we establish the iterated logarithm for its sojourn time and first passage time. However, we find it is difficult to obtain the upper bound of theHausdorff measure of the graph by this method. So we take advantage of themethod of solving fractional Brownian motion. According to the local asymptoticbehavior of dilation-stable, we cover the time [0,1] by some specifical squares. Themethod of studying the Packing measure of the range and graph mainly comefrom solving the corresponding problem of the strictly stable process.Secondly, we study the exact Hausdorff measure function for the productsets of the two independent stable processes. In 1994, Hu obtained the exactHausdorff measure function of the product of range of two independent stablesubordinators on R. In this paper, we consider the more general case by differentmethod.