The Study Of Level Sets Of Stable Processes And Multifractals Of Graph Directed Iterated Function System

There are two aspects of contexts in this thesis. One is on the study of the sample path property of multi-parameter stochastic processes. In precise, we obtain the exact Hausdorff measure functions for level sets of N-parameter symmetric stable processes. The other is about the multifractal properties of fractal generated via iterated function system. We investigate "the multifractal decomposition of directed multi-graph self-similar sets and self-conformal sets and the generalized dimension spectrum of self-conformal measures as well. We summarize them in the following.Firstly, we discuss the problem of level sets of N-parameter d-dimensional strictly stable processes in chapter 3. These kind of process are the natural generality in parameter space of one-parameter strictly stable processes. So, the background of it is clear, for example, Brownian sheets are the stochastic processes received much special interest, and we get the Brownian sheets when taking 2 in the character index a of symmetric stable processes. In addition, as one of multi-parameter stochastic processes, the good properties of stable processes make it popular processes. Therefore, there are theoretic and applied values in studying the sample path properties of multi-parameter stable processes. W. Ehm gave this stochastic processes in 1981, and he obtained most of the exact Hausdorff measure functions for the sets of the range and graph of the processes, excluding some special cases, (i.e., the cases N times is exactly equal to d). Another problem he did not answer is the Hausdorff dimension and the exact Hausdorff measure functions for level sets of this processes.Now we refer to some related stochastic processes, of which the exact Hausdorff measure functions for their level sets have been obtained. In 1996, S. J. Taylor & J. G. Wendel gave the exact Hausdorff measure functions for level sets of one-parameter stable processes. X. Zhou provided the answer to the one-dimensional Brownian sheets in 1993, and H. Lin generalized it to the cases of high dimension in 2001. A detailed comparison of all the measure functions above, we conjecture the expression of the exact Hausdorff measure functions for level sets of the N-parameter d-dimensional strictly stable processes, and we show the conjecture is at least true for the symmetric stable processes.To be precise, we first obtain the lower bounds of the Hausdorff measure functions for level sets of the strictly stable processes. The methods of the proof we used come mainly from that of solving the corresponding problem of the multi-dimensional Brownian sheets. That is, we introduce the Local times processes, which is well known when considering the level sets of stochastic processes. D. Geman & J. Horowitz (1980) gave us a comprehensive survey onivthe Local times, which sometimes known as occupation densities. Fortunately, W. Ehm has proved the existence of the Local times of the multi-parameter stable processes. Base on this, we construct an auxiliary stochastic processes, boundary processes, of the original stable processes, and by using the methods provided by W. Ehm, we also get the existence of Local times of the new stochastic processes. By formulating the growth of the Local times, we thus obtain the lower bounds of the Hausdorff measure functions for the level sets of strictly stable processes. On the other hand, when we are proving the upper bounds of the Hausdorff measure functions for the level sets, although we get the answer to the cases for the symmetric processes, we had to consider multi-integral of highly dimensional functions, the proof is quite complex. Therefore, we suggest that the proof for the corresponding problem needs some new ways. Ultimately, by the two aspects above, we have solved the exact Hausdorff measure functions for the level sets of symmetric multi-parameter stable processes in chapter 3.Secondly, we begin to study the multifractal properties of the directed multi-graph recursive sets from chapter 4. These sets are the fractals generated by iterating...