Random Recursive Sets And Random Walk In Random Environment

The research of random fractal, an overlapping subject of probability and fractal, is focused on two respects: the random recursive sets and the sample paths of stochastic processes. Random recursive sets are a class of typical random sets with particular construction and their fractal properties are researched mostly in this thesis. Stochastic processes in random environments have been the hot topics in the domain of stochastic processes in recent years, however, most mathematicians only focused on the limit theory, on the base of the model of random walk in time-random environment, some fractal properties of its range as well as the asymptotic behaviors are emphasized in this thesis. Therefore, The thesis will begin with these two respects.In the researches of random recursive sets, people begin with the randomizing and generalizing of classical Cantor set whose structure is simple but application is extensive, and the basic fractal problem including dimensions and measures were already solved. However, random recursive sets, a type of more extensive and complicated random sets are investigated in this thesis. The probability properties and fractal properties are discussed under the base of the general definition of random recursive sets. Meanwhile, more detailed researches about the typical and general random recursive sets such as statistically self-similar set, a.s. self-similar set, random sub-self-similar sets are developed. The central contents include convergence, Hausdorff dimension and measure. Although some results about these topics are given in this thesis, the present results are far from the complete settlement.The researches of the sample path of stochastic processes start with Brown motion, stable processes and then the more general Levy processes, etc. Naturally, the more general are the stochastic processes, the more superficial are the results. There has recently been considerable interest in random walk in random environment since the concept of random environment appeared thirty years ago. Random walks in space-random environments and Markov chains in time-random environments are combined in this thesis to give an total concept of random walks in time-random environments, moreover, the fractal properties for the range of this kinds of random walk is emphasized. Since the state space of random walks Zd is discrete, it is necessary to introduce the concept of discrete fractal. Under some reasonable conditions, the Hausdorff dimension and packing dimension of the range are developed, which implies that the range is a so-called fractal set. The asymptotic behaviors of the random walks are also discussed, as by-product, some results such as recurrenceand transience criterion, strong law of large numbers and center limit theorem are established under some reasonable conditions.There are seven chapters in this thesis.In the preface in chapter 1, some general situation of the research of random recursive sets and random walks in time-random environments are introduced and the author's main results in these two directions are summarized.As the basis of the thesis, measure theory and dimension concept including several technical lemmas are presented in the second chapter. Hausdorff dimension, Packing dimension and other dimension concepts in Kd are introduced, the relations between each other are also generalized. In order to characterize the size of any numerable sets, the concept of discrete fractal in Zd is introduced.The probability and fractal theories about the random recursive sets are investigated from chapter 3 to chapter 5. As a class of special sets, recursive sets have many good properties, especially, their fractal properties have become an important branch in the theory of fractal sets. The simplest Cantor sets, self-similar sets and sub-self similar sets can all be seen as particular examples of recursive sets.In chapter 3, the convergence, Hausdorff dimension and measure about random recursive sets are investigated after establishing a general definition of these sets. O...