Fractional Properties Of Some Stochastic Processes

Since twenty century, the standard Brownian motion has a rich golden mine. Plenty of important theorems have been found about it. Today, the Brownian motion is still studied by many mathematician, and it is indeed worthy of study.In 1929, Wiener studied the almost surely continuity for paths of the standard Brownian motion. After then, Lévy researched it deeply and obtained an account of important theory for Brownian motion, for example, the uniformly continuous modulus, et. al. All these make the path properties of stochastic process more rich and more attractive. Now, it has became one of the most vigourious and important branches in the basic research in probability, and has attracted attention of many authors.Up to now, the research on the path properties of the stochastic processes has been developed profoundly. The local properties such as the boundness, continuity modulus, non-differentiable modulus and iterated logarithm laws have been deepened to the exact global path properties like the large increment theorems. Most of these results were collected in the work of Cs(?)rg(?) and Révész named "Strong Approximations in Probability and Statistics" (1981). For the study from the standard Brownian motion to the general Gaussian processes, we refer to the works of Alder "The Geometry of Random Fields" (1981), Lin, Lu and Zhang "Path Properties of Gaussian Processes" (2001).With the development of the extensive applications of stochestic processes to other fields, furthermore, the permeation and consolidation of stochastic processes and other fields, the theoretical research on path properties is faced with new challenges and chances. And there are a large number of topics with strong application background and theoretical value, for example, the water discharge of rivers and the appearance of the mountain are altered continuously by the impact of the nature factors, the standard Brownian motion model seems no longer very perfect in characterizing these phenomena. Consequently, it is necessary to extend the research objects. And there mainly are three extension orientations: the first one is to extend the Brownian motion to the general Gaussian processes according to the distribution of the process, for example, fractional Brownian motion, multifractional Browniau motion, the second is to the general independent increment processes, i.e., the Lévy process according to the independent increment property, the third is to the compound processes, for example the iterated Brownian motion etc.The deep and beautiful theory about the paths of Brownian motion making by Lévy inspired Mandelbrot, who raised a new branch in mathematics: Fractral Geometry. The appearance of Mandelbrot's famous work "The Fractal Geometry of Nature" (1975), fractal properties of the stochastic processes attracted many authors and the theory was developed very quickly. To 1990's, there have been a lot of deep and perfect results about them. For example, the work of Goldman (1988) named" Movement BrownienàPiusiurs Paramètres: Measure de Hausdorff des Trajectories" and Facloner's work (1990) "Fractal Geometry-Mathematical Foundations and Applications" and so on. Now, fractal properties of the stochastic processes are still very active topics in the theoretical researches.In this thesis, we discuss the path properities and the fractral dimension of random set about the following five stochastic processes: multifractional Brownian motion, generalized iterated Brownian motion, infinite series of independent O-U processes, Markov process of O-U type and Lévy process.We say a set A is a level set of a process {X(t), t∈R⁺} at x provided A={t:X(t)=x}; a set B is a range set of a process {X(t), t∈R⁺} on Q provided B={x:x(t)=x, t∈Q}; a set C is a graph set of a process {X(t), t∈R⁺} on Q provided C={(t, x):X(t)=x, t∈Q}.In the first chapter, we discuss the Hausdorff dimension of the level set of the multifractional Brownian motion. Since the multifractional Brownian motion is a Gaussian process, we check that it has local nondeterminism at first. The concept of local nondeterminism was introduced by Berman (1973). A process is locally nondeterministic if a future observation is "relatively unpredictable" on the basis of a finite set of observation from the immediate past.Theorem 1.2.1. Let (?)(t) be a mBm, assume that |H_t-H_s|=o((1-s/t)^H_t(log t)^-1) as 0＜t-s→0, (1.2.7) then (?) (t) is locally nondeterministic on any open interval (0, T).For a Gaussian process, the local nondeterminism is closely related to the existence of a jointly continuous local time. In the section 2 of this chapter, we prove the existence and continuity of the local time of the multifractional Brownian motion.Theorem 1.3.1. Let (?) (t) be a mBm and satisfy condition (2.7), then it has a jointly continuous local time.In sections 3 and 4 of this chapter, we show the upper bound and the lower bound of the Hausdorff dimensions of the level sets.Theorem 1.4.1. Suppose that the mBm (?) (·) satisfies condition (1.2.7), then for any T∈R⁺ and almost all x∈R, the Hausdorff dimension of the level set E(x, T) dim_H E(x, T)≤1-(?) H(t) a.s.Theorem 1.4.2. Let (?) (t) be a mBm and satisfy condition (1.2.7), then for any T∈R and almost every x, dim_H E(x, T)≥1-(?) H_t a.s.We prove the upper bound by the uniformly continuous modulus, and the lower bound by the principle of energy integration. The results obtained by us generalize that of Taylor et.al, for the fractional Brownian motion dim_H E′(x, T)=1-H a.s. This is Corollary 1.4.1. In the second chapter, we consider the Hausdorff dimension of the level set of the generalized iterated Brownian motion.Theorem 2.1.1 Let E(x, T) be the level set of the generalized iterated Brownian motion {H(t), 0≤t＜∞}. With probability one, the Hansdorff dimension dim_H E(x, T)=3/4 a.s. for all 0＜T≤1 and every x in the interior of the range of {H(t), t∈[0, T]}.We prove it by the order of uniformly H(?)lder continuity of local time in the second chapter instead of the principle of energy integration in the first chapter. In section 4 of this chapter, we obtain the packing dimension of the level set of the generalized iterated Brownian motion.Theorem 2.1.2 Let E(x,T) be the level set of the generalized iterated Brownian motion {H(t), 0≤t＜∞}. With probability one, the packing dimension dim_P E(x,T)=3/4 a.s. (1.2) for all 0＜T≤1 and every x in the interior of the range of {H(t), t∈[0, T]}.These two dimensions are equal, so the level sets are regular.In the third chapter, we study the fractal property of the infinite series of independent O-U processes. By applying the main ideas of Geman and Horowitz et. al. (1984), we obtain the conditions of existence and continuity of the local time. Then from it we give the Hansdorff dimension of its level set. In second 5, we obtain the Hausdorff dimension of the graph set. The method is motivated by Facloner (1990), which is not widely used in the random fractal field.In the fourth chapter, we give a criterion for the existence and continuity of local time of the Markov process of O-U type {X(t), 0≤t＜∞}. The Hausdorff dimension of the range of X([0,1]) is given under some conditions. In general, there are some differences in tools and techniques for studying random fractals between Gaussian processes and other Markov processes. The processes discussed in chapters 1, 2 and 3 are not Markov processes in general, at each stage of studying, some special properties about Gaussian processes are used. Since the Markov process of O-U type is generated by the Lévy process, we discuss it by Taylor's ideas for Lévy processes, at the same time, we apply some Blumenthal and Getoor's results about Markov processes.In the last chapter, we answer an open question of Pruitt and Taylor (1996) (see also Xiao (2004)): Is it true thatγ₀=inf{α≥0:α^-αT(α, 1)→∞asα→0 a.s.}? What we use are the definition and properties of local time, and a inequality for the occupation time measure which is proved by using the Lévy inequality.The author make a progress during writing this thesis by the guide professor Lin Zhengyan.This dissertation is made up of some papers which were written by authors in the past three years. Some of these papers have been accepted, and the rest have been submitted to various journals of probability theory. Details are attached to the appendix. Due to my limited knowledge, errors may incur in this dissertation, so your criticism would be greatly appreciated.