Fractal Properties Of Independent Increment Random Fields

Random fractal, which involves probability, classical analysis and geometry, is a new mathematics branch. The fractal properties of the sample path of stochastic processes, which are an important component of the random fractal theory, have become one of the most important and active research fields of random fractal. With many favorable properties and structures emerging in some independent increment stochastic processes, it is necessary to generalize them and discuss their related fractal properties. The purpose of this thesis is to investigate the fractal properties of several classes of stochastic processes and the main contents are the following:· The properties are studied of the polar sets for N-parameter d-dimensional generalized Brownian Sheet. The necessary conditions and sufficient conditions for a set to be polar are listed. Finally, the infimum is acquired of Hausdorff dimensions of its polar sets by means of constructing a Cantor-type set to connect Hausdorff dimension and capacity.A problem proposed by Xiao about the dimensions of its polar sets is solved[145].· The characteristics about the polar-functions for N-parameter d-dimensional generalized Brownian Sheet, are discussed. The relationship between the class of continuous functions satisfying Lipschitz condition and the class of polar-functions of generalized Brownian Sheet is obtained. The Hausdorff dimension about the stable points and the inequality about the Kolmogorov's entropy index for N-parameter d-dimensional generalized Brownian Sheet are presented and a question proposed by LeGall about the existence of no-polar, continuous functions statisfying the Holder condition is solved[54].· The following are also discussed about the Hausdorff dimension, the Packing dimension and the existence of the positive Lebesgue measure and the interior points of m-term algebraic sum of the image for generalized Brownian Sheet. A problem proposed by Mountford about the existence of the interior points of the image for Brownian Sheet is also solved[122].· The relationship between the hitting probabilities for generalized Brownian Sheet and the capacity is studied by means of multiparameter martingales. A capacity estimate for its hitting probabilities is provided. Finally the connections between the Lebesgue measure for its image and Bessel-Ricsz capacity are discussed and an explicit Bessel-Riesz capacity estimate is obtained. The conclusions also include the solution to a problem proposed by J. P. Kahane about the Bessel-Riesz capacity of additive Brown motion [62].· The existence and joint continuity are studied for the self-intersection local time of N-parameter d-dimension generalized α-stable process. A sufficient condition is presented for the existence of square integrable and joint continuity of its self-intersection local time. The existence of its multiple points is proved by means of getting Holder condition of its self-intersection local time and the lower bounds are obtained about the Hausdorff dimension and measure of its multiple times.· The problem of the uniform dimension for image set and graph set of N-parameter d-dimensional generalized a-stable process, which may not possess the uniform stochastic Holder condition, is investigated. The uniform Hausdorff dimension and Packing dimension for its image set and graph set are obtained by means of distribution of the sojourn time and the tail probability distribution for the increment maximum of stochastic process over interval.· The study also covers the usual Hausdorff measure and Hausdorff-type measure for the image set and graph set of certain locally nondeterministic Gaussian random fields. The exact Hausdorff measure of its graph set is given. The exact Hausdorff-type measures of its image set and graph set are provided and it is proved that the usual Hausdorff measure functions for its image set and graph set are still correct measure functions. Under certain conditions, the positive and finite Hausdorff-type measures for its image set and graph set arc given by some...