Operator Scaling Random Fields

Random field is a family of random variables indexed by points in an Euclidian parameter space. Random fields are a useful tool for modeling spatial phenomenon like environmental fields, including for example, hydrology, geology, oceanography and medi-cal images,etc. Classical examples such as the fractional Brownian fields have stationary increments and are self-similar with Hurst index H. Moreover they are isotropic, that is invariant under rotations of the underlying parameter space.In many applications, for example the modeling of fractured rock,however,random fields should have an anisotropic nature in the sense that they have different geometric characteristic in different directions. For this reason, an increasing interest has been paid in defining a suitable concept for anisotropic self-similarity. Many authors have developed techniques to handle anisotropy in the scaling. A model of anisotropic self-similar random field is the class of Operator Scaling Random Fields (OSRF) introduced by Bierme, Meerschaert and Scheffler [1].In this paper we investigate in detail about operator scaling random fields. We present a limit theorem about operator scaling random fields. Then we investigate Lamperti characterization of some well known operator scaling random fields such as H-mss. Finally we focus on the Log-representation of operator scaling random fields and analyze the related path properties.This paper is organized as follows. First of all we concisely retrospect the current situation of the research about selfsimilar processes in Introduction. In Chapter 2 we introduce some necessary knowledge to prepare for the following chapters.In Chapter 3 we present a limit theorem about operator scaling random fields.Chapter 4 is devote the Lamperti transformation of some well-know operator scaling random fields.Finally in Chapter 5 we focus on the log-representation of operator scaling stable random fields and analyze the related sample path properties.