The Anomalous Diffusion Equations For Some Time-Changed Processes

This dissertation focuses on the anomalous diffusion equations of some subordinated processes. The main content of this paper is divided into four chapters. Chapters 1-3 discuss the uncoupled subordinated processes and the last chapter studies the coupled one. The parent processes in chapters 1-3 are the general Levy process or the stochastic processes driven by (fractional) Brownian motion, while the time processes are about the inverse tempered stable subordinator Sα,λ(t). In chapter 4 we construct a new kind of coupled subordinated process and analyze its governing equation.In the first chapter, the Levy measure of the tempered stable subordinator Tα,λ(t) is introduced. We construct a new operator Φt and analyze its prop-erties in detail. The Fokker-Planck type equation of the subordinated process with the tempered stable subordinator is developed, where the parent process is the stochastic process driven by (fractional) Brownian motion.In chapter 2, we obtain the Fokker-Planck type equation of the subordi-nated process with space-and-time-dependent force and diffusion.In chapter 3, we define a subordinator Tν，λ(t) determined by a Borel prob-ability measure v on (0,1). We prove the existence of this subordinator and give some properties of the process and its inverse. Then we apply the inverse process as a random time of a general Levy processor the stochastic process driven by fractional Brownian motion) to generate two kind of mixed tempered stable subordinated process. We discuss the governing equations of these two subordinated processes.In chapter 4, we construct a special kind of coupled subordinated process and study its governing equation. The expression of the governing equations of the coupled subordinated process by three different inverse subordinators are discussed. We also analyze the diffusion of the three coupled subordinated processes by the Fourier-Laplace transform of the probability density function of the processes.