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Cover -- Half Title -- Title Page -- Copyright Page -- Contents -- Preface -- CHAPTER 1: Statistical Observables -- 1.1. INTRODUCTION -- 1.1.1. Physical Models -- 1.1.1.1. Continuous time random walk -- 1.1.1.2. Langevin equation -- 1.1.2. Stochastic Processes -- 1.1.2.1. Levy process -- 1.1.2.2. Subordinator -- 1.1.2.3. Time-changed process -- 1.2. POSITION -- 1.2.1. Probability Density Function -- 1.2.2. Fokker-Planck Equation -- 1.2.2.1. Derivation from continuous time random walk -- 1.2.2.2. Derivation from Langevin equation -- 1.3. FUNCTIONAL -- 1.3.1. Derivation from Continuous Time Random Walk -- 1.3.1.1. Forward Feynman-Kac equation -- 1.3.1.2. Backward Feynman-Kac equation -- 1.3.2. Derivation from Langevin Equation -- 1.3.2.1. Forward Feynman-Kac equation -- 1.3.2.2. Backward Feynman-Kac equation -- 1.3.2.3. Coupled Langevin equation -- 1.3.3. Derivation from Ito Formula -- 1.4. MEAN SQUARED DISPLACEMENT -- 1.4.1. Green-Kubo Formula -- 1.4.2. Ergodic and Aging Behavior -- 1.5. MISCELLANEOUS ONES -- 1.5.1. Fractional Moments -- 1.5.1.1. Infinite density of rare fluctuations -- 1.5.1.2. Dual scaling regimes in the central part -- 1.5.1.3. Complementarity among different scaling regimes -- 1.5.1.4. Ensemble averages -- 1.5.2. First Passage Time and First Hitting Time -- 1.5.2.1. First passage time of Levy flight and Levy walk -- 1.5.2.2. First hitting time of Levy flight and Levy walk -- CHAPTER 2: Numerical Methods for the Governing Equations of PDF of Statistical Observables -- 2.1. NUMERICAL METHODS FOR THE TIME FRACTIONAL FOKKER-PLANCK SYSTEM WITH TWO INTERNAL STATES -- 2.1.1. Preliminaries -- 2.1.2. Equivalent Form of (2.1) and Some Useful Lemmas -- 2.1.3. First-Order Scheme and Error Analysis -- 2.1.3.1. Error estimates for the homogeneous problem -- 2.1.3.2. Error estimates for the inhomogeneous problem.