Distribution of Statistical Observables for Anomalous and Nonergodic Diffusions: From Statistics to Mathematics

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
		2.1.4. Second-Order Scheme and Error Analysis
			2.1.4.1. Error analysis
		2.1.5. Numerical Experiments
			2.1.5.1. One-dimensional cases
			2.1.5.2. Two-dimensional cases
	2.2. NUMERICAL METHODS FOR THE SPACE-TIME FRACTIONAL FOKKER-PLANCK SYSTEM WITH TWO INTERNAL STATES
		2.2.1. Regularity of the Solution
			2.2.1.1. Preliminaries
			2.2.1.2. A priori estimate of the solution
		2.2.2. Space Discretization and Error Analysis
		2.2.3. Time Discretization and Error Analysis
		2.2.4. Numerical Experiments
CHAPTER 3: Numerical Methods for the Stochastic Governing Equations of PDF of Statistical Observables
	3.1. ASSUMPTIONS AND GAUSSIAN PROCESSES
	3.2. NUMERICAL SCHEMES FOR STOCHASTIC FRACTIONAL DIFFUSION EQUATION
		3.2.1. Numerical Approximation of Stochastic Fractional Diffusion Equation with a Tempered Fractional Gaussian Noise
			3.2.1.1. Regularity of the solution
			3.2.1.2. Galerkin approximation for spatial discretization
			3.2.1.3. Fully discrete scheme
			3.2.1.4. Numerical experiments
	3.3. NUMERICAL SCHEMES FOR STOCHASTIC FRACTIONAL WAVE EQUATION
		3.3.1. Higher Order Approximation for Stochastic Space Fractional Wave Equation
			3.3.1.1. Regularity of the solution
			3.3.1.2. Galerkin approximation for spatial discretization
			3.3.1.3. Fully discrete scheme
			3.3.1.4. Numerical experiments
		3.3.2. Galerkin Finite Element Approximation of Stochastic Fractional Wave Equations
			3.3.2.1. Solution representation
			3.3.2.2. Galerkin finite element approximation
			3.3.2.3. Error estimates for the nonhomogeneous problem
			3.3.2.4. Numerical results
Bibliography
Index