Modeling Anomalous Diffusion: From Statistics to Mathematics

Contents
Preface
1. Stochastic Models
	1.1 Background Knowledge in Statistics and Probability
		1.1.1 Random Variables and Distributions
		1.1.2 Moments
	1.2 Algorithm for the Generation of Random Variables
	1.3 Continuous Time Random Walk and Lévy Process
		1.3.1 Continuous Time Random Walk
		1.3.2 Propagator Function
		1.3.3 Lévy Process
	1.4 Lévy Flight, Lévy Walk, and Subordinated Processes
		1.4.1 Lévy Flight
		1.4.2 Lévy Walk
		1.4.3 Subordinator
	1.5 Langevin Pictures for Lévy Flights
	1.6 Continuous Time Random Walk and Lévy Walk withMultiple Internal States
2. Fokker-Planck Equations
	2.1 Fractional Derivative and Integral
		2.1.1 Grünwald-Letnikov Fractional Derivative
		2.1.2 Riemann-Liouville Fractional Derivative
		2.1.3 Fractional Substantial Derivative
		2.1.4 Laplace Transform of Fractional Derivative
	2.2 Derivation of Fractional Fokker-Planck Equation
	2.3 Solution of Fractional Fokker-Planck Equation
		2.3.1 Integral Form of the Solution for Fokker-Planck Equation
		2.3.2 Solution for Force Free Fractional Diffusion
		2.3.3 Solution for Biased Fractional Wiener Process
		2.3.4 Solution Obtained by Separation of Variables
3. Feynman-Kac Equations
	3.1 Brownian Functionals
	3.2 Fractional Feynman-Kac Equations
		3.2.1 Forward Fractional Feynman-Kac Equation
		3.2.2 Backward Fractional Feynman-Kac Equation
		3.2.3 Distribution of Occupation Times
	3.3 Tempered Fractional Feynman-Kac Equations
		3.3.1 Model and Tempered Dynamics
		3.3.2 Tempered Fractional Feynman-Kac Equations of Random Walk on a One-Dimensional Lattice
		3.3.3 Tempered Fractional Feynman-Kac Equations of Random Walk with Forces
		3.3.4 Distribution of Occupation Time in Half Space
		3.3.5 Distribution of First Passage Time
		3.3.6 Distribution of Maximal Displacement
		3.3.7 Fluctuations of Occupation Fraction
	3.4 Feynman-Kac Equations Revisited: Langevin Picture
		3.4.1 Forward Feynman-Kac Equation
		3.4.2 Backward Feynman-Kac Equation
		3.4.3 Distribution of Occupation Time in Positive Half Space
		3.4.4 Distribution of First Passage Time
		3.4.5 Area under Random Walk Curve
4. Aging Fokker-Planck and Feynman-Kac Equations
	4.1 Aging CTRW
	4.2 Aging Renewal Theory
	4.3 ACTRW with Tempered Power Law Waiting Time
		4.3.1 MSD
		4.3.2 Propagator Function p(x, ta, t)
	4.4 Strong Relation between Fluctuation and Response
	4.5 Fokker-Planck Equations for Tempered ACTRW
	4.6 Derivations of Aging Feynman-Kac Equation
		4.6.1 Forward Feynman-Kac Equation with Discrete Step Length PDF
		4.6.2 Forward Feynman-Kac Equation with Continuous Step Length PDF
			4.6.2.1 Power Law Waiting Time
			4.6.2.2 Tempered Power Law Waiting Time
		4.6.3 Backward Feynman-Kac Equation with Discrete Step Length PDF
		4.6.4 Backward Feynman-Kac Equation with Continuous Step Length PDF
	4.7 Application
		4.7.1 Occupation Time in Half Space for ACTRW
		4.7.2 Fluctuation of Occupation Fraction
		4.7.3 Distribution of First Passage Time
5. Fokker-Planck and Feynman-Kac Equations with Multiple Internal States
	5.1 Model and Notations
	5.2 Fractional Fokker-Planck Equations for CTRW with Multiple Internal States
	5.3 Equations Governing Distribution of Functionals of Paths and Internal States of Process
	5.4 Some Applications of Feynman-Kac Equations and Governing Equations of Functionals of Internal States
	5.5 Lévy Walk with Multiple Internal States
	5.6 More Applications for CTRW and Lévy Walk with Multiple Internal States
6. Fractional Reaction Diffusion Equations and Corresponding Feynman-Kac Equations
	6.1 Fractional Reaction Diffusion Equations
		6.1.1 Reaction-Anomalous Diffusion Equations
		6.1.2 Non-Markovian Transport with Nonlinear Reactions
	6.2 Feynman-Kac Equations for Reaction and Diffusion Processes
		6.2.1 Forward Feynman-Kac Equations for Nonlinear Reaction Rate r(ρ(x, t))
		6.2.2 Forward Feynman-Kac Equations for Nonlinear Reaction Rate r(t)
		6.2.3 Forward Feynman-Kac Equations for Nonlinear Reaction Rate r(x)
		6.2.4 Derivation of Backward Feynman-Kac Equations
		6.2.5 Distribution of Occupation Time in Half Space and its Fluctuations
		6.2.6 Distribution of First Passage Time
		6.2.7 Distribution of Occupation Time in Half Interval
7. Renewal Theory for Fractional Poisson Process: Typical versus Rare
	7.1 Introduction
	7.2 Model
	7.3 Number of Renewals between 0 and t
		7.3.1 Number of Renewals between 0 and t with 0 < α < 1
		7.3.2 Number of Renewals between 0 and t with 1 < α < 2
	7.4 Forward Recurrence Time
		7.4.1 Forward Recurrence Time with 0 < α < 1
		7.4.2 Forward Recurrence Time with 1 < α < 2
	7.5 Backward Recurrence Time
		7.5.1 Backward Recurrence Time with 0 < α < 1
		7.5.2 Backward Recurrence Time with 1 < α < 2
	7.6 Time Interval Straddling t
		7.6.1 Time Interval Straddling t with 0 < α < 1
		7.6.2 Time Interval Straddling Time t with 1 < α < 2
	7.7 Occupation Time
		7.7.1 Occupation Time with 0 < α < 1
		7.7.2 Occupation Time with 1 < α < 2
	7.8 Some Properties of Stable Distribution
	7.9 Discussion
8. Governing Equation for Average First Passage Time and Transitions among Anomalous Diffusions
	8.1 Governing Equation for Average First Passage Time
	8.2 Transition among Anomalous Diffusions: CTRW Description
	8.3 Non-Negativity of Solution: Subordinated Approach, and Stochastic Representation
	8.4 MSD, Fractional Moments, and Multi-Scale
	8.5 Fractional Fokker-Planck Equation with Prabhakar Derivative
		8.5.1 Relaxation of Modes
		8.5.2 Harmonic External Potential
	8.6 A Brief Introduction of Three Parameter Mittag-Leffler Functions
Bibliography
Index