Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Introduction
1. Preliminaries
1.1. Markov processes
1.1.1. Brownian motion
1.1.2. Diffusion process
1.1.3. Poisson process
1.2. Random evolutions
1.3. Determinant theorem
1.4. Kurtzβs diffusion approximation theorem
1.5. Special functions
1.5.1. Bessel functions
1.5.2. Struve functions
1.5.3. Chebyshev polynomials
1.5.4. Chebyshev polynomials of two variables on Banach algebra
1.6. Hypergeometric functions
1.6.1. Euler gamma-function and Pochhammer symbol
1.6.2. Gauss hypergeometric function
1.6.3. Powers of Gauss hypergeometric function
1.6.4. General hypergeometric functions
1.7. Generalized functions
1.8. Integral transforms
1.8.1. Fourier transform
1.8.2. Laplace transform
1.9. Auxiliary lemmas
2. Telegraph Processes
2.1. Definition of the process and structure of distribution
2.2. Kolmogorov equation
2.3. Telegraph equation
2.4. Characteristic function
2.5. Transition density
2.6. Probability distribution function
2.7. Convergence to the Wiener process
2.8. Laplace transform of transition density
2.9. Moment analysis
2.9.1. Moments of the telegraph process
2.9.2. Asymptotic behaviour
2.9.3. Carleman condition
2.9.4. Generating function
2.9.5. Semi-invariants
2.10. Group symmetries of telegraph equation
2.11. Telegraph-type processes with several velocities
2.11.1. Uniform choice of velocities
2.11.2. Cyclic choice of velocities
2.12. Euclidean distance between two telegraph processes
2.12.1. Probability distribution function
2.12.2. Numerical example
2.13. Sum of two telegraph processes
2.13.1. Density of the sum of telegraph processes
2.13.2. Partial differential equation
2.13.3. Probability distribution function
2.13.4. Some remarks on the general case
2.14. Linear combinations of telegraph processes
2.14.1. Structure of distribution and system of equations
2.14.2. Governing equation
2.14.3. Sum and difference of two telegraph processes
3. Planar Random Motion with a Finite Number of Directions
3.1. Description of the model and the main result
3.2. Proof of the Main Theorem
3.2.1. System of equations and basic notations
3.2.2. Characters of a finite cyclic group and spectral decomposition of the unit matrix
3.2.3. Equivalent system of equations
3.2.4. Partial differential equation
3.3. Diffusion area
3.4. Polynomial representations of the generator
3.5. Limiting differential operator
3.6. Weak convergence to the Wiener process
4. Integral Transforms of the Distributions of Markov Random Flights
4.1. Description of process and structure of distribution
4.2. Recurrent integral relations
4.3. Laplace transforms of conditional characteristic functions
4.4. Conditional characteristic functions
4.4.1. Conditional characteristic functions in the plane R2
4.4.2. 4.4.2 Conditional characteristic functions in the space R4
4.4.3. Conditional characteristic functions in the space R3
4.4.4. Conditional characteristic functions in arbitrary dimension
4.5. Integral equation for characteristic function
4.6. Laplace transform of characteristic function
4.7. Initial conditions
4.8. Limit theorem
4.9. Random flight with rare switchings
4.10. Hyperparabolic operators
4.10.1. Description of the problem
4.10.2. Governing equation
4.10.3. Random flights in low dimensions
4.10.4. Convergence to the generator of Brownian motion
4.11. Random flight with arbitrary dissipation function
4.12. Integral equation for transition density
4.12.1. Description of process and the structure of distribution
4.12.2. Recurrent relations
4.12.3. Integral equation
4.12.4. Some particular cases
5. Markov Random Flight in the Plane R2
5.1. Conditional densities
5.2. Distribution of the process
5.3. Characteristic function
5.4. Telegraph equation
5.5. Limit theorem
5.6. Alternative derivation of transition density
5.7. Moments
5.8. Random flight with Gaussian starting point
5.9. Euclidean distance between two random flights
5.9.1. Auxiliary lemmas
5.9.2. Main results
5.9.3. Asymptotics and numerical example
5.9.4. Proofs of theorems
6. Markov Random Flight in the Space R3
6.1. Characteristic function
6.2. Discontinuous term of distribution
6.3. Limit theorem
6.4. Asymptotic relation for the transition density
6.4.1. Auxiliary lemmas
6.4.2. Conditional characteristic functions
6.4.3. Asymptotic formula for characteristic function
6.4.4. Asymptotic formula for the density
6.4.5. Estimate of the accuracy
6.5. Fundamental solution to Kolmogorov equation
7. Markov Random Flight in the Space R4
7.1. Conditional densities
7.2. Distribution of the process
7.3. Characteristic function
7.4. Limit theorem
7.5. Moments
8. Markov Random Flight in the Space R6
8.1. Conditional densities
8.2. Distribution of the process
9. Applied Models
9.1. Slow diffusion
9.1.1. Preliminaries
9.1.2. Slow diffusion condition
9.1.3. Stationary densities in low dimensions
9.2. Fluctuations of water level in reservoir
9.3. Pollution model
9.4. Physical applications
9.4.1. Transport processes
9.4.2. Relativity effects
9.4.3. Cosmic microwave background radiation
9.5. Option pricing
Bibliography
Index