Markov Random Flights
Markov Random Flights is the first systematic presentation of the theory of Markov random flights in the Euclidean spaces of different dimensions. Markov random flights is a stochastic dynamic system subject to the control of an external Poisson process and represented by the stochastic motion of a particle that moves at constant finite speed and changes its direction at random Poisson time instants. The initial (and each new) direction is taken at random according to some probability distribution on the unit sphere. Such stochastic motion is the basic model for describing many real finite-velocity transport phenomena arising in statistical physics, chemistry, biology, environmental science and financial markets. Markov random flights acts as an effective tool for modelling the slow and super-slow diffusion processes arising in various fields of science and technology.
- Provides the first systematic presentation of the theory of Markov random flights in the Euclidean spaces of different dimensions.
- Suitable for graduate students and specialists and professionals in applied areas.
- Introduces a new unified approach based on the powerful methods of mathematical analysis, such as integral transforms, generalized, hypergeometric and special functions.
Alexander D. Kolesnik is a professor, Head of Laboratory (2015–2019) and principal researcher (since 2020) at the Institute of Mathematics and Computer Science, Kishinev (Chișinău), Moldova. He graduated from Moldova State University in 1980 and earned his PhD from the Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev in 1991. He also earned a PhD Habilitation in mathematics and physics with specialization in stochastic processes, probability and statistics conferred by the Specialized Council at the Institute of Mathematics of the National Academy of Sciences of Ukraine and confirmed by the Supreme Attestation Commission of Ukraine in 2010. His research interests include: probability and statistics, stochastic processes, random evolutions, stochastic dynamic systems, random flights, diffusion processes, transport processes, random walks, stochastic processes in random environments, partial differential equations in stochastic models, statistical physics and wave processes. Dr. Kolesnik has published more than 70 scientific publications, mostly in high-standard international journals and a monograph. He has also acted as external referee for many outstanding international journals in mathematics and physics, being awarded by the "Certificate of Outstanding Contribution in Reviewing" from the journal "Stochastic Processes and their Applications." He was the visiting professor and scholarship holder at universities in Italy and Germany and member of the Board of Global Advisors of the International Federation of Nonlinear Analysts (IFNA), United States of America.
1. Preliminaries. 1.1. Markov processes. 1.2. Random evolutions. 1.3. Determinant theorem. 1.4. Kurtz’s diffusion approximation theorem. 1.5. Special functions. 1.6. Hypergeometric functions. 1.7. Generalized functions. 1.8. Integral transforms. 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. 2.11. Telegraph-type processes with several velocities. 2.12. Euclidean distance between two telegraph processes. 2.13. Sum of two telegraph processes. 2.14. Linear combinations of 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. 3.3. 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. 4.5. 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 switching. 4.10. Hyper-parabolic operators. 4.11. Random flight with arbitrary dissipation function. 4.12. Integral equation for transition density. 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. 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. 6.5. 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. 9.2. Fluctuations of water level in reservoir. 9.3. Pollution model. 9.4. Physical applications. 9.5 Option pricing.