Markov Random Flights

Alexander D. Kolesnik is the 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 received PhD from the Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev, Ukraine in 1991 and PhD Habilitation in mathematics and physics with the 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, wave processes. Dr Kolesnik has published more than 70 scientific publications, including those published in high-standard international journals and a monograph. He has also acted as external referee for many outstanding international journals in mathematics and physics. He was the visiting professor and scholarship holder in the universities in Italy and Germany and member of the Board of Global Advisors of the International Federation of Nonlinear Analysts (IFNA), U.S.A.

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.