Fading random evolution on a complex plane

Igor Samoilenko

Abstract


The article explores a generalization of the Goldstein-Kac model, specifically a model of random evolution on a complex plane, over time. This process simulates the motion of a particle in a force field, among other phenomena. Limit theorems describing the distribution of the absorbing point for this process have been derived. Additionally, nonlinear integral equations for functionals of the process have been obtained, and the existence and uniqueness of their solutions have been proven.

Full Text:

PDF

References


H. Bateman, A. Erdélyi, Higher Transcendental Functions, McGraw-Hill Book Company, 1954.

F. Cinque, E. Orsingher, Random motions in R3 with orthogonal directions, Stochastic Processes and their Applications 161 (2023), 173-200.

H. Gzyl, Lorentz covariant physical Brownian motion: classical and quantum, Annals of Physics 472 (2024), 169857.

M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mount. J. Math. 4 (1974), no. 3, 497-509.

A.D. Kolesnik, A note on planar random motion at finite speed, Journal of Applied Probability 44 (2007), no. 3, 838-842.

A.D. Kolesnik, Random motions at finite speed in higher dimensions, Journal of Statistical Physics 131 (2008), no. 6, 1039-1065.

A.D. Kolesnik, Weak convergence of a planar random evolution to the Wiener process, Journal of Theoretical Probability 14 (2001), no. 2, 485-494.

A.D. Kolesnik, E. Orsingher, A planar random motion with an infinite number of directions controlled by the damped wave equation, Journal of Applied Probability 42 (2005), no. 4, 1168-1182.

V.S. Korolyuk, A.F. Turbin, Mathematical Foundations of the State Lumping of Large Systems, Springer Dordrecht, 1993.

E. Orsingher, A planar random motion governed by the two-dimentional telegraph equations, Journal of Applied Probability 23 (1986), no. 2, 385-397.

E. Orsingher, Bessel functions of third order and the distribution of cyclic planar motions with three directions, Stochastics and Stochastics Reports 74 (2002), no. 3-4, 617-631.

E. Orsingher, A. De Gregorio, Random flights in higher spaces, Journal of Theoretical Probability 20 (2007), 769-806.

E. Orsingher, R. Garra, A.I. Zeifman, Cyclic random motions with orthogonal directions, Markov Processes and Related Fields 26 (2020), no. 3, 381-402.

E. Orsingher, A. Sommella, Cyclic random motion in R3 with four directions and finite velocity, Stochastics and Stochastics Reports 76 (2004), no. 2, 113-133.

M. Pinsky, Lectures on Random Evolutions, World Scientific, 1991.

A. Pogorui, A. Swishchuk, R.M. Rodríguez-Dagnino, Random Motions in Markov and SemiMarkov Random Environments 2: High-Dimensional Random Motions and Financial Applications, Wiley, 2021.

A.A. Pogorui, R.M. Rodríguez-Dagnino, Goldstein-Kac telegraph equations and random flights in higher dimensions, Applied Mathematics and Computation 361 (2019), 617-629.

I.V. Samoilenko, Markovian random evolution in Rn, Random Operators and Stochastic Equations 9 (2001), no. 2, 139-160.

I. Samoilenko, G. Verovkina, T. Samoilenko, Analytic solutions of equation for random evolution on a complex plane, Austrian Journal of Statistics 52 (2023), SI, 71-81.

A.F. Turbin, I.V. Samoilenko, A probability method for the solution of the telegraph equation with real-analytic initial conditions, Ukrainian Mathematical Journal 52 (2000), no. 8, 1292-1299.




DOI: https://doi.org/10.52846/ami.v52i2.2066