A planar random motion with an infinite number of directions controlled by the damped wave equation
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In this article, the authors considered the planar random motion of a particle that moves with constant finite speed c and changes its direction 0 with uniform law in [0, 27r] and derived the explicit probability law f(x, y, t) of (X(t), Y(t)), t > 0.Abstract:
We consider the planar random motion of a particle that moves with constant finite speed c and, at Poisson-distributed times, changes its direction 0 with uniform law in [0, 27r). This model represents the natural two-dimensional counterpart of the wellknown Goldstein-Kac telegraph process. For the particle's position (X (t), Y(t)), t > 0, we obtain the explicit conditional distribution when the number of changes of direction is fixed. From this, we derive the explicit probability law f(x, y, t) of (X(t), Y(t)) and show that the density p(x, y, t) of its absolutely continuous component is the fundamental solution to the planar wave equation with damping. We also show that, under the usual Kac condition on the velocity c and the intensity X of the Poisson process, the density p tends to the transition density of planar Brownian motion. Some discussions concerning the probabilistic structure of wave diffusion with damping are presented and some applications of the model are sketched.read more
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Fractional Poisson processes and related planar random motions
Luisa Beghin,Enzo Orsingher +1 more
TL;DR: In this article, three different fractional versions of the standard Poisson process and some related results concerning the distribution of order statistics and the compound poisson process are presented, and a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by fractional Poisson processes is presented.
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Random Flights in Higher Spaces
Enzo Orsingher,A. de Gregorio +1 more
TL;DR: In this article, the conditional characteristic function of the position of a particle after n changes of direction was obtained from this characteristic function and the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions.
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Random Motions at Finite Speed in Higher Dimensions
TL;DR: In this article, a general method of studying the transport process in the Euclidean space is presented, based on the analysis of the integral transforms of its distributions, which are connected with each other by a convolution-type recurrent relation.
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Random Evolutions Are Driven by the Hyperparabolic Operators
TL;DR: In this paper, it was shown that the multidimensional random evolutions are driven by the hyperparabolic operators composed of the telegraph operators and their integer powers, and that the only exception is the 2D random flight whose transition density is the fundamental solution to the two-dimensional telegraph equation.
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Fractional Klein–Gordon Equations and Related Stochastic Processes
TL;DR: In this paper, the fractional hyper-Bessel operator was converted into the Erdelyi-Kober integral operator and the distribution of fractional Klein-Gordon equations was analyzed.
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Table of Integrals, Series, and Products
TL;DR: Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integral Integral Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequality 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform
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