Stochastic Dynamics of Generalized Planar Random Motions with Orthogonal Directions

Fabrizio Cinque () and Enzo Orsingher ()
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Fabrizio Cinque: Sapienza University of Rome
Enzo Orsingher: Sapienza University of Rome

Journal of Theoretical Probability, 2023, vol. 36, issue 4, 2229-2261

Abstract: Abstract We study planar random motions with finite velocities, of norm $$c>0$$ c > 0 , along orthogonal directions and changing at the instants of occurrence of a nonhomogeneous Poisson process with rate function $$\lambda = \lambda (t),\ t\ge 0$$ λ = λ ( t ) , t ≥ 0 . We focus on the distribution of the current position $$\bigl (X(t), Y(t)\bigr ),\ t\ge 0$$ ( X ( t ) , Y ( t ) ) , t ≥ 0 , in the case where the motion has orthogonal deviations and where also reflection is admitted. In all the cases, the process is located within the closed square $$S_{ct}=\{(x,y)\in {\mathbb {R}}^2\,:\,|x|+|y|\le ct\}$$ S ct = { ( x , y ) ∈ R 2 : | x | + | y | ≤ c t } and we obtain the probability law inside $$S_{ct}$$ S ct , on the edge $$\partial S_{ct}$$ ∂ S ct and on the other possible singularities, by studying the partial differential equations governing all the distributions examined. A fundamental result is that the vector process (X, Y) is probabilistically equivalent to a linear transformation of two (independent or dependent) one-dimensional symmetric telegraph processes with rate function proportional to $$\lambda $$ λ and velocity c/2. Finally, we extend the results to a wider class of orthogonal-type evolutions.

Keywords: Planar motions with finite velocities; Telegraph processes; Motions with reflections; Partial differential equations; Bessel functions; Primary 60K99; 60G50 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01229-2

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