Marginals of the planar symmetric Markov random flight on long time intervals behave like the Goldstein-Kac telegraph process
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911049220882432 |
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| author | Kolesnik, Alexander D. |
| author_facet | Kolesnik, Alexander D. |
| contents | The planar symmetric Markov random flight $\bold X(t), \; t>0,$ is represented by the stochastic motion of a particle moving with constant finite speed $c>0$ in the Euclidean plane $\Bbb R^2$ and taking on its initial and each new directions at $λ$-Poisson ($λ>0$) distributed random time instants by choosing them at random according to the uniform distribution on the unit circumference. We consider the marginals of $\bold X(t)$, that is, the projection of this stochastic motion onto the axes. This projection onto the $x_1$-axis (respectively, onto the $x_2$-axis) represents a one-dimensional stochastic motion with random velocity $c \cosα$ (respectively, with random velocity $c \sinα$), where $α$ is a random variable distributed uniformly on the interval $[0, 2π)$. We prove that the density of the marginals of $\bold X(t)$ is asymptotically, as $t\to\infty$, equivalent to the density of the classical one-dimensional Goldstein-Kac telegraph process with parameters ($c, λ$). This unexpected and interesting result is confirmed by numerical calculations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_07525 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Marginals of the planar symmetric Markov random flight on long time intervals behave like the Goldstein-Kac telegraph process Kolesnik, Alexander D. Probability 60K35, 60J60, 62E20, 62F12, 82C41, 82C70 The planar symmetric Markov random flight $\bold X(t), \; t>0,$ is represented by the stochastic motion of a particle moving with constant finite speed $c>0$ in the Euclidean plane $\Bbb R^2$ and taking on its initial and each new directions at $λ$-Poisson ($λ>0$) distributed random time instants by choosing them at random according to the uniform distribution on the unit circumference. We consider the marginals of $\bold X(t)$, that is, the projection of this stochastic motion onto the axes. This projection onto the $x_1$-axis (respectively, onto the $x_2$-axis) represents a one-dimensional stochastic motion with random velocity $c \cosα$ (respectively, with random velocity $c \sinα$), where $α$ is a random variable distributed uniformly on the interval $[0, 2π)$. We prove that the density of the marginals of $\bold X(t)$ is asymptotically, as $t\to\infty$, equivalent to the density of the classical one-dimensional Goldstein-Kac telegraph process with parameters ($c, λ$). This unexpected and interesting result is confirmed by numerical calculations. |
| title | Marginals of the planar symmetric Markov random flight on long time intervals behave like the Goldstein-Kac telegraph process |
| topic | Probability 60K35, 60J60, 62E20, 62F12, 82C41, 82C70 |
| url | https://arxiv.org/abs/2507.07525 |