Exact formula for the 2-marginal second moment function of the multidimensional symmetric Markov random flight
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913072575152128 |
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| author | Kolesnik, Alexander D. |
| author_facet | Kolesnik, Alexander D. |
| contents | We consider the symmetric Markov random flight $\bold X(t), \; t>0,$ in the Euclidean space $\Bbb R^m, \; m\ge 3$, performed by a particle that moves in $\Bbb R^m$ with constant finite speed and changes its directions at Poisson-distributed random time instants by choosing the initial and each new direction at random according to the uniform distribution on the unit $(m-1)$-dimensional sphere. The 2-marginal second moment function $μ_{(2,2,0,\dots,0)}(t), \; t>0,$ of $\bold X(t)$, corresponding to the multi-index $(2,2,0,\dots,0)$, is examined. An explicit formula for function $μ_{(2,2,0,\dots,0)}(t)$ is obtained. This formula is also valid for all other 2-marginal second moment functions corresponding to any multi-indices of the form $(0,\dots,0,2,0,\dots,0,2,0,\dots,0)$. It is also shown that this moment function, under the standard Kac scaling condition, turns into the product of the variances of two coordinates of the $m$-dimensional homogeneous Brownian motion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26432 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact formula for the 2-marginal second moment function of the multidimensional symmetric Markov random flight Kolesnik, Alexander D. Probability 60K35, 60J60, 62E20, 62F12, 82C41, 82C70 We consider the symmetric Markov random flight $\bold X(t), \; t>0,$ in the Euclidean space $\Bbb R^m, \; m\ge 3$, performed by a particle that moves in $\Bbb R^m$ with constant finite speed and changes its directions at Poisson-distributed random time instants by choosing the initial and each new direction at random according to the uniform distribution on the unit $(m-1)$-dimensional sphere. The 2-marginal second moment function $μ_{(2,2,0,\dots,0)}(t), \; t>0,$ of $\bold X(t)$, corresponding to the multi-index $(2,2,0,\dots,0)$, is examined. An explicit formula for function $μ_{(2,2,0,\dots,0)}(t)$ is obtained. This formula is also valid for all other 2-marginal second moment functions corresponding to any multi-indices of the form $(0,\dots,0,2,0,\dots,0,2,0,\dots,0)$. It is also shown that this moment function, under the standard Kac scaling condition, turns into the product of the variances of two coordinates of the $m$-dimensional homogeneous Brownian motion. |
| title | Exact formula for the 2-marginal second moment function of the multidimensional symmetric Markov random flight |
| topic | Probability 60K35, 60J60, 62E20, 62F12, 82C41, 82C70 |
| url | https://arxiv.org/abs/2604.26432 |