多维对称马尔可夫随机飞行的2边际二阶矩函数的精确公式
Exact formula for the 2-marginal second moment function of the multidimensional symmetric Markov random flight
AI总结：
研究了多维对称马尔可夫随机飞行的2边际二阶矩函数，推导出其精确表达式，并证明该公式适用于所有形式为(0,…,0,2,0,…,0)的二边际二阶矩函数，在标准Kac缩放条件下转化为高斯布朗运动坐标方差乘积。
摘要：
我们考虑在欧几里得空间R^m（m≥3）中进行的对称马尔可夫随机飞行X(t)，t>0，由一个以恒定有限速度在R^m中运动并以泊松分布的时间间隔改变方向的粒子构成。该粒子在每次方向改变时，根据单位(m-1)维球面的均匀分布随机选择初始方向和新方向。研究了X(t)的2边际二阶矩函数μ_{(2,2,0,…,0)}(t)，t>0，对应的多索引(2,2,0,…,0)。推导出该函数的显式公式。该公式也适用于所有形式为(0,…,0,2,0,…,0)的二边际二阶矩函数。此外，证明在标准Kac缩放条件下，该矩函数转化为m维同质布朗运动两个坐标的方差乘积。
英文摘要：
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.
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