论文标题
$ \ mathbb {r}^3 $的随机动作带有正交方向
Random motions in $\mathbb{R}^3$ with orthogonal directions
论文作者
论文摘要
本文专门针对$ \ mathbb {r}^3 $中的三维动作的详细分析，其正交方向在Poisson Times切换，并以恒定速度$ C> 0 $移动。在支持表面上对随机位置$ t> 0 $的研究，形成八面体$ s_ {ct} $，完全在边缘$ e_ {ct} $上进行，并面对$ f_ {ct} $。特别是，通过转换将其减少为三个方向的平面随机运动，对面部的运动$ f_ {ct} $进行了分析。这使我们可以在$ f_ {ct} $上获得一阶贝塞尔函数产品的积分上的积分表示。在$ s_ {ct} $内部$ p = p = p = p（t，x，y，z）$的分布的调查暗示了管理$ p $的六阶部分偏微分方程（用三个d'Anembert运营商的产品表示）。许多结果，也以明确的形式，涉及每个方向上花费的时间以及每个坐标作为运动Devolpes所达到的位置。当传入方向与正在进行的方向正交时以及在每个泊松事件中都可以均匀地选择所有方向时，进行分析。如果开关受均匀的泊松过程控制，则获得许多明确的结果。
This paper is devoted to the detailed analysis of three-dimensional motions in $\mathbb{R}^3$ with orthogonal directions switching at Poisson times and moving with constant speed $c>0$. The study of the random position at an arbitrary time $t>0$ on the surface of the support, forming an octahedron $S_{ct}$, is completely carried out on the edges $E_{ct}$ and faces $F_{ct}$. In particular, the motion on the faces $F_{ct}$ is analysed by means of a transformation which reduces it to a three-directions planar random motion. This permits us to obtain an integral representation on $F_{ct}$ in terms of integral of products of first order Bessel functions. The investigation of the distribution of the position $p=p(t,x,y,z)$ inside $S_{ct}$ implied the derivation of a sixth-order partial differential equation governing $p$ (expressed in terms of the products of three D'Alembert operators). A number of results, also in explicit form, concern the time spent on each direction and the position reached by each coordinates as the motion devolpes. The analysis is carried out when the incoming direction is orthogonal to the ongoing one and also when all directions can be uniformely choosen at each Poisson event. If the switches are governed by homogeneus Poisson process many explicit results are obtained.
