标题： 具有空间变化速度的随机运动 显示英文标题

标题： Random motions with space-varying velocities

摘要： 在线上和平面上具有空间变化速度的随机运动在本文中被考虑和分析。 在线上，我们研究具有空间依赖速度的对称和非对称电报过程，并能够给出移动粒子的位置$\mathcal{T}(t)$,$t>0$的显式分布。 同时，分析了三种特定情况下由非齐次泊松过程（速率为$\lambda = \lambda(t)$）控制方向变化的情况。 对于特殊情况$\lambda(t)= \alpha/t$，我们得到了与欧拉-泊松-达布克斯（EPD）方程相关的随机运动，该方程推广了例如 Foong 和 Van Kolck（1992）、Garra 和 Orsingher（2016）以及 Rosencrans（1973）中处理的著名情况。 还考虑了一种 EPD 型分数阶方程，并得到了一个抛物型解（在维度$d=1$中具有概率密度的结构）。 最后，在第5节中分析了具有空间变化速度和无限方向的平面随机运动。 我们能够给出它们的显式分布，对于多项式类型的速变结构，我们得到了它们支撑集的超椭圆和次椭圆形式（我们提供了其图像）。 显示英文摘要

摘要： Random motions on the line and on the plane with space-varying velocities are considered and analyzed in this paper. On the line we investigate symmetric and asymmetric telegraph processes with space-dependent velocities and we are able to present the explicit distribution of the position $\mathcal{T}(t)$, $t>0$, of the moving particle. Also the case of a non-homogeneous Poisson process (with rate $\lambda = \lambda(t)$) governing the changes of direction is analyzed in three specific cases. For the special case $\lambda(t)= \alpha/t$ we obtain a random motion related to the Euler-Poisson-Darboux (EPD) equation which generalizes the well-known case treated e.g. in Foong and Van Kolck (1992), Garra and Orsingher (2016) and Rosencrans (1973). A EPD--type fractional equation is also considered and a parabolic solution (which in dimension $d=1$ has the structure of a probability density) is obtained. Planar random motions with space--varying velocities and infinite directions are finally analyzed in Section 5. We are able to present their explicit distributions and for polynomial-type velocity structures we obtain the hyper and hypo-elliptic form of their support (of which we provide a picture).