Duits Remco, Bekkers Erik J, Mashtakov Alexey
Department of Mathematics and Computer Science (CASA), Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands.
Entropy (Basel). 2019 Jan 8;21(1):38. doi: 10.3390/e21010038.
Fokker-Planck PDEs (including diffusions) for stable Lévy processes (including Wiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D case, where they have been obtained using Fourier transform on S E ( 2 ) . Here, we extend these approaches to 3D using Fourier transform on the Lie group S E ( 3 ) of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations R 3 ⋊ S 2 : = S E ( 3 ) / ( { 0 } × S O ( 2 ) ) as the quotient in S E ( 3 ) . In our construction, two group elements are equivalent if they are equal up to a rotation around the reference axis. On this quotient, we design a specific Fourier transform. We apply this Fourier transform to derive new exact solutions to Fokker-Planck PDEs of α -stable Lévy processes on R 3 ⋊ S 2 . This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for α = 1 (the diffusion kernel) to the kernel for α = 1 2 (the Poisson kernel). We set up stochastic differential equations (SDEs) for the Lévy processes on the quotient and derive corresponding Monte-Carlo methods. We verified that the exact probability kernels arise as the limit of the Monte-Carlo approximations.
在位置和方向的联合空间上，用于稳定 Lévy 过程（包括维纳过程）的福克 - 普朗克偏微分方程（包括扩散方程）在力学、机器人技术、图像分析、方向统计和概率论中起着重要作用。在二维情况下，通过在 SE(2) 上使用傅里叶变换已得到精确的解析设计和解决方案。在此，我们利用刚体运动的李群 SE(3) 上的傅里叶变换将这些方法扩展到三维。更确切地说，我们将三维位置和方向的齐性空间 R³⋊S² := SE(3)/({0}×SO(2)) 定义为 SE(3) 中的商空间。在我们的构造中，如果两个群元素绕参考轴旋转后相等，则它们是等价的。在这个商空间上，我们设计了一种特定的傅里叶变换。我们应用这种傅里叶变换来推导 R³⋊S² 上 α - 稳定 Lévy 过程的福克 - 普朗克偏微分方程的新精确解。这减少了经典分析计算，并提供了解的显式代数谱分解。我们将 α = 1（扩散核）的精确概率核与 α = 1/2（泊松核）的核进行比较。我们为商空间上的 Lévy 过程建立了随机微分方程（SDE）并推导了相应的蒙特卡罗方法。我们验证了精确概率核是蒙特卡罗近似的极限。
