Abstract
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 . Here, we extend these approaches to 3D using Fourier transform on the Lie group of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations as the quotient in . 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 -stable Lévy processes on . This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for (the diffusion kernel) to the kernel for (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.
Highlights
The Fourier transform has had a tremendous impact on various fields of mathematics including analysis, algebra and probability theory
We find the exact formulas for the kernels in the frequency domain relying on a spectral decomposition of the evolution operator
The considered Fourier transform acts on functions that are bi-invariant with respect to the action of subgroup H
Summary
The Fourier transform has had a tremendous impact on various fields of mathematics including analysis, algebra and probability theory. The degenerate (hypo-elliptic) diffusion kernel formula in terms of a Fourier series representation was generalized to the much more wide setting of unimodular Lie groups by Agrachev, Boscain, Gauthier and Rossi [23] This approach was pursued by Portegies and Duits to achieve explicit exact solutions to (non-)degenerate (convection-)diffusions on the particular SE(3) case (see [40]). We generalize the exact solutions to other PDE systems beyond the diffusion case: We simultaneously solve the Forward-Kolmogorov PDEs for α-stable Lévy processes on the homogeneous space of positions and orientations We address their relevance in the fields of image analysis, robotics and probability theory. Such investigations first require a good grip on the spectral decompositions of the PDE-evolution operators for α-stable Lévy processes via a Fourier transform on R3 o S2 , which is our primary focus here
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