Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fourier Transform on the Homogeneous Space of 3D Positions and
  Orientations for Exact Solutions to Linear Parabolic and (Hypo-)Elliptic PDEs   Fokker-Planck PDEs (incl. diffusions) for stable L\'{e}vy processes (incl.
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 $SE(2)$. Here we
extend these approaches to 3D using Fourier transform on the Lie group $SE(3)$
of rigid body motions. More precisely, we define the homogeneous space of 3D
positions and orientations $\mathbb{R}^{3}\rtimes S^{2}:=SE(3)/(\{\mathbf{0}\}
\times SO(2))$ as the quotient in $SE(3)$. 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
$\alpha$-stable L\'{e}vy processes on $\mathbb{R}^{3}\rtimes S^{2}$. This
reduces classical analysis computations and provides an explicit algebraic
spectral decomposition of the solutions. We compare the exact probability
kernel for $\alpha = 1$ (the diffusion kernel) to the kernel for
$\alpha=\frac12$ (the Poisson kernel). We set up SDEs for the L\'{e}vy
processes on the quotient and derive corresponding Monte-Carlo methods. We
verify that the exact probability kernels arise as the limit of the Monte-Carlo
approximations.

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The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
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The categories of this paper are: econ
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph