### Abstract

Fokker-Planck PDEs (including diffusions) for stable Lévy processes (includingWiener 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 ℝ^{3} ⋊ S^{2} := SE(3)/0 × 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 a-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.

Language | English |
---|---|

Article number | 38 |

Number of pages | 38 |

Journal | Entropy |

Volume | 21 |

Issue number | 1 |

DOIs | |

State | Published - 8 Jan 2019 |

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### Bibliographical note

This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century.### Keywords

- Fourier transform
- Homogeneous spaces
- Lie Groups
- Lévy processes
- Partial differential equations
- Rigid body motions
- Stochastic differential equations

### Cite this

*Entropy*,

*21*(1), [38]. DOI: 10.3390/e21010038

}

*Entropy*, vol. 21, no. 1, 38. DOI: 10.3390/e21010038

**Fourier transform on the homogeneous space of 3D positions and orientations for exact solutions to linear PDEs.** / Duits, Remco (Corresponding author); Bekkers, Erik J.; Mashtakov, Alexey.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Fourier transform on the homogeneous space of 3D positions and orientations for exact solutions to linear PDEs

AU - Duits,Remco

AU - Bekkers,Erik J.

AU - Mashtakov,Alexey

N1 - This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century.

PY - 2019/1/8

Y1 - 2019/1/8

N2 - Fokker-Planck PDEs (including diffusions) for stable Lévy processes (includingWiener 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 ℝ3 ⋊ S2 := SE(3)/0 × 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 a-stable Lévy processes on ℝ3 ⋊ S2. 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.

AB - Fokker-Planck PDEs (including diffusions) for stable Lévy processes (includingWiener 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 ℝ3 ⋊ S2 := SE(3)/0 × 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 a-stable Lévy processes on ℝ3 ⋊ S2. 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.

KW - Fourier transform

KW - Homogeneous spaces

KW - Lie Groups

KW - Lévy processes

KW - Partial differential equations

KW - Rigid body motions

KW - Stochastic differential equations

UR - http://www.scopus.com/inward/record.url?scp=85060402262&partnerID=8YFLogxK

U2 - 10.3390/e21010038

DO - 10.3390/e21010038

M3 - Article

VL - 21

JO - Entropy

T2 - Entropy

JF - Entropy

SN - 1099-4300

IS - 1

M1 - 38

ER -