Langevin-type dynamics and hypocoercivity on SE(3)
Abstract
We construct hypocoercive Kolmogorov-type operators on the Euclidean motion group SE(3)=ℝ3rtimesSO(3) using ZXZ Euler angles and study their long-time behaviour using the abstract hypocoercivity method. The operator L=S−A decomposes into a symmetric part S, corresponding to a degenerate diffusion on the rotation component SO(3), and an antisymmetric part A, which captures transport and torque effects induced by the Lie group structure. The associated invariant probability measure is μΦ=γΦ⊗ν3, where γΦ is a probability measure on ℝ3 with density Z(Φ)−1e−Φ with respect to the Lebesgue measure, and ν3 is the normalized Haar measure on SO(3). As a consequence, if Φ∈ C∞(ℝ3) is bounded from below, γΦ satisfies a Poincaré inequality with constant Λ>0, and there exists c<∞ such that |∇ξ2Φ(ξ)| ≤ c(1+|∇ξΦ(ξ)|), ξ∈ℝ3, then the semigroup generated by the closure of L converges exponentially fast to equilibrium in L2(SE(3),μΦ).
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BibTeX
@misc{grothaus2026langevintype,
title = {{Langevin-type dynamics and hypocoercivity on $\mathrmSE(3)$}},
author = {Martin Grothaus and Andrea V. Hurtado-Quiceno},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03005},
eprint = {2610.03005},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03005},
}
RIS
TY - UNPB
TI - Langevin-type dynamics and hypocoercivity on SE(3)
AU - Martin Grothaus
AU - Andrea V. Hurtado-Quiceno
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03005
UR - https://arxiv.org/abs/2610.03005
AB - We construct hypocoercive Kolmogorov-type operators on the Euclidean motion group SE(3)=ℝ^3rtimesSO(3) using ZXZ Euler angles and study their long-time behaviour using the abstract hypocoercivity method. The operator L=S−A decomposes into a symmetric part S, corresponding to a degenerate diffusion on the rotation component SO(3), and an antisymmetric part A, which captures transport and torque effects induced by the Lie group structure. The associated invariant probability measure is μ_Φ=γ_Φ⊗ν_3, where γ_Φ is a probability measure on ℝ^3 with density Z(Φ)^−1e^−Φ with respect to the Lebesgue measure, and ν_3 is the normalized Haar measure on SO(3). As a consequence, if Φ∈ C^∞(ℝ^3) is bounded from below, γ_Φ satisfies a Poincaré inequality with constant Λ>0, and there exists c<∞ such that |∇_ξ^2Φ(ξ)| ≤ c(1+|∇_ξΦ(ξ)|), ξ∈ℝ^3, then the semigroup generated by the closure of L converges exponentially fast to equilibrium in L^2(SE(3),μ_Φ).
ER -