On a Class of Non-Local Elliptic-Transport Systems in Score-Guided Probability Flow on SE(3): Score-Singularity Resolution, Space-Time ADN Shift, and Markov Mean-Field Games
Abstract
We establish mathematical foundations for optimal spatiotemporal vector fields governing continuous score-guided probability flows over bounded submanifolds M ⊂ SE(3) on the Special Euclidean group SE(3) ≅ ℝ3 × SO(3). Modeling dynamic density evolution via a score-perturbed Lie-algebraic continuity equation under zero-flux boundary conditions, we resolve the non-local score drift singularity s(g,t) = ∇SE(3) log ρ(g,t) as ρ→ 0+ via weighted parabolic flux dissipation bounds. Using left-invariant exponential group mollifiers, we derive boundary-preserving DiPerna--Lions commutator estimates tracking structure constants {cijk}. Second, interpreting the optimality system as a 7D space-time elliptic operator L7D = −ΔSE(3) − ∂t2 with zero-order corner compatibility on ∂ M × {0,T}, we establish Agmon--Douglis--Nirenberg Wp2(QT) shift estimates tracking Lie algebra constants, boundary geometry |∂ M|C2, and Ricci curvature |RicSE(3)|L∞, yielding a full C∞(Q̅T) Schauder bootstrap expansion. Third, we extend this system to non-cooperative Markov Mean-Field Games (MFGs) on SE(3), establishing score-guided Nash equilibria via Schauder--Tychonoff fixed points. Crucially, the constant diagonal Ricci curvature RicSE(3) = diag(0,0,0,1/2,1/2,1/2) from SO(3) induces a sharp Bochner--Weitzenböck spectral gap boost λ1(−ΔSE(3)) ≥ λ1(−Δℝ3) + 1/2 > 0. This positive rotational curvature enhances the Lie-algebraic Poincaré coercivity constant, demonstrating that intrinsic group curvature stabilizes score-guided flows against dispersion and expands the critical Nash uniqueness horizon TSE(3)* beyond flat Euclidean limits.
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BibTeX
@misc{lee2026class,
title = {{On a Class of Non-Local Elliptic-Transport Systems in Score-Guided Probability Flow on $SE(3)$: Score-Singularity Resolution, Space-Time ADN Shift, and Markov Mean-Field Games}},
author = {Pilhwa Lee},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03278},
eprint = {2611.03278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03278},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a Class of Non-Local Elliptic-Transport Systems in Score-Guided Probability Flow on SE(3): Score-Singularity Resolution, Space-Time ADN Shift, and Markov Mean-Field Games
AU - Pilhwa Lee
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03278
UR - https://arxiv.org/abs/2611.03278
AB - We establish mathematical foundations for optimal spatiotemporal vector fields governing continuous score-guided probability flows over bounded submanifolds M ⊂ SE(3) on the Special Euclidean group SE(3) ≅ ℝ^3 × SO(3). Modeling dynamic density evolution via a score-perturbed Lie-algebraic continuity equation under zero-flux boundary conditions, we resolve the non-local score drift singularity s(g,t) = ∇_SE(3) log ρ(g,t) as ρ→ 0^+ via weighted parabolic flux dissipation bounds. Using left-invariant exponential group mollifiers, we derive boundary-preserving DiPerna--Lions commutator estimates tracking structure constants {c_ij^k}. Second, interpreting the optimality system as a 7D space-time elliptic operator L_7D = −Δ_SE(3) − ∂_t^2 with zero-order corner compatibility on ∂ M × {0,T}, we establish Agmon--Douglis--Nirenberg W_p^2(Q_T) shift estimates tracking Lie algebra constants, boundary geometry |∂ M|_C^2, and Ricci curvature |Ric_SE(3)|_L^∞, yielding a full C^∞(Q̅_T) Schauder bootstrap expansion. Third, we extend this system to non-cooperative Markov Mean-Field Games (MFGs) on SE(3), establishing score-guided Nash equilibria via Schauder--Tychonoff fixed points. Crucially, the constant diagonal Ricci curvature Ric_SE(3) = diag(0,0,0,1/2,1/2,1/2) from SO(3) induces a sharp Bochner--Weitzenböck spectral gap boost λ_1(−Δ_SE(3)) ≥ λ_1(−Δ_ℝ^3) + 1/2 > 0. This positive rotational curvature enhances the Lie-algebraic Poincaré coercivity constant, demonstrating that intrinsic group curvature stabilizes score-guided flows against dispersion and expands the critical Nash uniqueness horizon T_SE(3)^* beyond flat Euclidean limits.
ER -