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 $M \subset SE(3)$ on the Special Euclidean group $SE(3) \cong \mathbb{R}^3 \times SO(3)$. 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 $s(g,t) = \nabla_{SE(3)} \log ρ(g,t)$ as $ρ\rightarrow 0^+$ 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 $L_{7D} = -Δ_{SE(3)} - \partial_t^2$ with zero-order corner compatibility on $\partial M \times \{0,T\}$, we establish Agmon--Douglis--Nirenberg $W_p^2(Q_T)$ shift estimates tracking Lie algebra constants, boundary geometry $\|\partial M\|_{C^2}$, and Ricci curvature $\|Ric_{SE(3)}\|_{L^\infty}$, yielding a full $C^\infty(\overline{Q}_T)$ 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 $Ric_{SE(3)} = \mathrm{diag}(0,0,0,1/2,1/2,1/2)$ from $SO(3)$ induces a sharp Bochner--Weitzenböck spectral gap boost $λ_1(-Δ_{SE(3)}) \ge λ_1(-Δ_{\mathbb{R}^3}) + 1/2 > 0$. 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.