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 $s(g,t)={\nabla }_{SE(3)}\log \rho (g,t)$ as $\rho \to 0^{+}$ 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 $L_{7D}=-{\Delta }_{SE(3)}-{\partial }_t^2$ 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 $\Vert \partial M{\Vert }_{C^2}$, and Ricci curvature $\Vert Ri{c}_{SE(3)}{\Vert }_{L^{\infty }}$, 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 $Ri{c}_{SE(3)}=\mathrm{diag}(0,0,0,1/2,1/2,1/2)$ from $SO(3)$ induces a sharp Bochner--Weitzenböck spectral gap boost ${\lambda }_1(-{\Delta }_{SE(3)})\ge {\lambda }_1(-{\Delta }_{{\mathbb{R}}^3})+1/2>0$. 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 $T_{SE(3)}^{\ast }$ beyond flat Euclidean limits.

arXiv comment20 pages