Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains

What the math gives to ML

The paper identifies a non-Euclidean geometry for kinetic diffusion: spatial variables x diffuse at degree 1, transported variables y have degree 3, and time has degree 2 under the intrinsic dilation. This is a transferable architectural prior for models of phase-space dynamics, where ordinary isotropic mixing combines variables at physically incompatible scales. The most direct implementation is a hypoelliptic transport-diffusion layer that performs local diffusion in x while shifting y along the characteristic direction x, with receptive fields obeying the 1:3:2 scaling. It is most promising for learned simulators, world models, and sequence models representing positions, velocities, or other coupled state variables.