Reconstruction of surface and stochastic dynamics from a planar projection of trajectories

We show how to reconstruct a two-dimensional surface, the drift field, and the diffusion tensor from a planar projection of trajectories of a diffusion process on the surface. The reconstruction is based on the stochastic differential equations of the projected motion, whose drift and diffusion tensor depend on the local curvature of the surface. The reconstruction process requires the solution of new partial differential equations that we derive. Our analysis can be used to distinguish between the effective drift due to local curvature and that of the stochastic differential equation on the surface. We provide numerical examples of reconstruction from statistical estimates of the drift field and diffusion tensor of the projections.