From Scattered Phase-Space Observations to an Empirical Fokker-Planck Equation: A Delaunay-Voronoi finite-volume construction

A dynamical phase plane trajectory contains more than a sequence of state data. It records where the system has been, which way it moved, and how much each step has varied. We use these three observations to build a Delaunay-Voronoi Fokker-Planck (DVFP) description of a dynamical system from sparse or irregular phase-space data. The observed states define Voronoi control volumes, successive observations give the local mean motion, or drift, and the scatter around that motion gives diffusion. Because probability changes inside a cell only by crossing one of its faces, these ingredients lead directly to a finite-volume Fokker-Planck balance. We can then properly apply the Sparse Identification of Nonlinear Dynamics (SINDy) framework afterward, to derive the equations of motion. Because the reconstruction supplies density, drift, diffusion, and the ingredients needed to estimate probability current from the observations, it also provides the ingredients needed for stochastic-thermodynamic analysis when such an interpretation is appropriate. The time-ordered Delaunay-Voronoi construction was introduced in our earlier market application. Here we separate it from that setting, state the probability balance in general form, and test the reconstruction on a stochastic damped oscillator whose dynamics are known in advance.