Gauge symmetry in Fokker–Planck dynamics

Using a Galilean metric approach, based on an embedding of the Euclidean space into a (4+1)-Minkowski space, we analyze a gauge invariant Lagrangian associated with a Riemannian manifold , with metric g. With a specific choice of the gauge condition, the Euler–Lagrange equations are written covariantly in , and then the Fokker–Planck equation is derived, such that the drift and the diffusion terms are obtained from g. The analysis is carried out for both, Abelian and non-Abelian symmetries, and an example with the su(2) symmetry is presented.