Photon migration in highly forward scattering random media can be described as a non-Euclidean diffusion (NED) on the velocity sphere. An exact path-integral solution of the corresponding NED equation in the photon five-dimensional phase space has been obtained. The solution leads to a ``generalized Fermat principle'' (GFP) for the most probable photon paths in turbid media: GFP requires the least mean-square curvature of the path. An explicitly analytic description of an ultrashort laser pulse propagation in random media based on NED equation is presented. Experiments have been performed to verify the NED theory. \textcopyright{} 1996 The American Physical Society.