Is it possible to produce good trajectories of advected fluid particles and then make accurate statistical analysis of the Lagrangian dynamics without basing on the Eulerian field of a given diagnostic function?
Yes this is the basic premise of Particle Image Velocimetry, but for similar statistical significance as you get from Eulerian measures you need as many particles, and probably much greater because the particle density may not follow variations in your quantity of interest, as you would Eulerian measurements. This need for oversampling is true for numerical as well as practical experiments