We consider a set of laws of semi-martingales whose characteristics satisfy a weak canonical Euler-Lagrange requirement for a suitable class of Lagrangian maps with gauge invariances. Those laws may represent solutions to specific stochastic systems, which, under conditions, may involve correlated Brownian motions. Within this approach, those probabilities are naturally characterised by a canonical least action principle, and a weak Noether's theorem naturally applies under conditions. In particular, our approach applies to specific dissipated systems, to specific classical forced parametric oscillators, and to specific Einstein-Langevin type diffusions. Within this framework, such probabilities might be built through a variational approach. In particular we investigate pumped, and eventually damped, Schrodinger Bridges.