Symmetry principles and analogous Noether's theorem for stochastic systems

In the case of Newtonian mechanics, taking a variational or Lagrangian or Maximum principle (MP) view, one can obtain the conservation laws (energy, linear momentum and angular momentum) by combining the MP with the symmetry principles (time, linear displacement and angular displacement). This is a key contribution by Noether.

The Fokker-Planck equation or the Master equation (called Liouville in the case of deterministic dynamics) governs density evolution of an Ito stochastic process. The intuition behind obtaining this mathematical description is the principle of the conservation of mass combined with a minimum principle. This was shown by Nelson in the work on the derivation of the Schrodinger equation from Newtonian mechanics.

Is it possible to combine other types of symmetry principles (time,
linear displacement and angular displacement) to obtain stochastic
versions of the laws of conservation of stochastic energy, linear
and angular momenta?