Symmetries in the path integral formulation of the Langevin dynamics

We study dissipative Langevin dynamics in the path integral formulation using the Martin-Siggia-Rose formalism. The effective action is supersymmetric and we identify the supercharges. In addition we study the transformations generated by superderivatives, which were recently included in the cohomological structure emerging in the dissipative systems. We find that these transformations do not generate Ward identities, which are explicitly broken; however, they lead to universal sum-rule-type identities, which we derive from Schwinger-Dyson equations. We confirm that the above identities hold in an explicit example of the Ornstein-Uhlenbeck process.