We present a comprehensive study of the symmetries of the generating functionals of generic Langevin processes with multiplicative colored noise. We treat both Martin–Siggia–Rose–Janssen–De Dominicis and supersymmetric formalisms. We summarize the relations between observables that they imply including fluctuation relations, fluctuation–dissipation theorems, and Schwinger–Dyson equations. Newtonian dynamics and their invariances follow in the vanishing friction limit.
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