Supersymmetries in non-equilibrium Langevin dynamics

Stochastic phenomena are often described by Langevin equations, which serve as a mesoscopic model for microscopic dynamics. It is known since the work of Parisi and Sourlas that reversible (or equilibrium) dynamics present supersymmetries (SUSYs). These are revealed when the path-integral action is written as a function not only of the physical fields, but also of Grassmann fields representing a Jacobian arising from the noise distribution. SUSYs leave the action invariant upon a transformation of the fields that mixes the physical and the Grassmann ones. We show that, contrarily to the common belief, it is possible to extend the known reversible construction to the case of arbitrary irreversible dynamics, for overdamped Langevin equations with additive white noise - provided their steady state is known. The construction is based on the fact that the Grassmann representation of the functional determinant is not unique, and can be chosen so as to present a generalization of the Parisi-Sourlas SUSY. Our approach is valid both for Martin-Siggia-Rose-Janssen-de Dominicis and for Onsager-Machlup actions. We show how such SUSYs are related to time-reversal symmetries and allow one to derive modified fluctuation-dissipation relations valid in non-equilibrium. We give as a concrete example the results for the Kardar-Parisi-Zhang equation.