# A note on symmetries in the path integral formulation of the Langevin dynamics

## TL;DR

This paper explores the supersymmetric structure of dissipative Langevin dynamics within the path integral framework, revealing universal identities from superderivative transformations that extend understanding of symmetries in stochastic systems.

## Contribution

It identifies the supersymmetric effective action and analyzes the role of superderivatives, showing they produce universal identities despite not generating Ward identities.

## Key findings

- Supercharges are identified in the supersymmetric formulation.
- Superderivative transformations lead to universal identities.
- Universal identities are confirmed in the Ornstein-Uhlenbeck process.

## Abstract

We study a 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 identities, which we derive from Schwinger-Dyson equations. We confirm that the above identities hold in an explicit example of Ornstein-Uhlenbeck process.