# Functionals in stochastic thermodynamics: how to interpret stochastic integrals

## TL;DR

This paper clarifies how to interpret stochastic integrals in stochastic thermodynamics, confirming the standard use of Stratonovich integrals for heat and work, and discusses anomalies in entropy production under temperature gradients.

## Contribution

It provides a systematic mathematical analysis validating the standard interpretation of stochastic integrals in thermodynamics and discusses known anomalies in entropy production.

## Key findings

- Stratonovich integrals are confirmed as the correct interpretation for heat and work functionals.
- The paper elucidates the dependence of stochastic integral values on discretization rules.
- It discusses known anomalies in entropy production when temperature gradients are present.

## Abstract

In stochastic thermodynamics standard concepts from macroscopic thermodynamics, such as heat, work, and entropy production, are generalized to small fluctuating systems by defining them on a trajectory-wise level. In Langevin systems with continuous state-space such definitions involve stochastic integrals along system trajectories, whose specific values depend on the discretization rule used to evaluate them (i.e. the "interpretation" of the noise terms in the integral). Via a systematic mathematical investigation of this apparent dilemma, we corroborate the widely used standard interpretation of heat- and work-like functionals as Stratonovich integrals. We furthermore recapitulate the anomalies that are known to occur for entropy production in the presence of temperature gradients.