Abstract Turbulent energy transfer in nearly collisionless plasmas can be conceptualized as a scale-to-scale Langevin process. Hence, the statistics of magnetic field fluctuations can be embedded in the framework of stochastic process theory. In this work, we investigate the statistical properties of the pristine solar wind as observed by Parker Solar Probe by defining the cascade trajectories of magnetic field increments and by estimating the stochastic entropy variation along them. Through the stochastic entropy, we can identify two regimes where fluctuations exhibit contrasting statistical properties. In the inertial range, the entropy production is associated with an increase of the flatness indicating the occurrence of intermittency. Otherwise, trajectories associated with an entropy consumption exhibit global scale invariance. In the transition region toward ion scales, the phenomenology switches: entropy-consuming trajectories exhibit a sudden flatness increase, associated with the presence of small-scale intermittency, while entropy-producing trajectories display a nearly constant flatness. Results are interpreted in terms of physical processes consistent with an accumulation of energy at ion scales.