Exploring space plasma fluctuations at kinetic scales through stochastic process theory

During the last decades, space missions provided in situ data of diverse space plasma environments with increasingly higher resolution. This enabled the possibility to investigate peculiar properties of fluctuations in the magnetic field and plasma parameters, transitioning from the magnetohydrodynamic (MHD) to the ion-kinetic regime. The ion-kinetic regime is characterized by a global self-similar scaling of fluctuations, in contrast to the local scale-invariance of the MHD ones. In a series of works, we developed a data-driven approach based on the Langevin equation in order to model statistical features of kinetic fluctuations. In practical terms, the stochastic process thus introduced represents the evolution of the magnetic field fluctuations as a function of the scale. As far as such fluctuations are of the Langevin type, their statistics evolve according to a Fokker-Planck equation. Studying the evolution of fluctuation statistics across the scales, e.g., structure functions, allows us to make predictions about global statistical properties, e.g., scaling exponents. In this work, we review recent results obtained by using data from the ESA/Cluster mission in near-Earth space. We give evidence that the dynamics of magnetic field increments at kinetic scales can be modeled as a stochastic process of the Langevin type, and that the correct scaling law of the structure functions can be obtained through the stochastic equation in the non-diffusive limit, by linking the drift term of the Langevin equation to the Hölder exponent. Finally, this model allows us to derive the asymptotic limit of individual sets of fluctuations, giving thus predictions on the trend expected at kinetic scales.