Assessing the suitability of the Langevin equation for analyzing measured data through downsampling

Abstract The measured time series from complex systems are renowned for their complex stochastic behavior, characterized by random fluctuations stemming from external influences and nonlinear interactions. These fluctuations take diverse forms, ranging from continuous trajectories reminiscent of Brownian motion to noncontinuous trajectories featuring jump events. The Langevin equation is a versatile framework for modeling stochastic systems, effectively describing the complex behavior of measured data that exhibit continuous stochastic variability and adhere to Markov properties. However, the traditional modeling framework of the Langevin equation falls short when it comes to capturing the presence of abrupt changes, particularly jumps, in trajectories that exhibit non-continuity. Such non-continuous changes pose a significant challenge for general processes and have profound implications for risk management. Moreover, the discrete nature of observed physical phenomena, measured with a finite sample rate, adds another layer of complexity. In such cases, data points often appear as a series of discontinuous jumps, even when the underlying trajectory is continuous. In this study, we present an analytical framework that goes beyond the limitations of the Langevin equation. Our approach effectively distinguishes between diffusive or Brownian-type trajectories and non-diffusive trajectories such as those with jumps. By introducing downsampling techniques, where we artificially lower the sample rate, we derive a set of measures and criteria to analyze the data and differentiate between diffusive and non-diffusive behaviors. To further demonstrate its versatility and practical applicability, we have applied our proposed method to real-world data in various scientific fields, such as trapped particles in optical tweezers, market price, neuroscience, turbulence and renewable energy. For real-world data that lack Markov properties, we estimate the functions and parameters using the generalized Langevin equation, which incorporates a memory kernel to account for non-Markovian dynamics.