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The measured time series from complex systems are renowned for theirintricate stochastic behavior, characterized by random fluctuations stemmingfrom external influences and nonlinear interactions. These fluctuations takediverse forms, ranging from continuous trajectories reminiscent of Brownianmotion to noncontinuous trajectories featuring jump events. The Langevinequation serves as a powerful tool for generating stochasticity and capturingthe complex behavior of measured data with continuous stochasticcharacteristics. However, the traditional modeling framework of the Langevinequation falls short when it comes to capturing the presence of abrupt changes,particularly jumps, in trajectories that exhibit non-continuity. Suchnon-continuous changes pose a significant challenge for general processes andhave profound implications for risk management. Moreover, the discrete natureof observed physical phenomena, measured with a finite sample rate, addsanother layer of complexity. In such cases, data points often appear as aseries of discontinuous jumps, even when the underlying trajectory iscontinuous. In this study, we present an analytical framework that goes beyondthe limitations of the Langevin equation. Our approach effectivelydistinguishes between diffusive or Brownian-type trajectories and trajectorieswith jumps. By employing downsampling techniques, where we artificially lowerthe sample rate, we derive a set of measures and criteria to analyze the dataand differentiate between diffusive and non-diffusive behaviors. To furtherdemonstrate its versatility and practical applicability, we have applied ourproposed method to real-world data in various scientific fields, turbulence,optical tweezers for trapped particles, neuroscience, renewable energy, andmarket price analysis.

Assessing the Suitability of the Langevin Equation for Analyzing Measured Data Through Downsampling