  The measured time series from complex systems are renowned for their
intricate 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 serves as a powerful tool for generating stochasticity and capturing
the complex behavior of measured data with continuous stochastic
characteristics. 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 trajectories
with jumps. By employing 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, turbulence,
optical tweezers for trapped particles, neuroscience, renewable energy, and
market price analysis.
