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Basic continuous-time signals include the unit step function, unit impulse function, and unit ramp function, collectively referred to as singularity functions. Singularity functions are characterized by discontinuities or discontinuous derivatives.
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In signal processing, a continuous-time signal can be sampled using an impulse-train sampling technique, followed by the zero-order hold method. Impulse-train sampling involves the use of a periodic impulse train, which consists of a series of delta functions spaced at regular intervals determined by the sampling period. When a continuous-time signal is multiplied by this impulse train, it generates impulses with amplitudes corresponding to the signal's values at the sampling points.
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Using a Real-Time Locating System to Measure Walking Activity Associated with Wandering Behaviors Among Institutionalized Older Adults
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Continuous-time random walk under time-dependent resetting.

V P Shkilev1

  • 1Chuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine, 17 General Naumov Street, 03164 Kyiv, Ukraine.

Physical Review. E
|January 20, 2018
PubMed
Summary

This study examines continuous-time random walks with random resets. Anomalous subdiffusion significantly influences the stationary state distribution of waiting times.

Area of Science:

  • Physics
  • Stochastic Processes
  • Statistical Mechanics

Background:

  • Continuous-time random walks (CTRWs) are fundamental models for anomalous diffusion.
  • Particle reset mechanisms introduce unique dynamics to stochastic processes.
  • Understanding waiting time distributions is crucial for characterizing CTRW behavior.

Purpose of the Study:

  • To analyze CTRWs with random resets to an initial position.
  • To investigate the impact of anomalous subdiffusion on the system's stationary state.
  • To derive the governing equation and calculate the mean first-passage time.

Main Methods:

  • Modeling the waiting time distribution as a sum of exponentials.
  • Establishing the governing equation for the stochastic process.

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  • Calculating the mean first-passage time to a specific position.

    • Main Results:

    • The waiting time distribution is represented by a sum of arbitrary exponentials.
    • The governing equation for this reset process is derived.
    • Anomalous subdiffusion is shown to significantly alter the stationary state's shape.

    Conclusions:

    • The interplay between random resets and anomalous subdiffusion leads to distinct stationary state characteristics.
    • The derived framework allows for quantitative analysis of first-passage times in reset CTRWs.
    • This research provides insights into complex particle dynamics in systems with resetting behavior.