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Area of Science:

  • Physics
  • Mathematics
  • Complex Systems

Background:

  • Stochastic processes are fundamental in modeling natural phenomena.
  • Understanding the impact of intermittent large changes is crucial for complex system dynamics.

Purpose of the Study:

  • To analyze the effects of power-law distributed resets on continuous stochastic processes.
  • To derive analytical expressions for static and dynamic quantities under these conditions.
  • To investigate the implications for first-passage time problems.

Main Methods:

  • Modeling stochastic processes using diffusion.
  • Incorporating large, abrupt resets to the initial condition.
  • Deriving exact closed-form expressions for system properties.
  • Analyzing correlations arising from power-law distributions.

Main Results:

  • Exact solutions obtained for static and dynamic quantities.
  • Observed a spectrum of long-time behaviors: ever-spreading (α<1) or time-independent (α>1) spatial distributions.
  • Demonstrated that the time to reach a target is significantly affected by reset dynamics.

Conclusions:

  • The interplay of diffusion and power-law resets leads to rich, tunable dynamics.
  • An optimal reset parameter (α) minimizes the mean first-passage time to a target.
  • Findings offer strategies for efficient target localization in complex, dynamic environments.