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Unified approach to reset processes and application to coupling between process and reset.

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This study introduces a unified framework for analyzing stochastic processes with resets, offering general solutions for observables. It reveals how reset mechanisms impact system dynamics, particularly in subdiffusion-decay processes, independent of specific microscopic details.

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Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Mathematical Modeling

Background:

  • Stochastic processes with resets are crucial in diverse fields, yet current analyses often lack generality.
  • Existing derivations focus on specific systems, hindering broad applicability of reset principles.

Purpose of the Study:

  • To develop a unified theoretical framework for analyzing observables in stochastic processes under reset.
  • To derive general solutions and conditions for stationary values with and without reset.
  • To investigate the impact of reset mechanisms on subdiffusion-decay processes with microscopic dependencies.

Main Methods:

  • Derivation of general solutions for observables dependent on the most recent renewal period.
  • Analysis of intermittent reset times and asymptotic behavior of observables.
  • Focus on subdiffusion-decay processes with time-dependent restart probabilities.

Main Results:

  • General solutions for observables under reset are derived.
  • Conditions for the existence and equality of stationary values are determined.
  • For intermittent resets, exact asymptotic power-law expressions for observables are obtained.
  • Memory effects due to ageing in subdiffusion-decay processes significantly alter decay rates and currents.

Conclusions:

  • The unified framework provides broad applicability to stochastic processes with resets.
  • Reset mechanisms can drastically alter system dynamics, especially in coupled processes, due to memory effects.
  • The impact of reset on occupation densities, spatial moments, and other quantities is general and independent of microscopic details.