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Jump processes with deterministic and stochastic controls.

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This study introduces a new theory for systems with controlled random jumps, offering an analytical steady-state solution. The findings aid in assessing environmental risks, such as crop failure from stochastic irrigation.

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Area of Science:

  • Physics
  • Environmental Science
  • Applied Mathematics

Background:

  • Many systems are influenced by both continuous drift and random jump processes.
  • Controlling these jumps is crucial for system stability and prediction.
  • Existing models often lack a unified framework for analyzing controlled jump dynamics.

Purpose of the Study:

  • To develop a general theory for one-dimensional systems with deterministic drift and two types of jump processes (uncontrolled and controlled).
  • To provide an analytical solution for the steady-state behavior of such systems.
  • To demonstrate the theory's application in environmental geophysics, specifically for crop-failure risk assessment.

Main Methods:

  • Formulation of a master equation using antecedent and posterior jump states.
  • Development of a general theoretical framework for analyzing system dynamics.
  • Application of the theory to model stochastic irrigation scenarios.

Main Results:

  • An analytical solution for the steady-state distribution of the system was obtained.
  • The theory provides a robust method for analyzing systems with controlled random resets.
  • The model successfully illustrates the assessment of crop-failure risk under stochastic irrigation.

Conclusions:

  • The developed theory offers a powerful tool for understanding and predicting the behavior of complex dynamical systems with controlled random events.
  • This framework has significant implications for environmental geophysics and risk assessment.
  • The study highlights the importance of incorporating controlled stochastic processes in system modeling.