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Semi-Markovian switching in a fluctuating harmonic trap: An age-structured formulation

Derek Frydel

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Phys. Rev. E 114, 024105 – Published 3 August, 2026

DOI: https://doi.org/10.1103/7pxv-tg3b

Abstract

We study a Brownian particle in a harmonic trap whose stiffness switches between two values with arbitrary waiting-time statistics, leading to semi-Markovian dynamics. To restore Markovianity, we formulate the problem in an enlarged age-structured state space. Using this framework, we obtain steady-state integral equations for the stationary distributions and exact expressions for stationary moments, injected power, and potential energy. In the second part of the paper, we focus on the stochastic-resetting limit, corresponding to a particle alternately released and trapped. By representing the stationary spatial distribution as a superposition of Gaussian states for different effective stiffness (or variance), the problem can be reformulated in variance space, leading to a simplified description amenable to direct analytical treatment.

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Phys. Rev. E 114, 024105– Published 3 August, 2026
  • Received 26 January 2026
  • Accepted 13 July 2026
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