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

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.