Active particle in a harmonic trap driven by a resetting noise: an approach via Kesten variables

We consider the statics and dynamics of a single particle trapped in a one-dimensional harmonic potential, and subjected to a driving noise with memory, that is represented by a resetting stochastic process. The finite memory of this driving noise makes the dynamics of this particle ``active''. At some chosen times (deterministic or random), the noise is reset to an arbitrary position and restarts its motion. We focus on two resetting protocols: periodic resetting, where the period is deterministic, and Poissonian resetting, where times between resets are exponentially distributed with a rate $r$. Between the different resetting epochs, we can express recursively the position of the particle. The random relation obtained takes a simple Kesten form that can be used to derive an integral equation for the stationary distribution of the position. We provide a detailed analysis of the distribution when the noise is a resetting Brownian motion. In this particular instance, we also derive a renewal equation for the full time dependent distribution of the position that we extensively study. These methods are quite general and can be used to study any process harmonically trapped when the noise is reset at random times.