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

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