  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.
