  We study the effects of stochastic resetting on geometric Brownian motion
(GBM), a canonical stochastic multiplicative process for non-stationary and
non-ergodic dynamics. Resetting is a sudden interruption of a process, which
consecutively renews its dynamics. We show that, although resetting renders GBM
stationary, the resulting process remains non-ergodic. Quite surprisingly, the
effect of resetting is pivotal in manifesting the non-ergodic behavior. In
particular, we observe three different long-time regimes: a quenched state, an
unstable and a stable annealed state depending on the resetting strength.
Notably, in the last regime, the system is self-averaging and thus the sample
average will always mimic ergodic behavior establishing a stand alone feature
for GBM under resetting. Crucially, the above-mentioned regimes are well
separated by a self-averaging time period which can be minimized by an optimal
resetting rate. Our results can be useful to interpret data emanating from
stock market collapse or reconstitution of investment portfolios.
