Geometric Brownian Motion under Stochastic Resetting: A Stationary yet Non-ergodic Process.

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 such that the dynamics is renewed intermittently. Quite surprisingly, although resetting renders GBM stationary, the resulting process remains non-ergodic. We observe three different long-time regimes: a quenched state, an unstable and a stable annealed state depending on the resetting strength. Crucially, the 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.