Brownian motion

Invariant measure

Physics

Mathematics

Multiplicative function

Knowledge management

Stochastic process

Markov chain

Geometric Brownian motion

Statistical physics

Ergodic theory

Statistics

Mathematical analysis

Stationary ergodic process

Computer science

Stationary state

Quantum mechanics

Stationary distribution

Innovation diffusion

Diffusion process

Physical review. E

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.

Geometric Brownian motion under stochastic resetting: A stationary yet nonergodic process