Instantons in Monte Carlo simulation of geometric Brownian motion in a confining quartic potential

This study examines the behavior of the geometric Brownian motion (GBM) in a static confining quartic potential. The potential adopts the form of a double-well, indicating two distinct stable states separated by a barrier of finite height. We used Markov chain Monte Carlo (MCMC) simulation to sample paths of the time series and investigate the resulting statistics of paths. This sampling follows a path-integral-like representation of the probability weights that permits us to constrain the paths to pass through desired values at particular times if needed.