Quantum mechanical approach to stochastic resetting

Édgar Roldán, Shamik Gupta

Published 2017-03-30Version 1

We study the dynamics of overdamped Brownian particles diffusing in force fields and undergoing stochastic resetting to a given location with a generic {\em space-dependent} rate of resetting. We introduce a novel quantum mechanical approach that allows to calculate in a systematic way analytical expressions for a variety of statistics of the dynamics, such as (i) the propagator prior to first reset; (ii) the distribution of the first-reset time, and, most interestingly, (iii) the spatial distribution of the particle at long times. A key to our accomplishment is the derivation of an equality relating the transition probability prior to first reset with the propagator of a suitable quantum mechanical problem. We apply our method to obtain {\em exact} results for a number of representative and hitherto unexplored examples of resetting dynamics, including a case of energy-dependent resetting, thereby unveiling the nontrivial effects of resetting.