The double barrier problem for Brownian motion with Poissonian resetting

Abstract

Let a < 0 < b be two fixed points. We consider a diffusive particle in one space dimension whose dynamics combines continuous-time Brownian motion with resetting at random Poisson times. We study the double barrier problem regarding the probability that starting from 0 the Brownian particle escapes (a, b) at the upper barrier b and compare how resetting modifies the exit probabilities. We also study the distribution of the corresponding exit time. We show that the resetting activity may either increase or decrease the meantime to exit a region. A precise condition involving the golden ratio separates both cases. Optimal resetting rates that minimize the mean escape time are considered.