Extremal statistics for first-passage trajectories of drifted Brownian motion under stochastic resetting

Abstract

We study the extreme value statistics of first-passage trajectories generated from a one-dimensional drifted Brownian motion subject to stochastic resetting to the starting point with a constant rate r. Each stochastic trajectory starts from a positive position x 0 and terminates whenever the particle hits the origin for the first time. We obtain an exact expression for the marginal distribution Pr(M|x0) of the maximum displacement M. We find that stochastic resetting has a profound impact on Pr(M|x0) and the expected value ⟨M⟩ of M. Depending on the drift velocity v, ⟨M⟩ shows three distinct trends of change with r. For v⩾0 , ⟨M⟩ decreases monotonically with r, and tends to 2x0 as r→∞ . For vc<v<0 , ⟨M⟩ shows a nonmonotonic dependence on r, in which a minimum ⟨M⟩ exists for an intermediate level of r. For v⩽vc , ⟨M⟩ increases monotonically with r. Moreover, by deriving the propagator and using a path decomposition technique, we obtain, in the Laplace domain, the joint distribution of M and the time tm at which the maximum M is reached. Interestingly, the dependence of the expected value ⟨tm⟩ of tm on r is either monotonic or nonmonotonic, depending on the value of v. For v>vm , there is a nonzero resetting rate at which ⟨tm⟩ attains its minimum. Otherwise, ⟨tm⟩ increases monotonically with r. We provide an analytical determination of two critical values of v, vc≈−1.69415D/x0 and vm≈−1.66102D/x0 , where D is the diffusion constant. Finally, numerical simulations are performed to support our theoretical results.