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 P r ( M | x 0 ) of the maximum displacement M. We find that stochastic resetting has a profound impact on P r ( M | x 0 ) 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 2 x 0 as r → ∞ . For v c < v < 0 , ⟨ M ⟩ shows a nonmonotonic dependence on r, in which a minimum ⟨ M ⟩ exists for an intermediate level of r. For v ⩽ v c , ⟨ 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 ⟨ t m ⟩ of tm on r is either monotonic or nonmonotonic, depending on the value of v. For v > v m , there is a nonzero resetting rate at which ⟨ t m ⟩ attains its minimum. Otherwise, ⟨ t m ⟩ increases monotonically with r. We provide an analytical determination of two critical values of v, v c ≈ − 1.694 15 D / x 0 and v m ≈ − 1.661 02 D / x 0 , where D is the diffusion constant. Finally, numerical simulations are performed to support our theoretical results.