Extreme value statistics of first-passage trajectories of resetting Brownian motion in an interval

In this work, we consider one-dimensional resetting Brownian motion confined in a unit interval with absorbing boundaries at both ends. The Brownian particle is reset to the starting position x 0 at random times but with a constant rate r , and the process terminates whenever the particle touches any of absorbing boundaries. We study the statistics of the span S , defined as the size of the visited domain during the first-passage process, and the time t m at which the maximum (minimum) displacement is achieved when the particle is absorbed by the left (right) end. We analytically obtain the distribution of S and its mean value. The distribution of S can be significantly altered by the resetting, and the mean value of S can increase, decrease, or remain insignificant as r increases, depending on the value of x 0 . Moreover, we find that the mean value ⟨ t m ⟩ has a highly nontrivial dependence on r . When x 0 is close to the boundary, ⟨ t m ⟩ can be reduced by the resetting, and a nonzero resetting rate r = r opt exists for which ⟨ t m ⟩ attains its minimum. Otherwise, ⟨ t m ⟩ has its minimum when the resetting is absent. We observe that the transition occurs at x 0 ≈ 0.180 67 (or x 0 ≈ 0.819 33 ). Such a transition is first-order, characterized by an abrupt jump of r opt from a nonzero value to zero at the transition point.