Optimization in First-Passage Resetting

We investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold. In an infinite domain, this process is non-stationary and its probability distribution exhibits rich features. In a finite domain, we define a non-trivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost.
Comment: 4 pages, 3 figures, revtex 4-1 format. Version 1 contains changes in response to referee comments. Version 2: A missing factor of 2 in an inline formula has been corrected