Thermodynamic Trade-off Relation in Stochastic Resetting Process

We consider a paradigm of overdamped Brownian particle undergoing a stochastic resetting process in a thermal environment. In contrast to the usual set-up of stochastic resetting, here the

position of the particle is reset using a potential trap in a finite amount of time. Due to finite time reset phase, the first passage time to reach a target also increases. Introducing Brownian functionals, we calculate the first passage time to reach a final target and the work done during the process. We find that the behavior of the average first passage time is convex as function of resetting rate implying the existence of optimal resetting rate for which the average first passage time is minimum. Average first passage time depends inversely on the strength of the trapping potential in the reset phase. Average first passage time for the instantaneous stochastic resetting x_R is achieved when the x₀ strength of the trapping potential is very large. This indicates a trade-off between the first passage time and the work done for such process.