First-passage Brownian functionals with stochastic resetting

We study the statistical properties of first-passage time functionals of a one dimensional Brownian motion in the presence of stochastic resetting. A first-passage functional is defined as $V=\int_0^{t_f} Z[x(\tau)]$ where $t_f$ is the first-passage time of a reset Brownian process $x(\tau)$, i.e., the first time the process crosses zero. In here, the particle is reset to $x_R>0$ at a constant rate $r$ starting from $x_0>0$ and we focus on the following functionals: (i) local time $T_{loc} = \int _0^{t_f}d \tau ~ \delta (x-x_R)$, (ii) residence time $T_{res} = \int _0^{t_f} d \tau ~\theta (x-x_R)$, and (iii) functionals of the form $A_n = \int _{0}^{t_f} d \tau [x(\tau)]^n$ with $n >-2$. For first two functionals, we analytically derive the exact expressions for the moments and distributions. Interestingly, the residence time moments reach minima at some optimal resetting rates. A similar phenomena is also observed for the moments of the functional $A_n$. Finally, we show that the distribution of $A_n$ for large $A_n$ decays exponentially as $\sim \text{exp}\left( -A_n/a_n\right)$ for all values of $n$ and the corresponding decay length $a_n$ is also estimated. In particular, exact distribution for the first passage time under resetting (which corresponds to the $n=0$ case) is derived and shown to be exponential at large time limit in accordance with the generic observation. This behavioural drift from the underlying process can be understood as a ramification due to the resetting mechanism which curtails the undesired long Brownian first passage trajectories and leads to an accelerated completion. We confirm our results to high precision by numerical simulations.