First-passage Brownian functionals with stochastic resetting

Abstract We study the statistical properties of first-passage time functionals a one dimensional Brownian motion in presence stochastic resetting. A functional is defined as V = ∫ 0 t f Z [ x stretchy="false">( τ stretchy="false">) stretchy="false">] where t f reset process x ( τ ), i.e., first crosses zero. In here, particle to R > 0 at constant rate r starting from and we focus on following functionals: (i) local ${T}_{\mathrm{l}\mathrm{o}\mathrm{c}}={\int }_{0}^{{t}_{\mathrm{f}}}\,\mathrm{d}\tau \delta (x-{x}_{R})$?> T mathvariant="normal">l mathvariant="normal">o mathvariant="normal">c mathvariant="normal">d δ − R , (ii) residence ${T}_{\mathrm{r}\mathrm{e}\mathrm{s}}={\int \theta mathvariant="normal">r mathvariant="normal">e mathvariant="normal">s θ (iii) form ${A}_{n}={\int {[x(\tau )]}^{n}$?> A n with n −2. For two functionals, analytically derive exact expressions for moments distributions. Interestingly, reach minima some optimal resetting rates. similar phenomena also observed . Finally, show that distribution large decays exponentially $\sim \!\!\mathrm{exp}\left(-{A}_{n}/{a}_{n}\right)$?> ∼ width="-0.20em" exp / a all values corresponding decay length estimated. particular, passage under (which corresponds = case) derived shown be exponential limit accordance generic observation. This behavioural drift underlying can understood ramification due mechanism which curtails undesired long trajectories leads an accelerated completion. confirm our results high precision by numerical simulations.