First-passage functionals for Ornstein Uhlenbeck process with stochastic resetting

TL;DR

This paper analyzes the effects of stochastic resetting on the first-passage functionals of an Ornstein-Uhlenbeck process, revealing how resetting influences the mean times and phase transitions depending on system parameters.

Contribution

It provides analytical expressions for the first moments of first-passage functionals in an OU process with resetting, highlighting phase transitions and optimal resetting rates.

Findings

Resetting can either increase or decrease mean residence and first passage times.

A continuous phase transition in resetting behavior is identified as trap stiffness varies.

Analytical results match numerical simulations very well.

Abstract

We study the statistical properties of first-passage Brownian functionals (FPBFs) of an Ornstein-Uhlenbeck (OU) process in the presence of stochastic resetting. We consider a one dimensional set-up where the diffusing particle sets off from $x_0$ and resets to $x_R$ at a certain rate $r$. The particle diffuses in a harmonic potential (with strength $k$) which is centered around the origin. The center also serves as an absorbing boundary for the particle and we denote the first passage time of the particle to the center as $t_f$. In this set-up, we investigate the following functionals: (i) local time $T_{loc} = \int _0^{t_f}d \tau ~ \delta (x-x_R)$ i.e., the time a particle spends around $x_R$ until the first passage, (ii) occupation or residence time $T_{res} = \int _0^{t_f} d \tau ~\theta (x-x_R)$ i.e., the time a particle typically spends above $x_R$ until the first passage and (iii) the first passage time $t_f$ to the origin. We employ the Feynman-Kac formalism for renewal process to derive the analytical expression for the first moment of all the three FPBFs mentioned above. In particular, we find that resetting can either prolong or shorten the mean residence and first passage time depending on the system parameters. The transition between these two behaviors or phases can be characterized precisely in terms of optimal resetting rates, which interestingly undergo a continuous transition as we vary the trap stiffness $k$. We characterize this transition and identify the critical -parameter \& -coefficient for both the cases. We also showcase other interesting interplay between the resetting rate and potential strength on the statistics of these observables. Our analytical results are in excellent agreement with the numerical simulations.