First-passage Brownian functionals with stochastic resetting

Abstract

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.

The first-passage area of Ornstein-Uhlenbeck process revisited

“…Remark 2.3 Proposition 2.1 was already proved in [26] in the case when X(t) is Brownian motion. Note that, for r = 0 (that is, when no resetting is allowed) one obtains Eq.…”

“…, can be written in terms of special functions, precisely the Scorer's and Airy functions ( [1]) (see Eqs. ( 56) and (57) of [26]). For r = 0, that is for BM without resetting, Eq.…”

“…x) given by (3.5) can be inverted, at least for x = x R (see Eq. ( 46) of [26] and the comments therein).…”

“…This paper regards the first-passage area (FPA) of a diffusion process with stochastic resetting; it integrates some other articles [2], [3], [4] and [5], in which we studied the FPA of jump-diffusions, drifted Brownian motion, Lèvy process, and Ornstein-Uhlenbeck process. Actually, here we consider a one-dimensional diffusion process in the presence of stochastic resetting, obtained from a underlying diffusion X(t); this kind of process, that we call Reset Diffusion (RD) process, was first introduced in [13], and considered afterwards in [6], [9], [12], [23], [24], [25], in particular the corresponding FPA was studied in [26], in the case when the underlying process is Brownian motion. Our aim is to study the statistical properties of the first-passage time (FPT) through zero of a RD process X(t), starting from x > 0, and its FPA, namely the area enclosed between the time axis and the path of the process X(t) up to the FPT through zero.…”

“…Really, in the case when the underlying diffusion X(t) is Brownian motion without drift, the FPA was already studied in [26], though the results found therein were obtained making use of special functions. In contrast, in the present article we utilize nothing but elementary functions: our arguments are based on classical results for one-dimensional diffusions; in fact, the study of the distributions of FPT and FPA is carried out via solutions of certain associated ODEs, also using some results of [26]. We focus on the case when the underlying diffusion X(t) is a Wiener process, that is, a Brownian motion with or without drift, but the results can be extended to other RD processes.…”