# Drift-diffusive resetting search process with stochastic returns: speed-up
beyond optimal instantaneous return

Arup Biswas<EMAIL_ADDRESS>The Institute of Mathematical Sciences, CIT
Campus, Taramani, Chennai 600113, India & Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar, Mumbai 400094, India  Ashutosh Dubey
<EMAIL_ADDRESS>The Institute of Mathematical Sciences, CIT Campus,
Taramani, Chennai 600113, India & Homi Bhabha National Institute, Training
School Complex, Anushakti Nagar, Mumbai 400094, India  Anupam Kundu
<EMAIL_ADDRESS>International Centre for Theoretical Sciences, TIFR,
Bangalore, India  Arnab Pal<EMAIL_ADDRESS>The Institute of
Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India & Homi
Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai
400094, India

###### Abstract

Stochastic resetting has emerged as a useful strategy to reduce the completion
time for a broad class of first passage processes. In the canonical setup, one
intermittently resets a given system to its initial configuration only to
start afresh and continue evolving in time until the target goal is met. This
is, however, an instantaneous process and thus less feasible for any practical
purposes. A crucial generalization in this regard is to consider a finite-time
return process which has significant ramifications to the first passage
properties. Intriguingly, it has recently been shown that for diffusive search
processes, returning in finite but stochastic time can gain significant speed-
up over the instantaneous resetting process. Unlike diffusion which has a
diverging mean completion time, in this paper, we ask whether this phenomena
can also be observed for a first passage process with finite mean completion
time. To this end, we explore the set-up of a classical drift-diffusive search
process in one dimension with stochastic resetting and further assume that the
return phase is modulated by a potential $U(x)=\lambda|x|$ with $\lambda>0$.
For this process, we compute the mean first passage time exactly and underpin
its characteristics with respect to the resetting rate and potential strength.
We find a unified phase space that allows us to explore and identify the
system parameter regions where stochastic return supersedes over both the
underlying process and the process under instantaneous resetting. Furthermore
and quite interestingly, we find that for a range of parameters the mean
completion time under stochastic return protocol can be reduced further than
the optimally restarted instantaneous processes. We thus believe that
resetting with stochastic returns can serve as a better optimization strategy
owing to its dominance over classical first passage under resetting.

###### pacs:

Valid PACS appear here

##  I Introduction

Search processes are ubiquitous in nature and appear in diverse contexts – for
example, a pigeon searching for its nest Mora  _et al._ (2004); Wallraff
(2001); Walcott (1996), a computer algorithm searching for the optimal
solution Luby  _et al._ (1993), drones for locating some specified targets
Mishra  _et al._ (2020), or an enzyme that searches for a particular substrate
on DNA to bind Chou and D’Orsogna (2014). In the context of statistical
physics, target finding processes are popularly known as first passage (FP)
processes. The time to complete a FP process is known as the first passage
time (FPT) Redner (2001). A major aspect of a FP process is concerned with
finding optimal search strategies to expedite and complete the process in a
minimal time span. Animals have adapted several search strategies to find
their nest or food Bartumeus  _et al._ (2005); O’brien  _et al._ (1990); He
_et al._ (2009). At the cellular level, the idea of facilitated diffusion has
been accepted to answer significantly short time scales involved in chemical
kinetics Mirny  _et al._ (2009); Bénichou  _et al._ (2011a). From a more
statistical mechanics viewpoint, intermittent search strategies have been
explored a lot as a potential way for facilitating FP processes Bénichou  _et
al._ (2011b); Lomholt  _et al._ (2008). The quest for better and better search
strategies still goes on.

Nearly a decade ago, Evan-Majumdar in their seminal work Evans and Majumdar
(2011) reintroduced the idea of stochastic resetting as one of such search
strategies. Here, one intermittently stops and restarts the search process
back from where it originally started. Although seemingly counter-intuitive,
resetting has been shown to immensely improve the performance of a search
process. A panorama of studies followed after that to investigate the effect
of stochastic resetting in a variety of systems Pal (2015); Gupta  _et al._
(2014); Huang and Chen (2021); Chen and Huang (2022); Jolakoski  _et al._
(2023); Kusmierz  _et al._ (2014); Evans and Majumdar (2018a); De Bruyne  _et
al._ (2022); Mercado-Vásquez  _et al._ (2020); Majumdar  _et al._ (2020);
Magoni  _et al._ (2020); Pal  _et al._ (2022); Basu  _et al._ (2019); Bonomo
and Pal (2021); Singh and Pal (2022); Pal and Reuveni (2017); Pal  _et al._
(2016); Kumar and Pal (2023); Olsen  _et al._ (2023); Mori  _et al._ (2023);
Bonomo  _et al._ (2022); Biswas  _et al._ (2023a); Ahmad  _et al._ (2019);
Bonomo and Pal (2021); García-Valladares  _et al._ (2023); Ray  _et al._
(2019); Bodrova and Sokolov (2020a); Ray (2020); Reuveni (2016); Blumer  _et
al._ (2024); Bressloff (2021a, b); Kundu and Reuveni (2024); Yin and Barkai
(2023); Stanislavsky and Weron (2023); Reuveni  _et al._ (2014). We refer to
Evans  _et al._ (2020); Pal  _et al._ (2023); Nagar and Gupta (2023) for a
detailed review of the subject. Notwithstanding the simplicity of the model,
the major hindrance turns out to be its practical applicability as it assumes
a zero time teleportation of the searcher to a resetting location. Any return
event of the searcher to its starting position must take a finite amount of
time. Previous studies have made such attempt either by adding an overhead
time after each reset Evans and Majumdar (2018b); Roy  _et al._ (2024);
García-Valladares  _et al._ (2024); Reuveni  _et al._ (2014) or making the
return process space-time correlated Radice (2022, 2021); Pal  _et al._
(2020); Gupta  _et al._ (2020); Pal  _et al._ (2019); Bodrova and Sokolov
(2020b); Masó-Puigdellosas  _et al._ (2019); Bressloff (2020); Stanislavsky
and Weron (2022); Sunil  _et al._ (2023); Bodrova and Sokolov (2020b). Since
the return motion of the searcher was deterministic and was always directed to
the resetting location, such protocols have always contributed addition time
penalty to the FPT in comparison to the classical instantaneous return
processes.

In contrast, in a recent work Biswas  _et al._ (2023b) it has been shown that
non-instantaneous resetting with stochastic returns can actually supersede
instantaneous return under certain conditions. Though looking counter-
intuitive at first glance this can indeed be achieved by leveraging a
stochastic component in the searcher’s motion during return. When the searcher
returns home following some random motion, there is always a chance that it
finds the target. This sometime may make the search process more efficient as
target detection is conducted both during search and return phase. Notice that
this was not possible for the instantaneous return protocol as there the
searcher is guaranteed to reach the origin in zero time. Consequently, there
is no target detection probability during return. It has to restart the whole
search process once a resetting event occurs. Despite its target detection
probability during the return phase, one can not ignore the fact that
stochastic return consumes a finite time whenever the searcher reaches the
origin after the return phase. A universal criterion was found in Biswas  _et
al._ (2023b), which can ensure that stochastic return can indeed be
advantageous over instantaneous resetting. For a freely diffusive Brownian
particle in one dimension searching for a target, the return was accomplished
by turning on a potential trap around the origin. A parameter space was found
in terms of the resetting rate and the return potential strength, where the
mean first passage time (MFPT) with stochastic return becomes smaller than the
MFPT of instantaneous return.

The MFPT of a free diffusive particle is infinite due to its heavy-tailed
distribution Feller (1991). Thus any finite amount of resetting helps in
making the MFPT a finite quantity by cutting off the diverging trajectories
going far away from the target. But what if the underlying reset-free process
itself has a finite MFPT? How resetting is supposed to perform there? This
question has been answered in Pal and Reuveni (2017) where it was shown that
whenever the ratio of fluctuation to mean of FPT, known as the coefficient of
variation ($CV$), is greater than unity, resetting with the instantaneous
return is guaranteed to help. Certainly, there will be regions where resetting
with instantaneous return can not be beneficial. This leaves us to wonder,
whether resetting with stochastic return may come to be an advantageous
strategy there. Moreover, in the regions where instantaneous resetting helps
in reducing the MFPT compared to the underlying process, can stochastic return
do even better? Can one find a parameter space where stochastic return proves
to be fruitful in not only decreasing the MFPT compared to that of
instantaneous return but also compared to the MFPT of the underlying process?
Understanding these aspects is the central goal of this paper. The simplest
case where MFPT is rendered finite is when a drift is added to the Brownian
particle towards the target Redner (2001). The drift-diffusion process also
bears a close resemblance to modelling several real-life systems. Some of them
include the motion of a particle under the flow field of fluid Risken and
Eberly (1985), charge transport Jacoboni  _et al._ (1977), neuron firing
Capocelli and Ricciardi (1971), price of stocks Laurent and Shi (2020) etc. In
this paper, we consider the drift-diffusive search process in one dimension
with intermittent stochastic return to the origin. The return is modulated by
turning on a potential trap around the origin. We analyze the mean first
passage time of the particle to reach the target. Our main focus will be to
probe its parameter space and find criteria for stochastic return to be better
than the underlying process as well as the instantaneous return case.

The paper is organized as follows: In section II we illustrate our drift-
diffusion model in detail and find the MFPT for any arbitrary resetting time
distribution. In section III we take the explicit case when the resetting
times are drawn from an exponential distribution. In section III.1 we find the
criterion for resetting with stochastic return to provide an advantage over
the underlying process. In section III.2 we find the criterion where
stochastic return can be more beneficial than the instantaneous return. In
section III.3 we merge both these criteria to have a unified parameter space
from where one can identify the utility of stochastic return for any given set
of parameters. Finally, in section IV we find the parameter space where the
MFPT obtained from stochastic return protocol can be further minimized than
the lowest MFPT obtained from optimal instantaneous return protocol.

Figure 1: Drift-diffusive search process with resetting. A drift of magnitude
$v$ (red arrow) always acts on the particle, biasing its motion towards the
target. The resetting is modulated by turning on a potential $U(x)=\lambda|x|$
with $\lambda>0$ at random times (shown by the blue strips). This potential
causes an additional drift of magnitude $\lambda$ towards the origin. Overall
an effective potential $U_{\text{eff}}(x)$ as in Eq. (7) acts on the particle
during the return. After the particle reaches the origin the potential is
turned off marking the completion of one resetting event. The search process
can end in two ways: either the target is found in the search phase (potential
off) or in the return phase (potential on).

##  II Drift-diffusive search process in one dimension

Consider a diffusive Brownian particle in one dimension (1-d) which starts its
motion from $x=0$. There is a constant drift of magnitude $v>0$ that acts on
the particle. This drift biases the particle’s motion towards the target at
$x=L>0$. Once the particle reaches the target at $x=L$ the motion is completed
and we note the corresponding first passage time denoted by the random
variable $T$. This problem is well studied in literature Redner (2001). For
completeness, we provide small derivation of the first passage probabilities
in Appendix A.

In addition to the free drift-diffusive motion, now at random times $R$
generated from a distribution $f_{R}(t)$, a potential $U(x)$, centred at the
origin, is turned on. This potential tries to bring the particle back towards
the origin. The moment the particle reaches the origin the potential is
switched off marking the completion of one resetting event. After that, it
follows again follows underlying drift-diffusive motion. The entire process is
repeated until and unless the target is found. The search can end in two ways:
either the particle reaches the target (at $x=L$) in the search phase i.e.
while the potential is off or in the return phase i.e. when the potential is
turned on. Our aim is to analyse the MFPT for this system.

For analytical simplicity, we take the simplest form of the return potential
which is a linear one given by

$\displaystyle U(x)=\lambda|x|.$ (1)

This form of potential imposes another drift $\lambda>0$ on the particle on
top of the underlying drift velocity $v$ but now towards the origin (in
contrast to the drift towards the target) during the return phase. Note that,
due to the constant nature of the drift velocity the return motion is now also
a drift-diffusive process similar to the underlying motion (search phase).
However, the direction and magnitude are now changed. For $x>0$, the net drift
now becomes

$\displaystyle\lambda_{+}=\lambda-v,$ (2)

towards the origin (assuming $\lambda>v$). Quite evidently, when $\lambda<v$
the net drift acts towards the target in the region $x>0$. On the other hand
for $x<0$ the drift always is

$\displaystyle\lambda_{-}=\lambda+v,$ (3)

towards the target (or origin, as both fall on the same side). In another way,
one can construct an effective potential $U_{\text{eff}}(x)$ as below

$\displaystyle\begin{array}[]{l}U_{\text{eff}}(x)=\left\\{\begin{array}[]{lll}\lambda_{+}x&\text{if
}x>0,\\\ -\lambda_{-}x&\text{if }x<0,\end{array}\right.\text{ }\end{array}$
(7)

which acts on the particle during the return phase. Fig. 1 depicts a
trajectory of the particle schematically. In summary, each resetting event now
has two major caveats: first, as the particle reaches the origin in a
stochastic fashion, it consumes a finite time and secondly, the particle has a
finite probability of reaching the target during the return. In the limit
$\lambda\to\infty$ one however, expects to recover the results for the
classical instantaneous return for a drift-diffusive system Ray  _et al._
(2019); Pal  _et al._ (2020).

### Mean first passage time

Let us denote by $T_{R}$ the random variable associated with the first passage
time of the entire process (search + return phase) described above. We are
interested in finding the MFPT $\langle T_{R}\rangle$ of the overall search
process. A general renewal formalism was presented in Biswas  _et al._ (2023b)
for the MFPT of such a search process for any arbitrary return motion. Here we
do not delve much into the formalism but present the general result for the
MFPT in 1-d obtained in Biswas  _et al._ (2023b). We simply write here

$\displaystyle\langle T_{R}\rangle=\frac{\langle
min(T,R)\rangle+\int_{-\infty}^{L}dx\widetilde{G}_{R}(x)\langle
t^{ret}(x)\rangle}{1-\int_{-\infty}^{L}dx~{}\widetilde{G}_{R}(x)\epsilon^{ret}_{O}(x)},$
(8)

where, the quantity $\widetilde{G}_{R}(x)=\int_{0}^{\infty}dtf_{R}(t)G(x,t)$
is the time-integrated propagator with $G(x,t)$ being the propagator of the
underlying process in the absence of resetting and $f_{R}(t)$ is the
distribution of the resetting time $R$. Other quantities are defined as
follows. Here

$\displaystyle\langle t^{ret}(x)\rangle=\theta(-x)\langle
t_{1}(x)\rangle+\theta(x)\langle t_{2}(x)\rangle,$ (9)

represents the mean return time either to the origin ($O$) or target ($A$)
(see Fig. 1) starting the return motion from $x$. $\langle t_{1}(x)\rangle$ is
the mean time to reach the origin during return from $x<0$ and $\langle
t_{2}(x)\rangle$ is the unconditional mean time to reach either the target or
the origin during the return from $x>0$. The quantity $\epsilon^{ret}_{O}(x)$,
given by

$\displaystyle\epsilon^{ret}_{O}(x)=\theta(-x)+\theta(x)\epsilon_{O}(x),$ (10)

represents the splitting probability to reach the origin during return.
$\epsilon_{O}(x)$ is the splitting probability to reach the origin while
returning from $x>0$. Note that starting from $x<0$, the particle will
definitely reach the origin before reaching the target at $L>0$. The minimum
of the two random variables $T$ and $R$ is denoted by $min(T,R)$ where recall
that $T$ denotes the random variable associated with the underlying first-
passage time and $R$ is the random waiting time when the return phase
commences since the start of the last search phase.

We would thus need the exact expressions for the unconditional MFPT and the
splitting probabilities of the return motion facilitated by the potential
$U_{\text{eff}}(x)$. In Appendix A we give detailed derivations of these
quantities. Here we just recall the main results which we shall be using in
the later parts of this paper. The underlying reset-free propagator $G(x,t)$,
and its Laplace transform
$\widetilde{G}(x,s)=\int_{0}^{\infty}dte^{-st}G(x,t)$, for the drift-diffusive
process in presence of a target in 1-d are computed in the textbook Redner
(2001). It is given by

$\displaystyle G(x,t)$ $\displaystyle=\frac{1}{\sqrt{4\pi
Dt}}\left(e^{-\frac{(x-tv)^{2}}{4Dt}}-e^{\frac{Lv}{D}}e^{-\frac{(2L+tv-x)^{2}}{4Dt}}\right),$
(11)

which after Laplace transforming with respect to $t$ looks like

$\displaystyle\widetilde{G}(x,s)$
$\displaystyle=\frac{e^{\frac{vx}{2D}}\left(e^{-\frac{|x|\sqrt{4Ds+v^{2}}}{2D}}-e^{-\frac{(2L-x)\sqrt{4Ds+v^{2}}}{2D}}\right)}{\sqrt{4Ds+v^{2}}}.$
(12)

The MFPT to reach either the target at $x=L$ or the origin at $x=0$ during the
return motion starting from $x>0$ is found to be

$\displaystyle\langle
t_{2}(x)\rangle=\frac{L(1-e^{\lambda_{+}x/D})+x(e^{\lambda_{+}L/D}-1)}{\lambda_{+}(e^{\lambda_{+}L/D}-1)}.$
(13)

The associated splitting probability to reach the target i.e.
$\epsilon_{L}(x)$ or to reach the origin i.e. $\epsilon_{O}(x)$, is given by

$\displaystyle\epsilon_{L}(x)=1-\epsilon_{O}(x)=\frac{1-e^{\lambda_{+}x/D}}{1-e^{\lambda_{+}L/D}}.$
(14)

When the return motion starts from $x<0$ the particle has to cross the origin
first implying the probability of reaching the target is zero so that

$\displaystyle\epsilon_{L}(x)=1-\epsilon_{O}(x)=0,$ (15)

while the corresponding MFPT to reach the origin is given by

$\displaystyle\langle t_{1}(x)\rangle=\frac{|x|}{\lambda_{-}}.$ (16)

With these quantities in hand, one only requires the specific return
(resetting) time distribution to make further progress. In what follows we
shall take the set-up where resetting times are drawn from an exponential
distribution. The choice of exponential resetting is primarily due to its
advantage in giving nice closed-form results. However, we emphasize that the
formalism presented till now can always be utilized for other resetting
protocols as well.

##  III Resetting at exponential times

Suppose the resetting/return times are drawn randomly from an exponential
distribution with rate $r$, given by

$\displaystyle f_{R}(t)=re^{-rt}.$ (17)

Recalling Eq. (8), we first proceed to find the quantity $\langle
min(T,R)\rangle$. By definition, this is found to be (see Appendix B for
detailed derivation)

$\displaystyle\langle min(T,R)\rangle=\int_{0}^{\infty}dtPr(T>t)Pr(R>t),$ (18)

with $f_{T}(t)=-\frac{dPr(T>t)}{dt}$ being the first passage time distribution
of the underlying reset-free process. For exponential distribution, we obtain
from Eq. (18)

$\displaystyle\langle min(T,R)\rangle$
$\displaystyle=\int_{0}^{\infty}dte^{-rt}\int_{t}^{\infty}dt^{\prime}f_{T}(t^{\prime})$
$\displaystyle=\frac{1}{r}-\frac{1}{r}\int_{0}^{\infty}dte^{-rt}f_{T}(t)$
$\displaystyle=\frac{1-\widetilde{T}(r)}{r},$ (19)

where $\widetilde{T}(r)=\int_{0}^{\infty}dte^{-rt}f_{T}(t)$ is the Laplace
transform of $f_{T}(t)$. For the drift-diffusive search process, the first
passage time density $f_{T}(t)$ and its Laplace transform $\widetilde{T}(s)$
are given by (see Appendix C)

$\displaystyle f_{T}(t)$
$\displaystyle=\frac{Le^{-\frac{(L-tv)^{2}}{4Dt}}}{2\sqrt{\pi}\sqrt{D}t^{3/2}},~{}~{}\text{and}$
(20) $\displaystyle\widetilde{T}(s)$
$\displaystyle=e^{\frac{L\left(v-\sqrt{4Ds+v^{2}}\right)}{2D}},$ (21)

respectively. The time-integrated propagator for the exponential resetting
distribution is simply its Laplace transform multiplied by $r$ as shown below

$\displaystyle\widetilde{G}_{R}(x)=\int_{0}^{\infty}re^{-rt}G(x,t)dt=r\widetilde{G}(x,r),$
(22)

where exact expression for $\widetilde{G}(x,r)$ is given in Eq. (12).

After using the results obtained from Eq. (19) and Eq. (22) to Eq. (8), we
have

$\displaystyle\langle
T_{R}(r,\lambda,v)\rangle=\frac{\frac{1-\widetilde{T}(r)}{r}+r\int_{-\infty}^{L}dx\widetilde{G}(x,r)\langle
t^{ret}(x)\rangle}{1-r\int_{-\infty}^{L}dx~{}\widetilde{G}(x,r)\epsilon^{ret}_{O}(x)},$
(23)

where we have omitted the explicit dependence on $D$ and $L$. Recall that
$\langle t^{ret}(x)\rangle,\epsilon^{ret}_{O}(x)$ are given by Eq. (9) and Eq.
(10) respectively. It is now a straightforward task to plug the results
obtained from Eq. (13)-(16) and Eq. (21) in the above equation to have the
full MFPT for the drift-diffusive system with exponential resetting. It turns
out that the overall MFPT can be written in a dimensionless form

$\displaystyle\langle\tau(\overline{r},\overline{\lambda},\overline{v})\rangle=\frac{D}{L^{2}}\langle
T_{R}(r,\lambda,v)\rangle,$ (24)

where we introduce the scaled quantities

$\displaystyle\overline{r}=rL^{2}/D,~{}\overline{\lambda}=\lambda
L/D,~{}\overline{v}=vL/D.$ (25)

The exact expression for
$\langle\tau(\overline{r},\overline{\lambda},\overline{v})\rangle$ is quite
cumbersome and provided in the Mathematica file in the GitHub link key . Fig.
2 shows its variation with respect to the potential strength
$\overline{\lambda}$ and resetting rate $\overline{r}$ for two distinct values
of $\overline{v}=0.2\text{ and }3$.

Figure 2: Variation of the MFPT
$\langle\tau(\overline{r},\overline{\lambda},\overline{v})\rangle$ with
potential strength $\overline{\lambda}$ (panel a,c) and resetting rate
$\overline{r}$ (panel b,d) for exponentially distributed resetting times.
Panel (a,b) are for $\overline{v}=0.2$ and panel (c,d) are shown for
$\overline{v}=3$. The limit $\overline{\lambda}\to 0$ gives the MFPT of the
underlying process i.e. $\frac{1}{\overline{v}}$ shown by the dashed
horizontal lines. Note the contrasting behaviour for different values of
$\overline{v}$. For lower value of $\overline{v}$ each plot shows a non-
monotonic behaviour implying the utility of resetting compared to the reset-
free process. However, as $\overline{v}$ becomes higher MFPT for a finite
value of $\overline{r}$ and $\overline{\lambda}$ is always greater than that
of the underlying process. Also in some of the curves in panel (a) and in each
of the curves in panel (c), there exists a range of $\overline{\lambda}$ where
MFPT becomes lower than the instantaneous return limit
($\overline{\lambda}\to\infty$).

The behaviour with respect to potential strength is quite intricate as can be
seen from Fig. 2(a,c). Let us first discuss two important limiting cases of
the MFPT $\langle\tau(\overline{r},\overline{\lambda},\overline{v})\rangle$
that we shall frequently refer to in the rest of the paper.

MFPT of the underlying (reset-free) process: In the limit
$\overline{\lambda}\to 0$ no effect of resetting potential is there and the
process effectively goes back to the simple reset-free drift-diffusion. One
can check that in this limit the MFPT is given by

$\displaystyle\langle\tau(\overline{r},\overline{\lambda}\to
0,\overline{v})\rangle=\frac{1}{\overline{v}},$ (26)

which the MFPT of the underlying drift-diffusive process Redner (2001). Note
that this is also equivalent to setting $\overline{r}\to 0$. Both the limits
correspond to the same underlying reset-free process.

MFPT with instantaneous return: On the other hand when
$\overline{\lambda}\to\infty$ the potential strength is so high that the
particle returns almost instantaneously to the origin. The resulting
expression of the MFPT in this limit indeed reduces the result for the
instantaneous return case

$\displaystyle\langle\tau(\overline{r},\overline{v})\rangle_{\text{inst}}=\langle\tau(\overline{r},\overline{\lambda}\to\infty,\overline{v})\rangle=\frac{e^{\frac{1}{2}\left(\sqrt{\overline{v}^{2}+4\overline{r}}-\overline{v}\right)}-1}{\overline{r}},$
(27)

as derived in Ray  _et al._ (2019). Here we have denoted the MFPT associated
with instantaneous return protocol by
$\langle\tau(\overline{r},\overline{v})\rangle_{\text{inst}}$.

Quantification of MFPT for intermediate values of $\overline{\lambda}$ is
highly non-trivial. Note that depending on the magnitude of the drift
$\overline{v}$ of the underlying process the plots show two distinct kinds of
behaviour. Let us first focus on Fig. 2(a) which is for $\overline{v}=0.2$.
For lower values of resetting rates (e.g. $\overline{r}=0.1,1$), the curves
monotonically go down and saturate to the instantaneous return MFPT given by
Eq. (27). On the other hand, when resetting rates are high it shows non-
monotonic behaviour with $\overline{\lambda}$. Particularly, for
$\overline{r}=10,20$ it is evident from the plots that, for a range of
$\overline{\lambda}$ MFPT can be reduced below that of both the underlying
reset-free process and the instantaneous return limit. In contrast, for the
value of $\overline{v}=3$, we see from Fig. 2(c) that MFPT never goes below
the MFPT of the underlying reset-free process for any values of
$\overline{\lambda}$. However, in this case also one can find the set of
$\overline{\lambda}$ values where MFPT is lower than that of instantaneous
resetting.

With resetting rate $\overline{r}$ MFPT shows a non-monotonic variation for
lower values of $\overline{v}$ (look at Fig. 2(b)). In the absence of
resetting $\overline{r}\to 0$ one gets

$\displaystyle\langle\tau(\overline{r}\to
0,\overline{\lambda},\overline{v})\rangle=\frac{1}{\overline{v}},$ (28)

the same as in Eq. (26) as they are physically equivalent. When resetting is
too frequent i.e. in the limit $\overline{r}\to\infty$, the potential is
almost always turned on, and the MFPT resembles that of a particle in the
potential $U_{\text{eff}}(x)$, given by

$\displaystyle\langle\tau(\overline{r}\to\infty,\overline{\lambda},\overline{v})\rangle=\frac{\overline{v}^{2}-\overline{\lambda}\left(-2e^{\overline{\lambda}-\overline{v}}+\overline{\lambda}+2\right)}{(\overline{\lambda}-\overline{v})^{2}(\overline{\lambda}+\overline{v})}.$
(29)

Once $\overline{v}$ is increased this non-monotonous behaviour goes away and
MFPT starts increasing with $\overline{r}$ for any value of
$\overline{\lambda}$ (see Fig. 2(d)).

From the above discussion, it is evident that resetting is not guaranteed to
expedite the search process for any arbitrary parameters. Fig. 2(a,b) are the
cases where it indeed helps. On the contrary, Fig. 2(c,d) depicts that any
amount of resetting is detrimental. In the next section, we delve deeper into
this and try to find a criterion for resetting to be helpful.

###  III.1 Speed up over underlying process - the CV criterion

We ask: given a certain set of parameters what kind of behaviour the MFPT of
our system is expected to show? In other words, will resetting be beneficial
or detrimental there in comparison to the reset-free process? Mathematically
the task is to find the parameter space where $\langle T_{R}\rangle<\langle
T\rangle$. It turns out that for the case of exponential resetting time
distribution one can indeed derive a simpler criterion which will guarantee
that resetting will be beneficial. Let us start with a region of parameter
space such that resetting indeed helps in reducing the MFPT compared to the
underlying reset-free process. Mathematically this is ensured if for any small
amount of resetting $\delta r\to 0$ one has

$\displaystyle\langle T_{R}(\delta r,\lambda,v)\rangle<\langle T\rangle,$ (30)

where $\langle T\rangle$ is the MFPT of the underlying reset-free process and
$\langle T_{R}(\delta r,\lambda,v)\rangle$ is given in Eq. (23). Here we
should emphasize that the above condition is sufficient for resetting to be
helpful but not a necessary one. An expansion of the LHS of the above equation
up to $\mathcal{O}(\delta r)$ and setting the first order correction term to
be less than zero, leads to the following interesting relation (detailed
derivation in Appendix D)

$\displaystyle CV^{2}>\frac{2}{\langle T\rangle}\overline{\langle
t^{ret}(x)\rangle}+2\overline{\epsilon^{ret}_{O}(x)}-1,$ (31)

where $CV$ is the coefficient of variation of the underlying reset-free
process defined as

$\displaystyle CV=\frac{\sqrt{\langle T^{2}\rangle-\langle
T\rangle^{2}}}{\langle T\rangle}.$ (32)

This is a measure of the relative fluctuation of the first passage time of the
underlying reset-free process. The overbar $\overline{(...)}$ in Eq. (31)
implies a average has to be taken with $G_{0}(x)$ so that
$\overline{f(x)}=\int_{-\infty}^{L}dx~{}G_{0}(x)f(x)$. The quantity $G_{0}(x)$
is defined as

$\displaystyle G_{0}(x)=\frac{1}{\langle T\rangle}\int_{0}^{\infty}dtG(x,t),$
(33)

which is normalized by construction so that $\int_{-\infty}^{L}G_{0}(x)dx=1$.
The condition in Eq. (31) is not limited to the drift-diffusive system
considered here, but applies to any arbitrary first passage system in 1-d. For
the drift-diffusive system considered in this paper, the mean and the second
moment are given, respectively, by $\langle T\rangle=L/v$ and $\langle
T^{2}\rangle=\frac{L(2D+Lv)}{v^{3}}$ (which can be found from Taylor series
expansion of Eq. (21)). The $CV$ thus becomes

$\displaystyle CV=\sqrt{\frac{2D}{Lv}}=\frac{1}{\sqrt{\text{Pe}}},$ (34)

where $\text{Pe}=\frac{Lv}{2D}=\frac{\overline{v}}{2}$ is the so-called Peclet
number. It is simply the ratio of the diffusion and drift time scales. Eq.
(31) is quite useful since without any prior knowledge of the resetting rates
one can infer from here if resetting will be helpful. From here one can also
recover the $CV$ criterion for instantaneous and deterministic return
protocols obtained in previous studies as shown below.

Instantaneous return limit - In the instantaneous return limit the particle is
guaranteed to reach the origin so that $\epsilon_{O}(x)=1$. On top of that it
returns instantaneously so that $\langle t^{ret}(x)\rangle=0$. Hence for
instantaneous return, we have

$\displaystyle CV>1,$ (35)

a classic result that was derived earlier in Pal and Reuveni (2017); Pal and
Prasad (2019); Pal  _et al._ (2023) valid for any arbitrary FP process. This
says that, with only information about the mean and fluctuation of the
underlying process in hand, one can deduce the utility of resetting. For the
drift-diffusive system particularly this relation implies from Eq. (34)

$\displaystyle\text{Pe}<1\implies\overline{v}<2,$ (36)

for resetting to be beneficial Ray  _et al._ (2019).

Figure 3: Phase space obtained from the $CV$ criterion in Eq. (31) for the
drift-diffusive system with return via $U_{\text{eff}}(x)$. The blue-shaded
region is where resetting will help to reduce the MFPT beyond the underlying
process. The black dashed line at $\overline{v}=2$ refers to the phase
boundary in the instantaneous limit ($\overline{\lambda}\to\infty$).

Deterministic return limit -  From Eq. (31) it is also possible to obtain the
criterion when the return is deterministic. In particular, suppose the
particle consumes a finite deterministic time $\tau^{ret}(x)$ to return to the
origin starting from $x$. Note that, in this case also as the particle is
guaranteed to reach the origin thus we have $\epsilon_{O}(x)=1$. Using Eq.
(31) one then obtains the criterion for deterministic return case as

$\displaystyle CV^{2}>1+\frac{2\overline{\tau^{ret}(x)}}{\langle T\rangle},$
(37)

as was obtained in Pal  _et al._ (2020).

Stochastic return case - We now come back to our problem where the particle is
taken back to the origin stochastically facilitated by an external potential
$U(x)$. One can find the RHS of Eq. (31) exactly by substituting results from
Eq. (13)-(16) and Eq. (21)-(22). The resulting expression is quite cumbersome
which we provide in the key . However one can generate the phase space spanned
over $\overline{\lambda}-\overline{v}$ plane as shown in Fig. 3. If the
parameters are chosen from the blue-shaded region, then resetting is bound to
decrease the MFPT of this process further below the underlying drift-diffusive
limit i.e. $1/\overline{v}$ (see Eq. (28)). The black dashed line (
$\text{Pe}=1$ or $\overline{v}=2$ )represents the phase boundary for the
instantaneous resetting limit as obtained previously. Note that, keeping a
fixed value of $\overline{\lambda}$ if one gradually increases $\overline{v}$,
then at some finite value $\overline{v}$ one crosses from the region where
resetting helps to where resetting does not. This behavioural transition is
known as restart transition and is of second order for our system. A more
detailed discussion on restart transition is provided in the Appendix E.

The physical origin behind these two distinct regions can be understood as
follows. For very low values of the drift $\overline{v}$ the particle can
disperse very far from the origin. These diverging trajectories in the
direction opposite to the target can lead to a very high value of the MFPT.
Return (any finite value of $\overline{\lambda}>0$) in those cases prohibits
the particle from wandering off too far away from the target consequently
reducing the MFPT. In contrast, when the drift $\overline{v}$ is sufficiently
high, then the MFPT of the underlying process is itself very short. The
diverging trajectories as mentioned previously are already very rare in this
case. Any kind of resetting in this case would only hinder the trajectories
which were supposed to find the target in a shorter time span and cause a
delay in the process completion.

An important observation emanating out of Fig. 3 is the existence of the blue-
shaded region even above $\overline{v}=2$. Above this cut-off, instantaneous
return can not facilitate the process beyond the underlying one. However,
stochastic return turns out to be beneficial here. We shall come back to this
discussion again in Sec. IV. In the next subsection, we first try to quantify
the criterion when stochastic return is better than instantaneous return for a
given rate $\overline{r}$.

Figure 4: The parameter space where resetting with stochastic return can
facilitate the drift-diffusive search process over the instantaneous return,
as generated from the SR criterion in Eq. (42). The colour density denotes the
magnitude (log) of the speed-up parameter $\gamma$ as defined in Eq. (43). The
higher it is the more benefits one can gain with the stochastic return over
the classical instantaneous return. The red dashed line is the separatrix
which is obtained by setting Eq. (42) to an equality i.e., $\gamma=1$.

###  III.2 Speed up over instantaneous resetting- the SR criterion

The stochastic mode of return has two major impacts on the search time of the
system compared to the classical instantaneous return. The first one is that,
here the searcher consumes a finite amount of time to return to the origin. In
contrast, in the instantaneous return case, the searcher takes zero time to
return. This is indeed what we observe in Fig. 2(a) for $\overline{r}=0.1,1$,
where any finite $\overline{\lambda}$ yields a higher value of MFPT than that
at $\overline{\lambda}\to\infty$ (instantaneous limit). However, since the
searcher is returning in a stochastic manner, it is possible once in a while,
that it actually finds the target while returning home (origin) and the search
process is completed. In the instantaneous return case, however, the searcher
always returns to the origin in no time and the whole search process is again
restarted. Thus in this case one may see certain advantages in the stochastic
return motion. In Fig. 2(a) for $\overline{r}=5,10,20$ note that there exists
a range of $\overline{\lambda}$ values where MFPT is lowered than that of the
instantaneous limit. This behaviour is also observed in all the curves in Fig.
2(c). Evidently, there exists a trade-off between the return time and the
target search probability. To understand this trade-off quantitatively we
simply compare the MFPT for these two modes of return and look at the
condition when MFPT with stochastic return is less than that obtained for
classical instantaneous return i.e.

$\displaystyle\langle T_{R}\rangle<\langle T^{inst}_{R}\rangle,$ (38)

where $\langle T^{inst}_{R}\rangle$ is the MFPT with instantaneous return. One
can find $\langle T^{inst}_{R}\rangle$ from Eq. (8) by noting that the return
times $\langle t_{1}(x)\rangle,\langle t_{2}(x)\rangle$ are zero here as the
particle reaches the origin instantaneously. On top of that, as the particle
reaches the origin for sure, the splitting probability to the origin is unity
i.e. $\epsilon_{O}(x)=1$. Combining both these results we obtain

$\displaystyle\langle T^{inst}_{R}\rangle=\frac{\langle
min(T,R)\rangle}{Pr(T<R)},$ (39)

where, $Pr(T<R)=1-\int_{-\infty}^{L}dx~{}\widetilde{G}_{R}(x)$ quantifies the
probability that the first passage occurs before any resetting event takes
place. The same result was obtained earlier in Pal and Reuveni (2017). For the
case when resetting times are exponentially distributed, this takes the form
(using Eq. (19))

$\displaystyle\langle
T^{inst}_{R}\rangle=\frac{1-\widetilde{T}(r)}{r\widetilde{T}(r)},$ (40)

where $\widetilde{T}(r)$ is given by Eq. (21). Note that $\langle
T^{inst}_{R}\rangle=\frac{L^{2}}{D}\langle\tau(\overline{r},\overline{v})\rangle_{\text{inst}}$,
which can be also be found from Eq. (27) for the drift-diffusive system
considered.

Starting from Eq. (38) a general criterion was derived in Biswas  _et al._
(2023b) to identify the phase space regime where stochastic return could be
more advantageous. The resulting criterion in 1-d for exponential resetting
takes the following form Biswas  _et al._ (2023b)

$\displaystyle\mathcal{T}\coloneqq\frac{\int_{-\infty}^{L}dx\widetilde{G}(x,r)\langle
t^{ret}(x)\rangle}{\int_{0}^{L}dx\widetilde{G}(x,r)\epsilon_{L}(x)}<\langle
T^{inst}_{R}\rangle,$ (41)

where $\epsilon_{L}(x)$ is the probability of reaching the target during
return motion from $x>0$ (the exact expression given in Eq. (14)). The
quantity $\mathcal{T}$ quantifies the trade-off between the return times and
the target search probability of the stochastic return motion. In particular,
the numerator of $\mathcal{T}$ (in Eq. (41)) takes care of the return times.
The higher the return times, the higher the $\mathcal{T}$, adverse to
satisfying the criterion in Eq. (41). The denominator takes care of the
probability that the target is found while returning. Thus higher it is the
lower becomes the $\mathcal{T}$, which aids in satisfying the criterion (41).
Both the RHS and the LHS of the above inequality can again be written in a
dimensionless form to obtain the criterion

$\displaystyle\overline{\mathcal{T}}<\langle\tau(\overline{r},\overline{v})\rangle_{\text{inst}}.$
(42)

where we denote the dimensionless quantity
$\overline{\mathcal{T}}=\frac{D}{L^{2}}\mathcal{T}$. To further quantify the
amount of speed-up gained with stochastic return over instantaneous return we
define the speed-up parameter

$\displaystyle\gamma=\frac{\langle T^{inst}_{R}\rangle}{\langle
T_{R}\rangle}=\frac{\langle\tau(\overline{r},\overline{v})\rangle_{\text{inst}}}{\langle\tau(\overline{r},\overline{\lambda},\overline{v})\rangle}.$
(43)

Clearly, when $\gamma>1$ stochastic return wins. The magnitude of gamma is an
indicator of how much gain can be achieved with stochastic return. The
separatrix between the two regimes where stochastic or instantaneous return is
better can be obtained by setting Eq. (41) to equality i.e. from $\gamma=1$.
The LHS of Eq. (41) can be exactly computed by substituting the elements from
Eq. (13)-(16), and Eq. (22). Again we do not provide the functional form here
due to its complex structure which is provided in key .

Figure 5: A unified phase diagram for the drift-diffusive search process for
exponential resetting with rate $\overline{r}=2$. Different regions in this
parameter space represent the regions where either of the criteria in Eq. (31)
and Eq. (42) are satisfied. The most important is region IV (the red-shaded
region) where both the CV and SR criterion is satisfied and stochastic return
facilitates the process beyond both the underlying process and also that of
resetting with instantaneous return. The behaviour of the MFPT for a sample
point (star) in each of these regions is shown in the corners. In these plots,
the horizontal solid lines represent the MFPT that with instantaneous return
and the dashed horizontal line represents the MFPT of the underlying reset-
free process.

Fig. 4 shows the parameter space ($\overline{r}-\overline{\lambda}$) generated
by the criterion in Eq. (42) for different values of $\overline{v}$. The
colour gradient represents the value of the speed-up parameter $\gamma$. The
region right to the separatrix (the red dashed line) is where criterion Eq.
(42) is satisfied and stochastic return diminishes the MFPT to a lower value
than that obtained through the instantaneous return protocol. Note that the
line $\gamma=1$ moves upwards as $\overline{v}$ is increased. This means for a
higher value of the drift, the regime where the efficacy of the stochastic
return can be realized, gets broadened.

Let us now try to understand the physical origin behind such behaviour of the
MFPT. First focus on the regime where $\overline{v}$ is low. In this limit, it
is evident from both Fig. 2(a) and Fig. 4(a) that, instantaneous return always
wins for relatively lower values of the resetting rate. As discussed in
section III.1, for relatively lower values of $\overline{v}$ the particle may
hover far away from the target and consume excessive time to find the target.
Resetting cuts off these detrimental trajectories by bringing them back to the
origin. However, the crucial point to note here is that stochastic return in
this case incurs a huge return time in bringing back the particles from a far
away distance. On the other hand, instantaneous return causes them to get back
to the origin instantaneously, inducing no extra time. Thus in the limit of
small $\overline{v}$ and $\overline{r}$, MFPT of stochastic return is higher
than that of instantaneous return. When the resetting rate becomes
sufficiently high (irrespective of the value of $\overline{v}$) instantaneous
resetting becomes detrimental as it does not allow the particle to even get
close to the target. On the other hand for the stochastic return case, the
potential is turned on so frequently that the particle essentially moves under
the return potential $U_{\text{eff}}(x,t)$ all the time. Evidently, the
particle has a chance of reaching the target while moving inside the
potential. Consequently, the MFPT is much lower compared to the instantaneous
return value.

###  III.3 A unified phase diagram

Till now we have found parameter space associated with the two regions,
$\langle T_{R}\rangle<\langle T\rangle$ which is found from the CV criterion
and $\langle T_{R}\rangle<\langle T^{inst}_{R}\rangle$ which is found from the
SR criterion. Now we are in a position to probe a more intricate aspect: For a
given value of $\overline{r}$, can stochastic return do better than both the
underlying search process and that with instantaneous return protocol,
simultaneously? This is equivalent to finding the parameter space where
$\langle T_{R}\rangle<\langle T\rangle<\langle T^{inst}_{R}\rangle$ or
$\langle T_{R}\rangle<\langle T^{inst}_{R}\rangle<\langle T\rangle$. One can
guess this will be the case when both the conditions in Eq. (31) and Eq. (41)
are satisfied. To verify this we plot the phase boundary in the
$\overline{v}-\overline{\lambda}$ (for a fixed $\overline{r}=2$) plane
obtained from these conditions in the central image of Fig. 5. The region left
to the solid (red) separatrix is where the $CV$ criterion is satisfied. On the
other hand, the region below the dashed separatrix is where the SR criterion
is satisfied. This immediately presents us with 4 distinct phases in the
parameter space. We mark them from I-IV and discuss each of them below in
detail.

Region I: In this regime (yellow shaded) $CV$ criterion is satisfied although
the SR criterion is not. Consequently, stochastic return is supposed to
expedite the search process over the underlying process. However, it can not
supersede the instantaneous return protocol. To verify, we pick an arbitrary
point from this region at $(\overline{v},\overline{\lambda})\equiv(1,60)$ (the
yellow star) and mark its associated MFPT in the plot at the top left corner
of Fig. 5. The dashed horizontal line shows the MFPT corresponding to the
underlying reset-free process and the MFPT associated to resetting with
instantaneous return protocol is shown by the solid horizontal line. We can
clearly see the MFPT at the yellow star lies above the dashed line
(instantaneous return) and below that of the solid line (reset-free process),
as expected.

Region II: Here (blue shaded) none of the criteria is satisfied resulting in
stochastic return being the most detrimental protocol. For a generic point in
this space $(\overline{v},\overline{\lambda})\equiv(20,60)$ (blue star) we see
from the MFPT plot at the top right corner of Fig. 5 that it lies well above
both the horizontal solid (instantaneous return) and dashed line (underlying
process).

Region III: This is where (green shaded) only the SR criterion holds but the
$CV$ criterion does not. Evidently, stochastic return is a better alternative
than the classical instantaneous return. Although, in this space resetting
with stochastic return can not expedite the search process over the underlying
process. The same is shown for a point in this space
$(\overline{v},\overline{\lambda})\equiv(20,2)$ (green star) which lies below
the solid horizontal line (instantaneous return) but above the dashed line
(underlying process) in the rightmost plot at the bottom of Fig. 5.

Region IV: It is the most noteworthy region (red-shaded) as the supremacy of
stochastic return is more pronounced here. Both the SR and $CV$ criteria are
satisfied in this space. Thus stochastic return turns out to be the best way
to reduce the search time of the underlying process and also that obtained
from instantaneous return protocol. This is also observed by taking a sample
point in this phase space $(\overline{v},\overline{\lambda})\equiv(1.5,1)$
(the red star). In the MFPT vs $\overline{\lambda}$ plot in the bottom left
corner of Fig. 5 we see it lies well below both the solid horizontal line
(MFPT due to instantaneous return) and the dashed horizontal line (MFPT of the
underlying reset-free process). Moreover, at this point note that
instantaneous return does actually increase the mean search time than the
underlying process.

To summarize, we mark stochastic return as one of the strategies with which
one can expedite a search process even in cases where instantaneous return
fails to do so. Note that the phase space as shown in Fig. 5 is only for the
fixed value of $\overline{r}=2$. However, with changing $\overline{r}$ the
phase boundaries will change its position and the regions marked by (I-IV)
will change.

##  IV Speed-up beyond optimal instantaneous return

Let us now take a step backwards and look at the motivation behind restarting
a first passage process. The central role of inducing resetting to a search
process was to increase the search efficiency i.e. to minimize the MFPT.
Performing instantaneous resetting on the original process is the first
protocol which was seen to mitigate the diverging trajectories going away from
the target and thus expediting the search process Evans and Majumdar (2011);
Evans  _et al._ (2020); Pal and Reuveni (2017). However, when the reset-free
process itself has a finite mean completion time $\langle T\rangle$, then
performing instantaneous resetting may not always reduce the mean completion
time. For example, performing instantaneous resetting too frequently is
definitely detrimental. In such a situation, stochastic return may turn out to
be helpful in improving the search process. Due to the finite target-finding
ability of the searcher during the return phase stochastic return gains the
upper hand there. Despite the utility of stochastic return for a given
resetting rate and at places where instantaneous return fails, we are still
left to wonder: Can stochastic return reduce the MFPT of the process even
below the lowest MFPT that can be obtained with instantaneous return protocol?

Figure 6: (a) MFPT with stochastic return i.e.
$\langle\tau(\overline{r},\overline{\lambda},\overline{v})\rangle$ (blue and
orange curves) and instantaneous return i.e.
$\langle\tau(\overline{r},\overline{v})\rangle_{\text{inst}}=\langle\tau(\overline{r},\overline{\lambda}\to\infty,\overline{v})\rangle$
(green curve) are plotted with respect to the resetting rate $\overline{r}$,
for a fixed drift velocity $\overline{v}=1.8$. Remarkably, for
$\overline{\lambda}=1.4$ we find that optimal stochastic return can lower the
MFPT (the blue circle) even beyond the optimal instantaneous return (the green
circle). (b) Phase space showing regions where stochastic (or instantaneous)
return is better at the optimal resetting rate (which is different in general
for both the protocols). The blue shaded region (obtained by setting the
speed-up parameter $\gamma^{*}$ as in Eq. (47), to be greater than unity) is
where stochastic return performs better than the optimal instantaneous return.

###  IV.1 MFPT at optimal resetting rate

Let us elaborate more. For a fixed value of drift $\overline{v}$, one observes
from Fig. 2(b) that for a particular value of the resetting rate
$\overline{r}$ the MFPT is lowest. This resetting rate is called optimal
resetting rate (ORR) denoted by $\overline{r}^{*}$. Mathematically this can be
found from the equation

$\displaystyle\left.\frac{\partial\langle\tau(\overline{r},\overline{\lambda},\overline{v})\rangle}{\partial\overline{r}}\right|_{\overline{r}=\overline{r}^{*}}=0.$
(44)

For a fixed value of $\overline{\lambda},\overline{v}$ the lowest MFPT that
can be obtained through stochastic return is
$\langle\tau(\overline{r}^{*},\overline{\lambda},\overline{v})\rangle$. Note
that ORR $\overline{r}^{*}$ is a function of $\overline{\lambda},\overline{v}$
in general. In the limit $\overline{\lambda}\to\infty$ we obtain the ORR for
the instantaneous case as

$\displaystyle\overline{r}^{*}_{\text{inst}}=\overline{r}^{*}(\overline{\lambda}\to\infty,\overline{v}),$
(45)

which can also be found from Eq. (27) as

$\displaystyle\left.\frac{\partial\langle\tau(\overline{r},\overline{v})\rangle_{\text{inst}}}{\partial\overline{r}}\right|_{\overline{r}=\overline{r}^{*}_{\text{inst}}}=0.$
(46)

Thus for a fixed value of $\overline{v}$ the lowest MFPT that can be obtained
through instantaneous resetting is
$\langle\tau(\overline{r}^{*}_{\text{inst}},\overline{v})\rangle_{\text{inst}}$.
However, keeping $\overline{v}$ fixed one can still vary the potential
strength $\overline{\lambda}$ for the stochastic return to find the $r^{*}$
associated with each value of $\lambda$. Can we find a parameters space in
$\overline{\lambda},\overline{v}$ where MFPT at this ORR can be compared?

We recall from Sec. III.1 that for $\overline{v}>2$ instantaneous return fails
to expedite the underlying process. Consequently, the $r^{*}_{\text{inst}}$ is
exactly zero there (see Appendix E for more discussion). On the other hand, a
careful look at Fig. 3 reveals that there is a certain blue shaded region even
at $\overline{v}>2$. This is where stochastic return is actually able to
enhance the search process compared to the underlying process, in spite of
instantaneous return failing to do so. Evidently, $r^{*}$ is non-zero in this
region. Interestingly, there also exists a set of values of $\overline{v}$
(which is less than 2) where despite having a finite $r^{*}_{\text{inst}}$,
stochastic return at $r^{*}$ can perform better. As an illustrative example
look at Fig. 6(a) where we plot the MFPT for both the protocols with respect
to $\overline{r}$ keeping $\overline{v}=1.8$. Note that the blue curve,
corresponding to the MFPT with stochastic return for $\overline{\lambda}=1.4$,
shows a minima at $\overline{r}^{*}$. The MFPT at this minima is lower than
the lowest MFPT obtained at optimal instantaneous return (the green curve) at
$\overline{r}^{*}_{\text{inst}}$. Although, for $\overline{\lambda}=3$ (the
orange curve) the lowest MFPT obtained with stochastic return is higher than
that with instantaneous return. In a nutshell, we have a set of parameters in
terms of $(\overline{v},\overline{\lambda})$ where stochastic return at
optimal resetting can indeed outperform the optimal instantaneous resetting.
To quantify this region we calculate the speed-up parameter $\gamma^{*}$ at
the optimal resetting rate, defined as

$\displaystyle\gamma^{*}=\frac{\langle\tau(\overline{r}^{*}_{\text{inst}},\overline{v})\rangle_{\text{inst}}}{\langle\tau(\overline{r}^{*},\overline{\lambda},\overline{v})\rangle}.$
(47)

The region $\gamma^{*}>1$ is where optimal stochastic return better than
optimal instantaneous return, shown by the blue shaded region in Fig. 6(b). In
the green shaded region ($\gamma^{*}<1$), instantaneous return prevails.
Whereas, in the white region underlying process already performs at its best
and any kind of resetting is detrimental here.

###  IV.2 A possible physical explanation of such behaviour

We now make an attempt to understand such speed-up over optimal instantaneous
resetting from the behaviour of the process on a trajectory level. We first
list the major effects imposed in the search process by stochastic return
motion.

  1. (a)

In the region $x<0$, the effective drift $\lambda_{-}=\lambda+v$ is always
stronger than the underlying drift velocity $v$, which aids in cutting the
diverging trajectories in the $x<0$ region appearing in the underlying reset-
free process.

  2. (b)

In the region $x>0$, the drift $\lambda_{+}=\lambda-v$ acts in opposite
direction of the target when $\lambda>v$ and reduces the net drift towards the
target when $\lambda<v$. In both scenarios, the search is hampered compared to
the underlying process.

  3. (c)

On the other hand if we compare to the instantaneous resetting, trajectories
return to the origin in zero time during the return phase, adding no
additional return time to the MFPT.

  4. (d)

At $x>0$, however, some trajectories reach the target during return phase due
to its stochastic dynamics. Whereas, for the instantaneous return the particle
reaches the origin in zero time and thus no target search is possible during
return. In this case, stochastic return turns out to be advantageous over the
instantaneous return for some choices of parameters.

It is the complex interplay amongst the above-mentioned effects that come into
picture for any given set of parameters, which determines whether stochastic
or instantaneous return will be the better policy. Let us now explore them
taking different limits of the drift $\overline{v}$.

1) Very high $\overline{v}\gg 2$: From Fig. 6(b) (or Fig. 3) it is clear that
for very high values of the drift velocity $\overline{v}$, neither of
stochastic or instantaneous return can expedite the search process. For very
high values of drift towards the target, most of the trajectories are expected
to be going towards the target in a very short time. Bringing back the
particle to the origin (instantaneously or even turning on the return
potential) only inhibits these trajectories, which leads to higher values of
the MFPT. Thus any kind of resetting is detrimental in those cases.

2) Intermediate value of $\overline{v}\gtrsim 2$: For intermediate values of
$\overline{v}\gtrsim 2$, from Fig. 6(b) we find there exists a region (shown
by the violet shaded region above the dashed vertical line at
$\overline{v}=2$) where instantaneous return does not help in expediting the
search process but stochastic return protocol does. This might be due to the
following reasons. The MFPT of the underlying process increases here from the
previous case as the number of trajectories going away from the target is
comparatively higher here. Yet, most of the trajectories again go towards the
target and restarting them instantaneously only delays the process. Clearly,
instantaneous return again turns out to be detrimental. On the other hand, for
the stochastic return case when the potential is turned on (at suitably chosen
value of $\overline{\lambda}$, which is not very high) the particle does have
a finite chance to reach the target rather than returning to the origin. This
target search probability suppresses the effect due to finite return time from
trajectories going away from the target. So effect (d) is more prominent here
compared to (c). Alternatively, now the trajectories going away from the
target $(x<0)$ experience a higher value of net drift towards the origin (as
$\lambda_{-}>\overline{v}$) which obviously helps in preventing trajectories
from wandering off for a long time compared to the underlying process. Here
effect (a) is more pronounced than effect (b). Consequently, stochastic return
turns out to be a better strategy here compared to the underlying process.
However, for $\overline{\lambda}\gg 1$ the target search probability during
return is significantly reduced and those trajectories contribute to a higher
mean return times. As a result effect (c) now becomes stronger to make the
stochastic return protocol inefficient compared to the instantaneous return
protocol.

3) Intermediate value of $\overline{v}\lesssim 2$: The blue region left to the
dashed line Fig. 6 is where optimal instantaneous return helps in reducing the
MFPT compared to the underlying process. However, stochastic return protocol
at optimal values of the resetting rate and suitable $\overline{\lambda}$ can
still minimize the MFPT beyond the optimal instantaneous return here. The
underlying physical reason can again be explained similarly to the previous
case. When drift is low enough ($\overline{v}<2$), a sufficient number of
diverging trajectories are affected by the optimal instantaneous return
reducing the MFPT. However, the finite target search probability during
stochastic return fashion (for suitably chosen values of $\overline{\lambda}$)
still dominates over effect (d). So stochastic return is still a better
strategy here. Moreover, here we see for comparatively lower values of
$\overline{v}$, when $\overline{\lambda}\ll 1$ stochastic return fails to
expedite the search compared to the instantaneous return. This is because for
low values of $\overline{v}$ the diverging trajectories incur higher return
times, so that effect (c) suppresses effect (d). Consequently, the stochastic
return becomes detrimental. As previously discussed, for
$\overline{\lambda}\gg 1$ the target search probability gets reduced so that
(c) is more effective than (d), and stochastic return loses its benefits.

4) Very low values of $\overline{v}\ll 2$: In this case, the number of
trajectories going away from the target increases significantly. Evidently,
the return times are huge even at optimality for the stochastic return mode.
Clearly, instantaneous return wins here.

##  V Discussion and outlook

In this paper, we studied the drift-diffusive search process in 1-d under
resetting with stochastic return. The return phase is modulated by turning on
a linear potential trap at the origin. When the trap strength is sufficiently
high, one recovers the limit of resetting with instantaneous return. We
particularly focus on two aspects of stochastic return motion: first, whether
it can reduce the MFPT beyond the underlying reset-free process and second,
whether it can also expedite the search over the classical instantaneous
return protocol for a given resetting rate. To answer the former, we derive
the $CV$ criterion which guarantees that resetting with stochastic return
indeed enhances the search process compared to the underlying process. For the
later one, we derive the SR criterion which ensures that the stochastic return
protocol does better than the instantaneous return protocol. Then, we combine
both these criteria to find a unified parameter space where stochastic return
turns out to be a potential strategy to expedite the FP process over both the
underlying process and also that with instantaneous return protocol. Finally,
we investigate the question: can stochastic return perform better than the
instantaneous return at their respective optimality? We, indeed, find a
parameter space where despite the existence of a non-zero optimal
instantaneous resetting rate, stochastic return is seen to perform better in
reducing the search completion time further.

It is indeed remarkable that in spite of an overhead time caused by space-time
coupled return protocol, stochastic return does a good job of decreasing the
MFPT. This is primarily due to the target-finding ability of the searcher
during the return phase. Besides, stochastic return provides a more realistic
description of a resetting event. For example, in experiments involving the
colloidal particles resetting is implemented by turning on a potential trap at
the origin. In those scenarios, there is always a chance that the particle
goes uphill and finds the target. Those trajectories are usually filtered out
in experiments as they are theoretically hard to take into account. However,
we show that this can actually be done and also greatly helps in expediting
the search process.

The study of stochastic return in the context of resetting processes is still
at its infancy and many generalizations can be proposed. A straightforward
extension would be to study the problem in higher dimensions. It would also be
interesting to look at its effects when more than one target is present. Apart
from that it still remains as an open question: what is the optimal choice for
the resetting time distribution $f_{R}(t)$ so that the mean completion time
can be globally minimized? Finally, we would like to conclude by emphasising
that stochastic return motion can indeed be beneficial for other search
processes that go beyond diffusion and it would be tempting to know how an
analogous SR criterion can be manifested in those systems. With the parameters
space derived from those criteria, one can easily navigate between the best
choice of search strategy to facilitate the process.

##  VI Acknowledgement

AK acknowledges the support of the core research grant CRG/2021/002455 and the
MATRICS grant MTR/2021/000350 from the SERB, DST, Government of India. AK also
acknowledges support of the Department of Atomic Energy, Government of India,
under Project No. RTI4001. AP gratefully acknowledges research support from
the Department of Science and Technology, India, SERB Start-up Research Grant
Number SRG/2022/000080 and Department of Atomic Energy, Government of India.

## Appendices

##  Appendix A Calculations of MFPT and splitting probabilities of drift-
diffusive search in 1-d

In this section, we shall provide the detailed calculations for the basic
observables (as in Eq. (13)-(16)) of a 1-d drift-diffusive search process.
Note that when the return phase starts from a position $x_{0}>0$ then the
origin acts as a virtual absorbing boundary. This is because, in this case,
the return phase is terminated whenever the particle makes a first passage to
the origin along with the target at $x=L$. Let us consider a Brownian particle
that diffuses in the interval $x\in[0,L]$. The boundaries at $x=0,L$ are
purely absorbing in nature. In addition to its diffusive motion, it
experiences drift of magnitude $\lambda_{d}>0$ towards the boundary at
$x=L>0$. The drift $\lambda_{d}$ can stand form both $\lambda_{+}=\lambda-v$
(for $x>0$) or $\lambda_{-}=\lambda+v$ (for $x<0$) as in Eq. (7). One can
write the associated drift-diffusion equation for the probability density
$G_{ret}(x,t)$ as

$\displaystyle\frac{\partial G_{ret}(x,t)}{\partial
t}-\lambda_{d}\frac{\partial G_{ret}(x,t)}{\partial
x}=D\frac{\partial^{2}G_{ret}(x,t)}{\partial x^{2}},$ (48)

with the initial condition

$\displaystyle G_{ret}(x,t=0)=\delta(x-x_{0}),$ (49)

where $x_{0}$ is the particle’s position at $t=0$. The boundary conditions
read

$\displaystyle G_{ret}(x=0,t)=G_{ret}(x=L,t)=0.$ (50)

In Laplace space, this equation takes the form

$\displaystyle D\frac{\partial^{2}\widetilde{G}_{ret}(x,s)}{\partial
x^{2}}+\lambda_{d}\frac{\widetilde{G}_{ret}(x,s)}{\partial
x}-s\widetilde{G}_{ret}(x,s)=-\delta(x-x_{0}),$ (51)

where $\widetilde{G}_{ret}(x,s)=\int_{0}^{\infty}G_{ret}(x,t)e^{-st}dt$. The
above equation can be solved exactly to get a closed form expression for
$\widetilde{G}_{ret}(x,s)$ as

$\displaystyle\widetilde{G}_{ret}(x,s)=$
$\displaystyle-\frac{e^{\frac{\lambda_{d}(x_{0}-x)}{2D}}\text{csch}\left(mL\right)}{2mD}\Bigg{[}\theta(x-x_{0})\Big{(}\cosh\left(m(L+x-x_{0})\right)$
$\displaystyle\hskip 130.88284pt-\cosh\left(m(L-x+x_{0})\right)\Big{)}$
$\displaystyle\hskip 28.45274pt+\cosh\left(m(L-x-
x_{0})\right)-\cosh\left(m(L+x-x_{0})\right)\Bigg{]},$ (52)

where, $m=\frac{\sqrt{\lambda_{d}^{2}+4Ds}}{2D}$ and $\theta(x)$ is the
Heaviside step function.

###  A.1 Mean first passage times

To find the mean first passage times of the particle we need to find the
survival probability which can be computed as

$\displaystyle Q(t)=\int_{0}^{L}~{}dx~{}G_{ret}(x,t),$ (53)

which in Laplace space reads

$\displaystyle\widetilde{Q}(s)=\int_{0}^{\infty}dte^{-st}Q(t)=\int_{0}^{L}\widetilde{G}_{ret}(x,s)dx.$
(54)

From here the mean first passage time of the particle to reach either of the
boundaries is given by

$\displaystyle\langle t_{2}(x_{0})\rangle$ $\displaystyle=\lim_{s\to
0}\widetilde{Q}(s)=\frac{L(1-e^{\lambda_{d}x_{0}/D})+x_{0}(e^{\lambda_{d}L/D}-1)}{\lambda_{d}(e^{\lambda_{d}L/D}-1)}.$
(55)

With $\lambda_{d}=\lambda_{+}=\lambda-v$ we obtain Eq. (13) in the main text
i.e. the MFPT when the return phase is started from $x_{0}>0$. However, when
the return is started from $x_{0}<0$, there is no target detection possible
and in this case, the required result can be obtained by taking limit
$L\to\infty$ in Eq. (55), and one finds

$\displaystyle\langle t_{1}(x_{0})\rangle=\frac{|x_{0}|}{\lambda_{d}},$ (56)

Setting $\lambda_{d}=\lambda_{-}=\lambda+v$ one obtains Eq. (16) of the main
text.

###  A.2 Splitting probabilities

The probability flux through each of the boundaries in Laplace space is given
by Redner (2001)

$\displaystyle j_{L}(x_{0},s)$
$\displaystyle=-D\frac{\partial\widetilde{G}_{ret}(x,s)}{\partial
x}\bigg{|}_{x=L},$ (57) $\displaystyle j_{O}(x_{0},t)$
$\displaystyle=D\frac{\partial\widetilde{G}_{ret}(x,s)}{\partial
x}\bigg{|}_{x=0}.$ (58)

Using these one can obtain the splitting probabilities for each of the targets
as

$\displaystyle\epsilon_{L}(x_{0})=j_{L}(x_{0},s\to 0)$
$\displaystyle=\frac{1-e^{\lambda_{d}x_{0}/D}}{1-e^{\lambda_{d}L/D}},$ (59)
$\displaystyle\epsilon_{O}(x_{0})=j_{O}(x_{0},s\to 0)$
$\displaystyle=1-\epsilon_{L}(x_{0})$
$\displaystyle=\frac{e^{\lambda_{d}x_{0}/D}-e^{\lambda_{d}L/D}}{1-e^{\lambda_{d}L/D}},$
(60)

as announced Eq. (14) of the main text. Here
$\lambda_{d}=\lambda_{+}=\lambda-v$ i.e. when return commences from $x_{0}>0$.
For $x_{0}<0$, the target detection is not possible and the associated
splitting probabilities are simply given by Eq. (15).

##  Appendix B Derivation of $\langle min(T,R)\rangle$ in Eq. (18)

Let $Z$ represent the random variable

$\displaystyle Z=min(T,R),$ (61)

with the associated density $f_{Z}(t)$. We thus have

$\displaystyle\langle min(T,R)\rangle=\int_{0}^{\infty}tf_{Z}(t)dt.$ (62)

The density $f_{Z}(t)$ can be found from the cumulative distribution of $Z$ as

$\displaystyle f_{Z}(t)=-\frac{d}{dt}Pr(Z>t).$ (63)

Now the cumulative distribution $Pr(Z>t)$ can easily be found by noting that
the condition $Z=min(T,R)>t$ holds only when both of $T$ and $R$ are
simultaneously greater than $t$. That in turn implies

$\displaystyle P(Z>t)=Pr(T>t)Pr(R>t).$ (64)

One can now do an integration by parts of the integral in Eq. (62) to have

$\displaystyle\langle min(T,R)\rangle$
$\displaystyle=\int_{0}^{\infty}tf_{Z}(t)dt$ $\displaystyle=\left[t\int
f_{Z}(t)dt\right]_{0}^{\infty}-\int_{0}^{\infty}dt\left(\int
dtf_{Z}(t)\right).$ (65)

From Eq. (63) we have $\int f_{Z}(t)dt=-Pr(Z>t)$ which upon inserting to the
above equation we have

$\displaystyle\langle min(T,R)\rangle$
$\displaystyle=\lim_{t\to\infty}tPr(Z>t)+\int_{0}^{\infty}dtPr(Z>t).$ (66)

Assuming $Pr(Z>t)$ to decay faster than $t$ as $t\to\infty$ (note that this
condition holds trivially for the drift-diffusive system with exponential
resetting as both of $Pr(T>t)$ and $Pr(R>t)$ decay exponentially, which can be
verified from the respective distributions Eq. (20) and Eq. (17)), we finally
have

$\displaystyle\langle min(T,R)\rangle=\int_{0}^{\infty}dtPr(Z>t),$ (67)

which upon substituting result form Eq. (64) yields

$\displaystyle\langle min(T,R)\rangle=\int_{0}^{\infty}dtPr(T>t)Pr(R>t).$ (68)

This is the same result as in Eq. (18) of the main text.

##  Appendix C First passage time distribution of drift-diffusion process in
1-d

In this section, we derive the first passage time distribution of a 1-d drift-
diffusive process in presence of a single absorbing boundary at $x=L$, as
given in Eq. (20). If $f_{T}(t)$ denotes the first passage time distribution
then it is related to the survival probability $Q(t)$ as

$\displaystyle f_{T}(t)=-\frac{\partial Q(t)}{\partial t}.$ (69)

where the survival probability $Q(t)$ is given by

$\displaystyle Q(t)=\int_{-\infty}^{L}dx~{}G(x,t)~{}.$ (70)

The drift-diffusive propagator $G(x,t)$ is given by Eq. (11). Inserting this
expression of $G(x,t)$ in Eqs. (70, 69) and performing the integral, one can
obtain the first passage time distribution as

$\displaystyle f_{T}(t)$
$\displaystyle=\frac{Le^{-\frac{(L-tv)^{2}}{4Dt}}}{\sqrt{4\pi Dt^{3}}},$ (71)

as was written in Eq. (20).

##  Appendix D Derivation of the CV criterion for resetting with stochastic
return

In this section, we shall derive the $CV$ criterion as in Eq. (31). We start
by considering a small resetting rate $\delta r\to 0$ limit of Eq. (23). The
reason for doing so is that whenever the first order term $\mathcal{O}(\delta
r)$ is negative, the MFPT of the reset process is guaranteed to be smaller
than that of the underlying process. For reader’s convenience, we rewrite Eq.
(23) here

$\displaystyle\langle
T_{R}(r,\lambda,v)\rangle=\frac{\frac{1-\widetilde{T}(r)}{r}+r\int_{-\infty}^{L}dx\widetilde{G}(x,r)\langle
t^{ret}(x)\rangle}{1-r\int_{-\infty}^{L}dx~{}\widetilde{G}(x,r)\epsilon^{ret}_{O}(x)}.$
(72)

For this let us first expand Eq. (72) with respect to $\delta r\to 0$. Note
that in this limit we have

$\displaystyle\widetilde{T}(\delta r\to 0)=1-\delta r\langle
T\rangle+\frac{\delta r^{2}}{2}\langle T^{2}\rangle+\mathcal{O}(\delta
r^{3}).$ (73)

Substituting this in Eq. (72) we obtain

$\displaystyle\langle T_{R}(\delta r,\lambda,v)\rangle$
$\displaystyle=\left(\langle T\rangle-\frac{\delta r}{2}\langle
T^{2}\rangle+\delta r\int_{-\infty}^{L}dx\widetilde{G}(x,0)\langle
t^{ret}(x)\rangle\right)$ $\displaystyle\hskip 36.98866pt\times\left(1+\delta
r\int_{-\infty}^{L}dx~{}\widetilde{G}(x,0)\epsilon^{ret}_{O}(x)\right).$ (74)

With slight rearrangement, we obtain

$\displaystyle\langle T_{R}(\delta r,\lambda,v)\rangle$ $\displaystyle=\langle
T\rangle+\delta r\Bigg{(}-\frac{\langle
T^{2}\rangle}{2}+\int_{-\infty}^{L}dx\widetilde{G}(x,0)\langle
t^{ret}(x)\rangle$ $\displaystyle\hskip 56.9055pt+\langle
T\rangle\int_{-\infty}^{L}dx~{}\widetilde{G}(x,0)\epsilon^{ret}_{O}(x)\Bigg{)}$
(75) $\displaystyle+\mathcal{O}(\delta r^{2}).$

The first term is the MFPT of the underlying reset-free process. Thus any
infinitesimal $\delta r$ is supposed to decrease the MFPT beyond that only
when the term under the parentheses is less than zero. In that case, we obtain

$\displaystyle\Bigg{(}-\frac{\langle
T^{2}\rangle}{2}+\int_{-\infty}^{L}dx\widetilde{G}(x,0)\langle
t^{ret}(x)\rangle$ $\displaystyle\hskip 73.97733pt+\langle
T\rangle\int_{-\infty}^{L}dx~{}\widetilde{G}(x,0)\epsilon^{ret}_{O}(x)\Bigg{)}<0$
$\displaystyle\implies\langle
T^{2}\rangle>2\int_{-\infty}^{L}dx\widetilde{G}(x,0)\langle t^{ret}(x)\rangle$
$\displaystyle\hskip 85.35826pt+2\langle
T\rangle\int_{-\infty}^{L}dx~{}\widetilde{G}(x,0)\epsilon^{ret}_{O}(x)$
$\displaystyle\implies\langle T^{2}\rangle-\langle
T\rangle^{2}>2\int_{-\infty}^{L}dx\widetilde{G}(x,0)\langle t^{ret}(x)\rangle$
$\displaystyle\hskip 56.9055pt+2\langle
T\rangle\int_{-\infty}^{L}dx~{}\widetilde{G}(x,0)\epsilon^{ret}_{O}(x)-\langle
T\rangle^{2}$ $\displaystyle\implies\frac{\langle T^{2}\rangle-\langle
T\rangle^{2}}{\langle T\rangle^{2}}>\frac{2}{\langle
T\rangle^{2}}\int_{-\infty}^{L}dx\widetilde{G}(x,0)\langle t^{ret}(x)\rangle$
$\displaystyle\hskip 85.35826pt+\frac{2}{\langle
T\rangle}\int_{-\infty}^{L}dx~{}\widetilde{G}(x,0)\epsilon^{ret}_{O}(x)-1.$
(76)

Noting that that
$\widetilde{G}(x,r)=\int_{0}^{\infty}dt~{}e^{-rt}G(x,t)\implies\widetilde{G}(x,0)=\int_{0}^{\infty}dt~{}G(x,t)$
we have

$\displaystyle\frac{\langle T^{2}\rangle-\langle T\rangle^{2}}{\langle
T\rangle^{2}}>\frac{2}{\langle
T\rangle^{2}}\int_{-\infty}^{L}dx\int_{0}^{\infty}dt~{}G(x,t)\langle
t^{ret}(x)\rangle$ $\displaystyle\hskip 56.9055pt+\frac{2}{\langle
T\rangle}\int_{-\infty}^{L}dx~{}\int_{0}^{\infty}dt~{}G(x,t)\epsilon^{ret}_{O}(x)-1$
$\displaystyle\implies\frac{\langle T^{2}\rangle-\langle T\rangle^{2}}{\langle
T\rangle^{2}}$ $\displaystyle\hskip 42.67912pt>\frac{2}{\langle
T\rangle}\int_{-\infty}^{L}dx\left(\frac{1}{\langle
T\rangle}\int_{0}^{\infty}dt~{}G(x,t)\right)\langle t^{ret}(x)\rangle$
$\displaystyle\hskip 42.67912pt+2\int_{-\infty}^{L}dx~{}\left(\frac{1}{\langle
T\rangle}\int_{0}^{\infty}dt~{}G(x,t)\right)\epsilon^{ret}_{O}(x)-1.$ (77)

Finally identifying the coefficient of variation ($CV$) as
$CV=\frac{\sqrt{\langle T^{2}\rangle-\langle T\rangle^{2}}}{\langle T\rangle}$
we have the following the $CV$ criterion

$\displaystyle CV^{2}>\frac{2}{\langle T\rangle}\overline{\langle
t^{ret}(x)\rangle}+2\overline{\epsilon^{ret}_{O}(x)}-1,$ (78)

as written in Eq. (31) of the main text. Recall that we use the notation
$\overline{f(x)}=\int_{0}^{\infty}dxf(x)G_{0}(x)$ with $G_{0}(x)$ being
defined as

$\displaystyle G_{0}(x)=\frac{1}{\langle T\rangle}\int_{0}^{\infty}dtG(x,t).$
(79)

Figure 7: Variation of the optimal resetting rate (ORR) $\overline{r}^{*}$
with respect to the underlying drift velocity $\overline{v}$ for different
values of the potential strength $\overline{\lambda}$. The ORR shows a second-
order phase transition with respect to $\overline{v}$.

##  Appendix E Restart transition

In this section, we investigate the restart transition as briefly mentioned in
section II of the main text. When $\overline{v}$ is low enough (recall Fig.
2(b)), note that the MFPT becomes the lowest at some finite value of
$\overline{r}=\overline{r}^{*}$. This resetting rate $\overline{r}^{*}$ is
formally known as the optimal resetting rate (ORR). Mathematically, this can
be found by setting

$\displaystyle\left.\frac{\partial\langle\tau(\overline{r},\overline{\lambda},\overline{v})\rangle}{\partial\overline{r}}\right|_{\overline{r}=\overline{r}^{*}}=0.$
(80)

A non-zero value of $\overline{r}^{*}$ ensures that resetting can benefit the
search process in lowering the MFPT to its lowest. In contrast, one observes
in Fig. 2(d), that for a higher value of $\overline{v}$ resetting only
increases the MFPT. Consequently, the reset-free process performs at its best
and the optimal resetting rate $\overline{r}^{*}$ is zero. Solving Eq. (80)
numerically we can now probe this behavioural transition as depicted in Fig.
7. We find that $\overline{r}^{*}$ undergoes a second-order transition as
$\overline{v}$ is gradually increased, for any values of $\overline{\lambda}$.
For very high potential strength ($\overline{\lambda}=200$) the transition
occurs at $\overline{v}=2$, commensurate with the instantaneous return limit
Ray  _et al._ (2019).

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