# Extremal statistics for stochastic resetting systems

Prashant Singh1 and Arnab Pal2<EMAIL_ADDRESS><EMAIL_ADDRESS>1 International Centre for Theoretical Sciences, Tata
Institute of Fundamental Research, Bengaluru 560089, India 2 School of
Chemistry, The Center for Physics and Chemistry of Living Systems, Tel Aviv
University, Tel Aviv 6997801, Israel

###### Abstract

While averages and typical fluctuations often play a major role to understand
the behavior of a non-equilibrium system, this nonetheless is not always true.
Rare events and large fluctuations are also pivotal when a thorough analysis
of the system is being done. In this context, the statistics of extreme
fluctuations in contrast to the average plays an important role, as has been
discussed in fields ranging from statistical and mathematical physics to
climate, finance and ecology. Herein, we study Extreme Value Statistics (EVS)
of stochastic resetting systems which have recently gained lot of interests
due to its ubiquitous and enriching applications in physics, chemistry,
queuing theory, search processes and computer science. We present a detailed
analysis for the finite and large time statistics of extremals (maximum and
arg-maximum i.e., the time when the maximum is reached) of the spatial
displacement in such system. In particular, we derive an exact renewal formula
that relates the joint distribution of maximum and arg-maximum of the reset
process to the statistical measures of the underlying process. Benchmarking
our results for the maximum of a reset-trajectory that pertain to the Gumbel
class for large sample size, we show that the arg-maximum density attains to a
uniform distribution regardless of the underlying process at a large
observation time. This emerges as a manifestation of the renewal property of
the resetting mechanism. The results are augmented with a wide spectrum of
Markov and non-Markov stochastic processes under resetting namely simple
diffusion, diffusion with drift, Ornstein-Uhlenbeck process and random
acceleration process in one dimension. Rigorous results are presented for the
first two set-ups while the latter two are supported with heuristic and
numerical analysis.

##  I INTRODUCTION

In many situations of physical relevance, extreme events are of tremendous
importance despite they occur rarely. Starting from stock market crashes or
large insurance losses in finance, records in Olympics to natural calamities
such as earthquake, heat waves, extreme events or tsunamis – all are typical
examples of extreme events. Extreme Value Statistics (EVS) sits on the heart
of a branch of statistics which deals with the probabilities generated by
random processes responsible for such unusual extreme events EVS-review-1 ;
EVS-review-2 ; EVS-review-3 ; EVS-review-4 ; EVS-review-5 ; EVS-review-6 ;
EVS-review-7 ; EVS-review-cor . The study of EVS has been extremely important
in the field of disordered systems disorder-1 ; disorder-2 , fluctuating
interfaces KPZ-1 ; KPZ-2 , interacting spin systems spin , stochastic
transport models EVS-con-1 ; EVS-STR , random matrices RM-1 ; RM-2 ; RM-3 ,
ecology ecology , in binary search trees tree  and related computer search
algorithms CS-1 ; CS-2  and even in material science EVS-material . We refer
to these extensive reviews which provide detailed account of recent
theoretical and application based progresses of EVS in science. In EVS, the
famous Gnedenko’s classical law of extremes provides statistics of the maximum
(or minimum) of a set of uncorrelated random variables (see e.g. EVS-review-1
; EVS-review-2 ; EVS-review-3 ; EVS-review-4 ; EVS-review-5 ; EVS-review-6 ;
EVS-review-7 ; EVS-review-cor ). However, there exists a myriad of systems for
which the underlying random variables can be weakly or strongly correlated due
to the correlations EVS-correlated-2 ; EVS-correlated-3 ; EVS-correlated-4  or
a global conservation EVS-con-3 ; EVS-con-4 ; EVS-con-5 ; EVS-con-6  among the
random variables, see EVS-review-cor  for a comprehensive review. One of the
central goals of the subject is then to understand the statistics of extremes
i.e., the maximum $M(t)$ of a given trajectory $x(t)$ which is observed upto
time $t$ and the time $t_{m}$ to reach the maximum for such correlated systems
namely Brownian motion and its generalizations Levy ; Andersen ; tmax-1 ;
tmax-2 ; tmax-3 ; tmax-CTRW ; tmax-RAP ; tmax-CTRW , run and tumble motion
tmax-RTP-1 ; Mori2020 , fractional Brownian motion tmax-FBM-1 ; tmax-FBM-2 ;
tmax-FBM-3 , random acceleration tmax-RAP , anomalous walker tmax-anamolous
and fluctuating interfaces tmax-interface-growth . Surprising enough, there
are only a very limited exact results known on renewal processes (e.g., see
EVS-con-3  where EVS for the longest waiting interval was analyzed and not the
quantities of our interest). Moreover, prediction of limiting extreme value
distributions in renewal processes has been extremely challenging (also see
Restart1 ; review  and the discussion below). In this paper, we set out to
understand in details the extremal statistics in a recently popularized
renewal process namely stochastic resetting.

Stochastic resetting is a renewal process in which the dynamics repeats by
itself after random intervals controlled externally review . The subject has
recently gained considerable attention due to its vigorous applications in
statistical physics Restart1 ; Restart2 ; Restart3 ; Restart4 ; Restart5 ;
Restart6 ; PalJphysA ; confining , stochastic process SP-0 ; SP-1 ; SP-4 ;
SP-5 ; SP-6  and other cross-disciplinary fields such as chemical and
biological process ReuveniEnzyme1 ; bio-1 ; bio-2 ; bio-3 , computer science
Luby ; algorithm  and search theory HRS ; Montanari ; bressloff . Brownian
motion with resetting introduced by Evans and Majumdar in Restart1 ; Restart2
is the paradigmatic example of the subject which essentially captures two
central features: emergence of a non-equilibrium steady state with a non-zero
probability current Restart1 ; Restart2 ; Restart3 ; Restart4 ; Restart5 ;
Restart6 ; PalJphysA  and expedition of a first passage time process
PalReuveniPRL17 ; ReuveniPRL16 ; branching ; Belan ; Chechkin ; interval ;
Peclet ; space ; interval-v . Over the years, a large volume of work has been
done to extend this simple model beyond diffusion to others such as
underdamped underdamped  and scaled scaled  Brownian motion, random
acceleration process RAP , and active particles RTP-0 ; RTP-1 ; RTP-2 .
Rigorous efforts have also been made to understand non-Poissonian strategies
review ; PalJphysA ; PalReuveniPRL17 ; Chechkin . The subject has also found
interesting applications in stochastic thermodynamics sth-0 ; sth-1 ; sth-2 ,
quantum systems quantum-1 , many particle systems SEP ; TASEP ; Ising , and
nonlinear systems dynamical . Very recently, resetting has also seen advances
in single particle experiments using optical traps expt-1 ; expt-2 . We refer
to this recent review review  and references therein for more details on the
subject.

Notwithstanding that the subject has seen tremendous progress in statistical
physics, exact results on EVS of stochastic resetting have been very limited.
It has been understood that the maximum $M$ of a Brownian trajectory under
stochastic resetting (i.e., Poissonian resetting) belongs to the Gumbel class
for large sample size review ; Restart1 . But only recently exact expressions
for the first two moments of the maximum $M(t)$ have been obtained by Majumdar
et al in MajumdarMori2020 . Using these, the mean perimeter and the mean area
of a convex hull of the 2D Brownian motion with resetting were computed. On
the other hand, when resetting times are taken from a power law density (i.e.,
non-exponential or non-Poissonian resetting), it was shown that the
distribution of the maximum of the reset process is given by the Fréchet law
when appropriately centred and scaled Villarroel JStatMech . A Weibull limit
law was derived for first passage time under restart with branching branching
. But to the best of our knowledge, not much is known on the statistics for
the arg-max i.e., the time $t_{m}$ to reach this maximum which is also a very
important statistical measure (recall the famous arcsine law of Lévy in
classical probability theory Levy ). Also exact results for maximum beyond
simple diffusion are not available at this moment. This paper exactly aims to
bridge this void. First, we derive a renewal formula (5) for the joint
distribution of $M(t)$ and $t_{m}(t)$ in the presence of restart in terms of
their underlying joint distribution and other statistical quantities. This
formula is valid for any underlying stochastic process (Markov or non-Markov)
as long as the memory is erased after each resetting event. Secondly,
utilizing the renewal formula, we derive exact and asymptotic $n$-th order
moments of $M(t)$ for diffusion and drift-diffusion process. We also show the
convergence to the Gumbel limit law for both these cases. Third and
importantly, we obtain exact expression for the moment generating function for
$t_{m}$ from which we show that the density of $t_{m}$ pertains to a universal
$1/t$ form independent of the underlying process at large time with sub-
leading process dependent corrections. We first demonstrate this result
exactly for diffusion and drift-diffusion process and then generalize to
arbitrary stochastic process.

The remainder of the paper is structured as follows. We derive the joint
distribution of $M(t)$ and $t_{m}(t)$ in the Laplace space of $t~{}(\to s)$
and $t_{m}~{}(\to k)$ in Sec. II. This renewal formula becomes instrumental to
investigate the statistics of $M(t)$ for simple diffusion and drift-diffusion
process in Sec. III. In particular, we compute the exact moments for the
maximum $M(t)$ and then present the asymptotic limiting distributions. In Sec.
IV, we present our results for arg-max $t_{m}(t)$ for simple diffusion and
drift-diffusion process. We compute the moments and analyze the large time
behavior for the arg-max density. The large time limiting density is shown to
be universal which is further proven in Sec. V for generic stochastic process
augmented with numerical simulations. We summarize our results in Sec. VI. For
brevity, many supplemented derivations of our results have been reserved to
the Appendix.

##  II Renewal formula for JOINT DISTRIBUTION OF $M$ and $t_{m}$

We begin with the derivation of the joint distribution for $M$ and $t_{m}$ in
the presence of resetting. Consider a typical trajectory $x(\tau)$ of a
particle governed by some stochastic law of motion and observed upto a fixed
time $t$ (see Fig. 1). Motion of the particle is also subjected to resetting
that brings it back to the origin at a constant rate $r$. This essentially
means that the waiting time between any two resetting events is taken from the
distribution $p(\tau)=re^{-r\tau}$. Note that, in a fixed time window $t$, the
number of resetting undergone by the particle is a random variable and varies
from trajectory to trajectory. Let us assume that the trajectory is divided
into $N$-intervals out of which the particle has experienced $N-1$ resetting
events and further denote the waiting times in these intervals as $\tau_{i}$
with $i=1,2,...,N-1$. However, in the last intervals $\tau_{N}$, the particle
does not experience any resetting event: the probability of which is given by
$\int_{\tau_{N}}^{\infty}~{}p(\tau)d\tau=e^{-r\tau_{N}}$. Since the
observation time is fixed, $N$ is a random variable and varies from trajectory
to trajectory. Also, without any loss of generality, we assume the starting
point as a resetting event which gives $N\geq 1$.

Figure 1: Schematic of the maximum distance $M$ and the time $t_{m}$ to reach
$M$ for a trajectory $x(\tau)$ of a given stochastic process observed upto
time $t$. The vertical solid lines indicate the resetting events which take
place at times $\tau_{1},~{}\tau_{2},\cdots$ taken from a distribution
$p(\tau)$. The resetting coordinate is same as the initial condition which is
set to be the origin here. Total number of resetting intervals in this
realization is $4$, and the maximum occurs during the third interval.

We now proceed to compute the joint distribution of $M,~{}t_{m}$ and $N$ which
is denoted by $\mathcal{P}_{r}\left(M,t_{m},N|t\right)$. Note that the maximum
distance $M$ can be attained during any of the $N$ time intervals
$\\{\tau_{i}\\}=\\{\tau_{1},\tau_{2},...,\tau_{N}\\}$. For example, if the
maximum $M$ is attained at time $\tau$ during the the first interval then the
contribution to $\mathcal{P}_{r}\left(M,t_{m},N|t\right)$ is given by

$\displaystyle\mathcal{C}_{1}=$
$\displaystyle\left(\prod_{i=1}^{N}\int_{0}^{\infty}d\tau_{i}\right)~{}~{}\int_{0}^{\infty}~{}d\tau~{}P_{0}(M,\tau|\tau_{1})p(\tau_{1})$
$\displaystyle\times\left[\prod_{i=2}^{N-1}~{}S_{0}(0,\tau_{i}|M)p(\tau_{i})\right]\times\left[e^{-r\tau_{N}}S_{0}(0,\tau_{N}|M)\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\delta(\tau-
t_{m})~{}\delta(t-\sum_{i=1}^{N}\tau_{i}),$ (1)

where we introduce $P_{0}(M,t_{m}|t)$ as the joint distribution of $M$ and
$t_{m}$ for the underlying process (without resetting) till time $t$.
Moreover, $S_{0}(x_{0},\tau_{j}|M)$ is the probability that the particle
starting from $x_{0}$ always stays below $M$ upto time $\tau_{j}$ in the
absence of resetting. In other words, this is the survival probability that a
particle, starting from $x_{0}$, survives an absorbing boundary at $M(>x_{0})$
till time $\tau_{j}$. Rationale behind various terms in $\mathcal{C}_{1}$ can
be understood in the following way: the maximum $M$ is attained by the
underlying process in the first interval that lasts for time $\tau_{1}$ which
gives rise to $P_{0}(M,\tau|\tau_{1})p(\tau_{1})$ term in the first line of
Eq. (1). Essentially, in the remaining $N-1$ intervals, the trajectory always
stays below $M$. As a consequence, we get $N-1$ survival probabilities
appropriately weighted in time in the second line. Finally, the first
$\delta$-function asserts the condition $t_{m}=\tau$ which is true here while
the second one ensures that the total observation time is $t$.

Similarly, we can write the contribution to
$\mathcal{P}_{r}\left(M,t_{m},N|t\right)$ when the maximum $M$ is attained in
the second interval at time $\tau$ while the particle remains below $x=M$
during the other intervals. This contribution is given by

$\displaystyle\mathcal{C}_{2}=$
$\displaystyle\left(\prod_{i=1}^{N}\int_{0}^{\infty}d\tau_{i}\right)~{}\int_{0}^{\infty}~{}d\tau~{}P_{0}(M,\tau|\tau_{2})p(\tau_{2})$
$\displaystyle\times\left[\prod_{i=1,i\neq
2}^{N-1}~{}S_{0}(0,\tau_{i}|M)p(\tau_{i})\right]\times\left[e^{-r\tau_{N}}S_{0}(0,\tau_{N}|M)\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\times\delta(t-\sum_{i=1}^{N}\tau_{i})~{}\delta(\tau_{1}+\tau-
t_{m}).$ (2)

Following the same physical argument, one can also write the contributions
$\mathcal{C}_{3},\mathcal{C}_{4}..$ for the maximum to be in the second,
third,… interval respectively. In particular, when the maximum is in the last
reset-free interval $\tau_{N}$, we have

$\displaystyle\mathcal{C}_{N}=$
$\displaystyle\left(\prod_{i=1}^{N}\int_{0}^{\infty}d\tau_{i}\right)~{}\int_{0}^{\infty}~{}d\tau~{}P_{0}(M,\tau|\tau_{N})e^{-r\tau_{N}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left[\prod_{i=1}^{N-1}~{}S_{0}(0,\tau_{i}|M)p(\tau_{i})\right]~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\delta(t-\sum_{i=1}^{N}\tau_{i})~{}\delta(\sum_{i=1}^{N-1}\tau_{i}+\tau-
t_{m}).$ (3)

Thus, the joint distribution $\mathcal{P}_{r}\left(M,t_{m},N|t\right)$, can be
obtained by summing over all the contributions
$\mathcal{C}_{1},~{}\mathcal{C}_{2},...,\mathcal{C}_{N}$. Performing the sum,
$\mathcal{P}_{r}\left(M,t_{m},N|t\right)$ can be formally written as

$\displaystyle\mathcal{P}_{r}\left(M,t_{m},N|t\right)$
$\displaystyle=\frac{1}{r}\sum_{j=1}^{N}\int_{0}^{\infty}d\tau_{j}~{}d\tau~{}P_{0}\left(M,\tau|\tau_{j}\right)p(\tau_{j})$
$\displaystyle\times\left(\prod_{j^{\prime}=1,j^{\prime}\neq
j}^{N}\int_{0}^{\infty}d\tau_{j^{\prime}}S_{0}(0,\tau_{j^{\prime}}|M)p(\tau_{j^{\prime}})d\tau_{j^{\prime}}\right)$
$\displaystyle\times\delta\left(\sum_{i=1}^{j-1}\tau_{i}+\tau-
t_{m}\right)\delta(t-\sum_{i=1}^{N}\tau_{i}).$ (4)

To proceed further, it is only natural to take the Laplace transformations
with respect to $t~{}(\to s)$ and $t_{m}~{}(\to k)$. Denoting the Laplace
transform of $\mathcal{P}_{r}\left(M,t_{m},N|t\right)$ by
$\mathcal{Z}_{r}\left(M,k,N|s\right)$ and further performing the sum over all
the intervals $N$, we find (see Appendix A)

$\displaystyle\mathcal{Z}_{r}\left(M,k|s\right)$
$\displaystyle=\frac{\mathcal{Z}_{0}\left(M,k|r+s\right)}{\left[1-r\bar{S}_{0}(0,s+r|M)\right]~{}\left[1-r\bar{S}_{0}(0,s+r+k|M)\right]},$
(5)

where $\mathcal{Z}_{r}\left(M,k|s\right)$ is the Laplace transform of the
joint distribution $P_{r}(M,t_{m}|t)$ (subscript $0$ will indicate the same
without resetting) and
$\bar{S}_{0}(0,s|M)=\int_{0}^{\infty}~{}dt~{}e^{-st}S_{0}(0,t|M)$ is the
Laplace transformation of the underlying survival probability. Eq. (5) is the
first central result of our paper. Such a renewal formula is very important
since it relates the joint distribution of $M$ and $t_{m}$ with resetting to
the underlying joint distribution and the survival probabilities. For Markov
processes, one can also use the path-decomposition method tmax-1 ; tmax-2 ;
tmax-3 ; MajumdarMori2020  to arrive at Eq. (5). However, our derivation is
more robust since it holds even when underlying process is non-Markov while
the path-decomposition method strictly relies on the Markov property as was
illustrated for Brownian motion by Majumdar et al in MajumdarMori2020 . The
only assumption that goes into the derivation is that the process does not
retain memory between the resetting intervals. Finally, we remark that
although Eq. (5) has been derived under a one-dimensional framework, it is
valid also in higher dimensions. Also see Mori2020  where a similar approach
has been used to study the persistent properties of run and tumble particles
in arbitrary dimensions.

In what follows, we use the renewal formula in Eq. (5) to study statistics of
$M$ and $t_{m}$ for simple diffusion and drift-diffusion process in one
dimension, respectively described by

$\displaystyle\frac{dx}{d\tau}$ $\displaystyle=\eta(\tau)~{},$ (6)
$\displaystyle\frac{dx}{d\tau}$ $\displaystyle=v+\eta(\tau)~{},$ (7)

where $\eta(\tau)$ is the Gaussian white noise with mean
$\langle\eta(\tau)\rangle=0$ and correlation
$\langle\eta(\tau)\eta(\tau^{\prime})\rangle=2D\delta(\tau-\tau^{\prime})$,
and further assume the drift-velocity $v>0$. Here, $D$ is the diffusion
constant (which will be set to $\frac{1}{2}$ without any loss of generality
for the rest of the paper). We also consider that the particle starts from the
origin at $t=0$ and is reset to the origin at random times drawn from the
distribution $p(\tau)=re^{-r\tau}$.

Figure 2: Comparison of moments for the maximum $\langle M^{n}(t)\rangle$ in
Eq. (15) for the simple diffusion against the numerical simulations for (a)
$n=1,$ (b) $n=2$ and (c) $n=3$. We have fixed $r=1$ for all the simulations.

##  III Statistics of maximum $M$

In this section, we use Eq.(5) to analyse the statistical properties of $M$
for the above-mentioned stochastic processes. To compute the distribution of
$M$, one needs to integrate $P_{r}\left(M,t_{m}|t\right)$ over all $t_{m}$.
This is equivalent to putting $k=0$ in $\mathcal{Z}_{r}\left(M,k|s\right)$ in
Eq. (5). Let us consider the case of simple diffusion first.

###  III.1 Simple diffusion

It is instructive to first review some known results on $M$ and $t_{m}$
without resetting which will be useful for subsequent studies. For simple
diffusion, the joint distribution of $M$ and $t_{m}$ is given by tmax-CTRW

$\displaystyle P_{0}\left(M,t_{m}|t\right)=\frac{M}{\pi
t_{m}^{3/2}\sqrt{t-t_{m}}}e^{-\frac{M^{2}}{2t_{m}}}.$ (8)

Integrating over $M$, we get the distribution of $t_{m}$ as
$P_{0}(t_{m}|t)=\frac{1}{\pi\sqrt{t_{m}(t-t_{m})}}$ or equivalently the
cumulative distribution

$\displaystyle\text{Prob}[t_{m}\leq
T]=\int_{0}^{T}dt_{m}^{\prime}~{}P_{0}(t_{m}^{\prime}|t)=\frac{2}{\pi}\sin^{-1}\left[\sqrt{\frac{T}{t}}\right],$
(9)

which is the celebrated ‘Arc-sine’ law for the Brownian motion due to Lévy
Levy . To use Eq.(5), we need to specify the Laplace transformations
$\bar{S}_{0}(0,s|M)$ and $\mathcal{Z}_{0}\left(M,k|s\right)$. The latter can
be computed by taking the Laplace transformation of Eq. (8) with respect to
$t$ and $t_{m}$ and this gives

$\displaystyle\mathcal{Z}_{0}\left(M,k|s\right)=\sqrt{\frac{2}{s}}e^{-\sqrt{2(s+k)}M}.$
(10)

On the other hand, survival probability for a Brownian particle is a canonical
result due to Lévy which reads
$S_{0}(0,t|M)=\text{Erf}\left(\frac{M}{\sqrt{4Dt}}\right)$, where
$\text{Erf}(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}~{}e^{-y^{2}}dy$ is the error-
function. Taking the Laplace transform, one gets

$\displaystyle\bar{S}_{0}(0,s|M)=\frac{1}{s}\left(1-e^{-\sqrt{2s}M}\right).$
(11)

Inserting Eqs. (10) and (11) in the renewal Eq. (5) yields the Laplace
transformation $\mathcal{Z}_{r}\left(M,k|s\right)$ from which one gets

$\displaystyle\mathcal{Z}_{r}\left(M,k=0|s\right)=\frac{\sqrt{2}\left(r+s\right)^{3/2}~{}e^{-\sqrt{2(r+s)}M}}{\left(s+re^{-\sqrt{2(r+s)}M}\right)^{2}}.$
(12)

Note that this was also obtained in MajumdarMori2020  using the path
decomposition method. To get the distribution of $M$ in the time domain, one
has to perform the inverse Laplace transform of
$\mathcal{Z}_{r}\left(M,k=0|s\right)$ with respect to $s$. Before that, we
look at the moments of $M$ to get the effect of resetting on $M$. The $n$-th
moment can be written in terms of $\mathcal{Z}_{r}\left(M,k=0|s\right)$ as

$\displaystyle\int_{0}^{\infty}dt~{}e^{-st}\langle M^{n}(t)\rangle$
$\displaystyle=\int_{0}^{\infty}dM\mathcal{Z}_{r}\left(M,k=0|s\right),$
$\displaystyle=-\frac{n!}{2^{\frac{n}{2}}r(r+s)^{\frac{n-2}{2}}}~{}\text{Li}_{n}\left(-\frac{r}{s}\right),$
$\displaystyle=-\frac{n!}{2^{\frac{n}{2}}r(r+s)^{\frac{n-2}{2}}}~{}\sum_{k=1}^{\infty}\frac{(-r)^{n}}{k^{n}~{}s^{k+1}},$
(13)

where $\text{Li}_{n}\left(-\frac{r}{s}\right)$ is the PolyLog function in the
second line Table  and while going to the third line, we have used the series
representation
$\text{Li}_{n}\left(-y\right)=\sum_{k=1}^{\infty}\frac{(-y)^{k}}{k^{n}}$.
Next, to get the moments in the time domain, we use the following inverse
Laplace transformation:

$\displaystyle\mathcal{L}_{s\to
t}^{-1}\left[\frac{1}{(r+s)^{\frac{n-2}{2}}~{}s^{k+1}}\right]=\frac{{}_{1}\bar{F}_{1}\left(-1+\frac{n}{2},k+\frac{n}{2},-rt\right)}{t^{1-k-\frac{n}{2}}},$
(14)

where ${}_{1}\bar{F}_{1}(a,b,z)$ stands for the regularized hypergeometric
function Table . Inserting this in Eq. (13), we find that $\langle
M^{n}(t)\rangle$ possesses the scaling form

$\displaystyle\langle M^{n}(t)\rangle=\frac{1}{(\sqrt{2r})^{n}}H_{n}(rt),$
(15)

with the scaling function $H_{n}(z)$ given by

$\displaystyle
H_{n}(z)=(-1)^{-\frac{n}{2}}n!\sum_{k=1}^{\infty}\frac{~{}_{1}\bar{F}_{1}\left(-1+\frac{n}{2},k+\frac{n}{2},-z\right)}{k^{n}(-z)^{1-k-\frac{n}{2}}}.$
(16)

Note again that the first two moments $(n=1,2)$ were recently obtained in
MajumdarMori2020 . Our results are consistent with that in MajumdarMori2020
and further extend to obtain exact expressions for all moments of $M$. In
Figure 2, we have plotted $\langle M^{n}(t)\rangle$ for $n=1,2,3$ and compared
them against the numerical simulation. We observe excellent match between
them. To illustrate the effect of resetting, it is instructive to look at the
asymptotic behaviours of the scaling function $H_{n}(z)$ for which we analyse
$\mathcal{Z}_{r}\left(M,k=0|s\right)$ in Eq. (12) in various limits of
$sr^{-1}$ and then appropriately use Eq. (13) (first line) to obtain the
moments of $M$. For the continuity of the presentation, we have relegated this
derivation to appendix B and present only the final results here. The
asymptotic forms read

$\displaystyle H_{n}(z)$
$\displaystyle\simeq\log^{n}z+O\left(\log^{n-1}z\right),~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}z\to\infty,$ (17)
$\displaystyle\simeq\frac{n!}{\Gamma\left(1+\frac{n}{2}\right)}z^{\frac{n}{2}}+\mathcal{B}_{n}z^{\frac{n}{2}+1},~{}~{}~{}~{}~{}\text{as
}z\to 0,$ (18)

where the exact expression of $\mathcal{B}_{n}$ is given in Eq. (89). Finally,
we insert Eqs. (17) and (18) in Eq. (15) to get the behaviour of $\langle
M^{n}(t)\rangle$ at large and short times as

$\displaystyle\langle M^{n}(t)\rangle$
$\displaystyle\simeq\frac{\log^{n}rt}{(\sqrt{2r})^{n}}+O\left[\log^{n-1}rt\right],~{}~{}~{}~{}\text{as
}t\gg\frac{1}{r},$ (19)
$\displaystyle\simeq\frac{n!~{}t^{\frac{n}{2}}}{2^{\frac{n}{2}}\Gamma\left(1+\frac{n}{2}\right)}+\frac{\mathcal{B}_{n}rt^{\frac{n}{2}+1}}{2^{n/2}},~{}~{}~{}\text{as
}t\ll\frac{1}{r}.$ (20)

For $r=0$, the moments in Eq.(20) match, as expected, with that of the free
Brownian motion tmax-CTRW . It is worth noting that there is a crossover in
$\langle M^{n}(t)\rangle$ from $\sim t^{n/2}$ behaviour to $\sim\log^{n}rt$
behaviour at time scale $t\sim\frac{1}{r}$ with the crossover function given
exactly in Eq. (16). Although a diffusing particle approaches a non-
equilibrium steady state in the presence of resetting Restart1 , the maximum
$M(t)$ still increases with time but rather slowly (with logarithmic growth)
for $t\gg\frac{1}{r}$. This same observation was also made recently by
Majumdar et al in MajumdarMori2020 . The logarithmic growth of the maximum in
presence of resetting can also be understood heuristically from the extreme
value statistics of weakly correlated variables which we illustrate later.

After looking at the moments, we now consider the distribution of $M$ for
which we have to invert $\mathcal{Z}_{r}\left(M,k=0|s\right)$ in Eq. (12) with
respect to $s$. Performing inversion for arbitrary $t$ turns out to be
challenging. However for large $t$ (or equivalently small $s$), we can make
some analytic progress. For $s\ll r$, we approximate $r+s\simeq r$ and use
this in Eq. (12) to get

$\displaystyle\mathcal{Z}_{r}\left(M,k=0|s\right)\simeq\frac{\sqrt{2}r^{3/2}~{}e^{-\sqrt{2r}M}}{\left(s+re^{-\sqrt{2r}M}\right)^{2}}.$
(21)

To get the distribution in the time domain, we use the inverse Laplace
transform $\mathcal{L}_{s\to t}^{-1}\left[\frac{1}{(s+a)^{2}}\right]=te^{-at}$
for $a\geq 0$ in Eq. (21) which yields

$\displaystyle
P_{r}(M|t)\simeq\sqrt{2r^{3}}~{}te^{-\sqrt{2r}M}\text{exp}\left(-rte^{-\sqrt{2r}M}\right).$
(22)

We emphasize that the approximate equality in Eq. (22) indicates that this
equation is valid only for $t\gg\frac{1}{r}$ when the effect of resetting is
highest. In Fig. 3 (top panel), we have compared $P_{r}(M|t)$ in Eq. (22) with
the same obtained from the numerical simulations. We observe an excellent
agreement between them. To understand Eq. (22) heuristically, we remark that
the survival probability $S_{r}(0,t|M)$ for Brownian motion under reset
possesses the Gumbel form at large times. Based on the extreme value
statistics of weakly correlated variables, it was shown that
$S_{r}(0,t|M)\simeq e^{-rte^{-\sqrt{2r}M}}$ for $t\gg\frac{1}{r}$ Restart1 .
Since $S_{r}(0,t|M)$ is the cumulative distribution of $M$, one can
appropriately differentiate it with respect to $M$ to get the distribution
$P_{r}(M|t)$ in Eq. (22). Furthermore, one can easily check that that using
$P_{r}(M|t)$ from Eq. (22), we get the same form of moments at large time as
given in Eq. (19).

Figure 3: Top panel: plot for the distribution of maximum $P_{r}(M|t)$ in Eq.
(22) (in solid line) for the simple diffusion against the numerical
simulations (in marker). For this plot, we fix: $r=2$ and $t=20$. Bottom
panel: comparison between the asymptotic form of $P_{r}(t_{m}|t)$ in Eq. (49)
for the simple diffusion with with the numerical distribution. Since the
theory works only in the large $t_{m}$ limit, we see a deviation when $t_{m}$
is small. Parameters set for this plot are: $r=1,~{}t=5$.

Figure 4: Comparison of $\langle M^{n}(t)\rangle$ in Eq. (29) for the drift-
diffusion process against the numerical simulations for (a) $n=1,$ (b) $n=2$
and (c) $n=3$. For all the plots, we have chosen $r=1,~{}v=0.5$.

###  III.2 Diffusion with drift

We now consider a particle diffusing in one dimension in presence of a
constant drift $v~{}(\geq 0)$. Herein, our aim is to analyze the statistics of
$M$ for this process with dynamics in Eq. (7). To this aim, we begin with
$\mathcal{Z}_{r}\left(M,k|s\right)$ in Eq. (5) for which we need the following
two quantities tmax-3 :

$\displaystyle\bar{S}_{0}(0,s|M)=\frac{1}{s}\left(1-e^{-M\left(\sqrt{v^{2}+2s}-v\right)}\right),$
(23)
$\displaystyle\mathcal{Z}_{0}\left(M,k|s\right)=\frac{\sqrt{v^{2}+2s}-v}{s}e^{-M\left(\sqrt{v^{2}+2(s+k)}-v\right)}.$
(24)

Substituting these two equations in Eq. (5) gives us
$\mathcal{Z}_{r}\left(M,k|s\right)$ which can then be suitably used to compute
the distribution of $M$ and $t_{m}$.

Let us first look at the statistics of $M$ for drifted diffusion for which we
put $k=0$ in $\mathcal{Z}_{r}\left(M,k|s\right)$ in Eq. (5) along with
$\bar{S}_{0}(0,s|M)$ and $\mathcal{Z}_{0}\left(M,k|s\right)$ in Eqs. (23) and
(24) respectively. This yields

$\displaystyle\mathcal{Z}_{r}\left(M,k=0|s\right)$
$\displaystyle=\frac{(s+r)[\sqrt{v^{2}+2(s+r)}-v]}{[s+re^{-M\left(\sqrt{v^{2}+2(s+r)}-v\right)}]^{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}\times
e^{-M\left(\sqrt{v^{2}+2(s+r)}-v\right)}.$ (25)

As done for the Brownian motion, we first analyse the moments of $M(t)$
followed by the distribution. Using Eq. (25), it is easy to see that the
$n$-th order moment is given by

$\displaystyle\int_{0}^{\infty}dte^{-st}\langle M^{n}(t)\rangle$
$\displaystyle=\int_{0}^{\infty}dM\mathcal{Z}_{r}\left(M,k=0|s\right),$ (26)
$\displaystyle=-\frac{n!(s+r)~{}\text{Li}_{n}\left(-\frac{r}{s}\right)}{sr[\sqrt{v^{2}+2(s+r)}-v]^{n}}~{},$
(27)
$\displaystyle=-\frac{n!(s+r)~{}\sum_{k=1}^{\infty}\frac{(-r)^{k}}{k^{n}~{}s^{k}}}{sr[\sqrt{v^{2}+2(s+r)}-v]^{n}},$
(28)

where $\text{Li}_{n}\left(-\frac{r}{s}\right)$ is the PolyLog function and its
series representation
$\text{Li}_{n}\left(-y\right)=\sum_{k=1}^{\infty}\frac{(-y)^{k}}{k^{n}}$ has
been used in Eq. (27). The Laplace transform in Eq. (28) can be inverted
exactly as illustrated in the appendix F. The $n$-th moment obeys the scaling
form

$\displaystyle\langle
M^{n}(t)\rangle=\left(\frac{t}{2}\right)^{n/2}\mathbb{H}_{n}\left(rt,-\frac{v\sqrt{t}}{\sqrt{2}}\right),$
(29)

where the scaling function $\mathbb{H}_{n}(z,y)$ is given by

$\displaystyle\mathbb{H}_{n}(z,y)=\frac{(-1)^{n}n}{z}\sum_{k=1}^{\infty}\frac{(-z)^{k}}{k^{n}}\int_{0}^{1}dwe^{-\left(z+y^{2}\right)w}\frac{(1-w)^{k-1}}{\Gamma(k)}$
$\displaystyle\left[1+z\frac{1-w}{k}\right]\left[\frac{\delta_{n,1}}{\sqrt{\pi
w}}-\frac{d^{n-1}}{dy^{n-1}}\left(ye^{y^{2}w}\text{Erfc}(y\sqrt{w})\right)\right],$
(30)

where $\text{Erfc}(z)=1-\text{Erf}(z)$ is the complementary error function.
The scaling function $\mathbb{H}_{n}(z,y)$ can be simplified further for some
values of $n$. For example, one can perform the summation over $k$ for $n=1$
to get

$\displaystyle\mathbb{H}_{1}(z,y)$
$\displaystyle=\frac{|y|}{z}\left[\gamma_{E}+\Gamma\left(0,z\right)+\log
z\right]$
$\displaystyle+\int_{0}^{1}\frac{dw}{z}\frac{\mathcal{J}_{w}(z,y)}{(1-w)}\left[1-e^{(1-w)z}\right],~{}~{}~{}\text{with
}$ (31) $\displaystyle\mathcal{J}_{w}(z,y)$
$\displaystyle=\frac{e^{-(z+y^{2})w}}{\sqrt{\pi
w}}+\sqrt{z+y^{2}}~{}\text{Erf}\left(\sqrt{(z+y^{2})w}\right).$ (32)

However, for arbitrary $n$, the scaling function is given by Eq. (30). In
Figure 4, we have plotted the first three moments of $M(t)$ and also compared
with the numerical simulations to find an excellent match. Next, we look at
the asymptotic forms of $\langle M^{n}(t)\rangle$ to study the effect of
resetting. For small $z$, we consider Eq. (30) and perform a direct expansion
in $z$ to obtain the behaviour of $\mathbb{H}_{n}(z,y)$. On the other hand,
for large $z$, the scaling function goes as
$\mathbb{H}_{n}(z,y)\simeq\log^{n}z$ and thus, the asymptotic forms read

$\displaystyle\mathbb{H}_{n}(z,y)$ $\displaystyle\simeq
n(-1)^{n}\mathbb{C}_{n}(y)+O(z),~{}~{}~{}~{}~{}\text{as }z\to 0,$ (33)
$\displaystyle\simeq\frac{\log^{n}z}{\left(\sqrt{y^{2}+z}-y\right)^{n}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}z\to\infty.$ (34)

The function $\mathbb{C}_{n}(y)$ in Eq. (33) is given by

$\displaystyle\mathbb{C}_{n}(y)$
$\displaystyle=\int_{0}^{1}dwe^{-y^{2}w}\frac{d^{n-1}}{dy^{n-1}}\left[ye^{y^{2}w}\text{Erfc}(y\sqrt{w})\right],$
(35)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}n\neq 1,$
$\displaystyle=y~{}\text{Erfc}(y)-\frac{e^{-y^{2}}}{\sqrt{\pi}}-\frac{\text{Erf}(y)}{2y},~{}~{}~{}\text{for
}n=1.$ (36)

Inserting the forms of $\mathbb{H}_{n}(z,y)$ from Eqs. (33) and (34) in Eq.
(29), we find that $\langle M^{n}(t)\rangle$ has the following asymptotic
forms

$\displaystyle\langle M^{n}(t)\rangle$
$\displaystyle=(-1)^{n}n\mathbb{C}_{n}\left(-v\sqrt{\frac{t}{2}}\right)t^{n/2}+O(rt),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t\ll r^{-1},$ (37)
$\displaystyle=\frac{\log^{n}rt}{\left(\sqrt{v^{2}+2r}-v\right)^{n}}+O\left(\log^{n-1}rt\right),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t\gg r^{-1}.$ (38)

At leading order, $\langle M^{n}(t)\rangle$ for $t\ll r^{-1}$ matches with the
drift-diffusion without resetting tmax-3 . Also, for $t\gg r^{-1}$ we find
that the maximum scales logarithmically with time as $M(t)\sim\log(rt)$. Thus
even though the position density reaches a non-equilibrium steady state at
late times Restart4 , the maximum $M(t)$ keeps growing albeit slowly in
logarithmical scale. This behavior can also be understood heuristically from
the extreme value statistics. To illustrate this, we look at the distribution
of $M(t)$ at $t\gg r^{-1}$ for which we analyse
$\mathcal{Z}_{r}\left(M,k=0|s\right)$ in Eq. (25) for $r\gg s$. Approximating
$s+r\simeq r$ in Eq. (25), we get

$\displaystyle\mathcal{Z}_{r}\left(M,k=0|s\right)\simeq$
$\displaystyle\frac{r[\sqrt{v^{2}+2r}-v]e^{-M\left(\sqrt{v^{2}+2r}-v\right)}}{[s+re^{-M\left(\sqrt{v^{2}+2r}-v\right)}]^{2}}.$
(39)

Performing the inverse Laplace transformation with respect to $s$ gives the
distribution of $M$ at large $t$ as

$\displaystyle P_{r}(M|t)=rt\alpha
e^{-M\alpha}~{}\text{exp}\left(-rte^{-M\alpha}\right),~{}~{}~{}~{}\text{for
}t\gg r^{-1},$ (40)

which is a Gumbel distribution and $\alpha=\sqrt{v^{2}+2r}-v$. In Fig. 5 (top
panel), we have compared the distribution $P_{r}(M|t)$ in Eq. (40) with the
same obtained from the numerical simulations. We observe an excellent
agreement. Note that using this form of $P_{r}(M|t)$, it is straightforward to
reproduce the $n$-th moment as given in Eq. (38). Again, the appearance of the
Gumbel distribution for $M(t)$ in Eq. (40) can be understood from the EVS as
done for the simple diffusion.

##  IV Statistics of arg-max $t_{m}$

In this section, we will present the results for statistics of the arg-max
i.e., the time $t_{m}$ at which the maximum $M$ occurs. The starting point
would be again to consider the joint distribution $\mathcal{Z}_{r}(M,k|s)$ of
$M$ and $t_{m}$ in Laplace space given in Eq. (5). Next, we would integrate
out $M$ to obtain an expression for the marginal distribution
$\mathcal{Z}_{r}(k|s)$. We will analyze this quantity to characterize $t_{m}$
for simple diffusion and then for drift-diffusion process.

###  IV.1 Simple diffusion

We now look at the statistical properties of $t_{m}$ for Brownian motion with
resetting. For free Brownian motion, the distribution of $t_{m}$ is
$P_{0}(t_{m}|t)=\frac{1}{\pi\sqrt{t_{m}(t-t_{m})}}$ whose cumulative exhibits
the ‘Arc-sine’ law (see Eq. (9)). To investigate how the statistical
properties are influenced due to resetting, we first consider
$\mathcal{Z}_{r}\left(M,k|s\right)$ in Eq. (5) and integrate it over $M$. The
resultant function
$\mathcal{Z}_{r}(k|s)=\int_{0}^{\infty}dM\mathcal{Z}_{r}\left(M,k|s\right)$
gives the double Laplace transformation of the distribution $P_{r}(t_{m}|t)$
with respect to $t_{m}~{}(\to k)$ and $t~{}(\to s)$. Inserting
$\mathcal{Z}_{0}\left(M,k|r+s\right)$ and $\bar{S}_{0}(0,s|M)$ from Eqs. (10)
and (11) in $\mathcal{Z}_{r}\left(M,k|s\right)$ in Eq. (5) and performing the
integration over $M$, we get

$\displaystyle\mathcal{Z}_{r}\left(k|s\right)=$
$\displaystyle\int_{0}^{\infty}dM\left[\frac{\sqrt{2(r+s)}e^{-\sqrt{2(r+s+k)}M}}{\left(s+re^{-\sqrt{2(r+s)}M}\right)}\right.$
$\displaystyle\left.\times\frac{(r+s+k)}{\left(s+k+re^{-\sqrt{2(r+s+k)}M}\right)}\right].$
(41)

Doing a change of variable: $w=e^{-\sqrt{2(r+s+k)}M}$ in the above equation
gives

$\displaystyle\mathcal{Z}_{r}\left(k|s\right)=\int_{0}^{1}dw\frac{\sqrt{(r+s)(r+s+k)}}{\left(s+rw^{\phi}\right)\left(s+k+rw\right)},$
(42)

where $\phi=\sqrt{\frac{r+s}{r+s+k}}$. We next use
$\mathcal{Z}_{r}\left(k|s\right)$ from Eq. (42) to compute the moments of
$t_{m}$ explicitly at all time. The $n$-th order moment $\langle
t_{m}^{n}(t)\rangle$ can be written in terms of
$\mathcal{Z}_{r}\left(k|s\right)$ as

$\displaystyle\int_{0}^{\infty}dt~{}e^{-st}~{}\langle
t_{m}^{n}(t)\rangle=(-1)^{n}\left[\frac{d^{n}}{dk^{n}}\mathcal{Z}_{r}\left(k|s\right)\right]_{k=0}.$
(43)

Inserting $\mathcal{Z}_{r}\left(k|s\right)$ from Eq. (42) in Eq. (43), one can
analyse the individual moments of $t_{m}(t)$ although getting a closed form
for the $n$-th order moment turns out to be difficult. Below, we provide the
exact expression for the first two moments of $t_{m}$:

$\displaystyle\langle t_{m}(t)\rangle$
$\displaystyle=\frac{1}{4r}\left[2rt+e^{-rt}-1+\gamma_{E}+\Gamma\left(0,rt\right)+\log(rt)\right],$
(44) $\displaystyle\langle t_{m}^{2}(t)\rangle$
$\displaystyle=\frac{1}{24r^{2}}\Bigg{[}e^{-rt}\left(9+5rt\right)-9+4rt\left(1+2rt\right)+4-3\gamma_{E}-3\Gamma\left(0,rt\right)-3\log\left(rt\right)\Bigg{.}$
$\displaystyle\left.+4rt\Big{\\{}-1+\gamma_{E}+\Gamma\left(0,rt\right)+\log(rt)\Big{\\}}+e^{-rt}\Big{\\{}-4-3\gamma_{E}+3\text{Chi}(rt)+3\text{Shi}(rt)-3\log(rt)\Big{\\}}\right.$
$\displaystyle\Bigg{.}+2e^{-rt}\int_{0}^{rt}~{}dy~{}y~{}e^{y}~{}_{3}F_{3}\Big{(}\\{1,1,1\\},\\{2,2,2\\},\\{-rt\\}\Big{)}\Bigg{]},$
(45)

where $\text{Chi}(z)$ and $\text{Shi}(z)$ stands for hyperbolic cosine
integral and hyperbolic sine integral respectively and
$~{}_{3}F_{3}\Big{(}\\{1,1,1\\},\\{2,2,2\\},\\{z\\}\Big{)}$ is the generalised
hypergeometric function Table . Expressions of $\langle t_{m}(t)\rangle$ and
$\langle t^{2}_{m}(t)\rangle$ are derived in Appendix C. In Figure 6, we have
plotted the first two moments of $t_{m}(t)$ and compared them against the same
obtained from the numerical simulations. An excellent match is observed.

Figure 5: Top panel: comparison of the distribution $P_{r}(M|t)$ in Eq. (40)
for the drift-diffusion process against the numerical simulations. For this
plot, we have taken: $r=2,~{}v=1$ and $t=20$. Bottom panel: here we have
compared the distribution $P_{r}(t_{m}|t)$ in Eq. (61) for the drift-diffusion
to the numerical simulations. As before, the data fits the theory perfectly in
the large $t_{m}$ limit. Parameters chosen for this simulation: $r=1.5$ and
$t=5$.

To analyse the consequences of resetting, we study the moments of $t_{m}(t)$
at short and large times. It turns out that for these times, one can write a
closed form for the $n$-th order moment. Starting from
$\mathcal{Z}_{r}\left(k|s\right)$ in Eq. (42), we perform simplifications for
the cases $sr^{-1}\to 0$ and $sr^{-1}\to\infty$ which in the time domain
correspond to $t\gg r^{-1}$ and $t\ll r^{-1}$ respectively. We then use Eq.
(43) to compute the moments for these cases. In order to avoid discontinuity
of the presentation, we have relegated these detailed calculations to appendix
D and present only the final results here. The moments read

$\displaystyle\langle t_{m}^{n}(t)\rangle$
$\displaystyle\simeq\frac{(2n-1)!t^{n}}{n!(n-1)!2^{2n-1}}+\mathcal{A}_{n}\gamma
t^{n+1},~{}\text{for }t\ll\frac{1}{r}$ (46)
$\displaystyle\simeq\frac{t^{n}}{n+1}+\frac{t^{n-1}\log(rt)}{2r(n+1)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t\gg\frac{1}{r},$ (47)

where $\mathcal{A}_{n}$ in the first line is given in Eq. (109). For
$t\ll\frac{1}{r}$, we recover the results of the free Brownian motion Levy .
Moreover, we observe that the leading order scaling of the $n$-th order moment
with respect to $t$ for both large and small times are same i.e. $\langle
t_{m}^{n}(t)\rangle\sim t^{n}$. However, the corresponding prefactors are
different. As we illustrate later that the leading order behaviour of $\langle
t_{m}^{n}(t)\rangle$ at $t\gg\frac{1}{r}$ is independent of the underlying
stochastic process as long as every reset event renews the process.
Contrarily, the sub-leading terms are sensitive to the underlying stochastic
process and resetting rate.

In the remaining part of this section, we analyse the distribution of $t_{m}$
which we denote by $P_{r}(t_{m}|t)$. To compute this distribution, we have to
perform the double inverse Laplace transformation of
$\mathcal{Z}_{r}\left(k|s\right)$ in Eq. (42) with respect to $k$ and $s$.
Performing inversion for arbitrary values of $s$ and $k$ turns out to be
difficult. However, one can make some analytic progress by analysing
$\mathcal{Z}_{r}\left(k|s\right)$ in various limits of $kr^{-1}$ and
$sr^{-1}$. For $r\gg s$ and $r\gg k$, we approximate $\phi\simeq 1$ in the
expression of $\mathcal{Z}_{r}\left(k|s\right)$ in Eq. (42) to get

$\displaystyle\mathcal{Z}_{r}\left(k|s\right)\simeq\int_{0}^{1}dw\frac{\sqrt{(r+s)(r+s+k)}}{\left(s+rw\right)\left(s+k+rw\right)}.$
(48)

Fortunately, one can now invert this double Laplace transform to get the
distribution $P_{r}(t_{m}|t)$ [see appendix E]. Performing the inverse Laplace
transformation, we get

$\displaystyle P_{r}(t_{m}|t)$ $\displaystyle\simeq
e^{-rt}\frac{\left[g(rt_{m})+g(r(t-t_{m}))-1\right]}{\pi\sqrt{t_{m}(t-t_{m})}}$
$\displaystyle+re^{-rt}\int_{0}^{1}~{}dw~{}w~{}e^{rtw}~{}\text{Erf}\left(\sqrt{rwt_{m}}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\text{Erf}\left(\sqrt{rw(t-t_{m})}\right),$
(49)

where $g(z)=~{}_{2}F_{2}\left(\\{1,1\\},\\{1/2,2\\},z\right)$. Note that this
expression is valid only in the limits $rt\to\infty$ and $rt_{m}\to\infty$. In
Fig. 3 (bottom panel), we have plotted $P_{r}(t_{m}|t)$ and compared it
against the numerical simulations. While our analytic result is consistent
with the simulation data at large $t_{m}$, the match is poor at small $t_{m}$.
This deviation stems from the fact that $\mathcal{Z}_{r}\left(k|s\right)$ in
Eq. (48) is valid only for $r\gg k$ which translates to $t_{m}\gg r^{-1}$ in
the time domain.

We now look at $P_{r}(t_{m}|t)$ when $t_{m}\neq t$ for which Eq. (49) can be
simplified further. Approximating
${}_{2}F_{2}\left(\\{1,2\\},\\{1/2,2\\},z\right)\simeq\sqrt{\frac{\pi}{z}}e^{z}$
and $\text{Erf}(z)\simeq 1$ for $z\to\infty$ and using them in (49), we find
that the leading order behaviour of $P_{r}(t_{m}|t)\simeq\frac{1}{t}$. To get
the sub-leading terms of $P_{r}(t_{m}|t)$, one has to consider higher order
terms on $rs^{-1}$ while performing simplifications in
$\mathcal{Z}_{r}\left(k|s\right)$ in Eq. (48). We refer to appendix D [see Eq.
(113)] for the derivation of the sub-leading corrections and present the
results here such that

$\displaystyle P_{r}(t_{m}|t)\simeq\frac{1}{t}+\frac{\log(rt)}{2rt^{2}}.$ (50)

Using this form of $P_{r}(t_{m}|t)$, it is easy to verify that the $n$-th
order moment of $t_{m}(t)$ is indeed given by Eq. (47). It is worth remarking
that the $\frac{1}{t}$ form of $P_{r}(t_{m}|t)$ is quite different than the
form of distribution of $t_{m}$ without reset which is given by
$P_{0}(t_{m}|t)=\frac{1}{\pi\sqrt{t_{m}(t-t_{m})}}$. We later show that
$\frac{1}{t}$ form of $P_{r}(t_{m}|t)$ under resetting is independent of the
underlying stochastic process as long as the process forgets about its prior
history after every reset event.

###  IV.2 Diffusion with drift

We start again by inserting $\mathcal{Z}_{0}(M,k|r+s)$ and
$\bar{S}_{0}(0,s|M)$ for drift diffusion process from Eqs. (23)-(24) in Eq.
(5) to obtain the following expression for $\mathcal{Z}_{r}(M,k|s)$ given by

$\displaystyle\mathcal{Z}_{r}\left(M,k|s\right)=\frac{\mathcal{B}_{s+r}(r+s+k)~{}e^{-M\mathcal{B}_{s+r+k}}}{\left[s+re^{-M\mathcal{B}_{s+r}}\right]\left[s+k+re^{-M\mathcal{B}_{s+k+r}}\right]},$
(51)

where $\mathcal{B}_{s}=\sqrt{v^{2}+2s}-v$. Now performing the integration over
$M$ and after some algebraic simplifications, we find

$\displaystyle\mathcal{Z}_{r}\left(k|s\right)$
$\displaystyle=\frac{\sqrt{v^{2}+2(s+r)}-v}{\sqrt{v^{2}+2(s+k+r)}-v}$
$\displaystyle\times\int_{0}^{1}~{}dw~{}\frac{(r+s+k)}{(s+rw^{\phi_{d}})(s+k+rw)}~{},$
(52)

with $\phi_{d}=\frac{\sqrt{v^{2}+2(s+r)}-v}{\sqrt{v^{2}+2(s+k+r)}-v}$. We can
now use Eq. (43) to get the expression for the moments in the Laplace space.
However, the inversion process to obtain the moments in real time becomes
quite tedious. Here, we just present the first moment for brevity. The first
moment in Laplace space reads (using Eq. (52) in Eq. (43))

Figure 6: Comparison of the first two moments of $t_{m}$ in Eqs. (44) and (45)
against the same obtained from the numerical simulations for the simple
diffusion. We have set $r=1$ for the simulation purpose.

$\displaystyle\int_{0}^{\infty}~{}dt~{}e^{-st}\langle
t_{m}(t)\rangle=\tilde{I}_{1}(s)+\tilde{I}_{2}(s),~{}~{}~{}~{}\text{with}$
(53)
$\displaystyle\tilde{I}_{1}(s)=\frac{2r+s+s\frac{v}{\sqrt{v^{2}+2(r+s)}}}{4s^{2}(r+s)},$
(54)
$\displaystyle\tilde{I}_{2}(s)=\left[1+\frac{v}{\sqrt{v^{2}+2(r+s)}}\right]\frac{1}{4rs}\log\left(\frac{r+s}{s}\right).$
(55)

To get $\langle t_{m}(t)\rangle$, we use the following inverse Laplace
transformation:

$\displaystyle I_{1}(t)$ $\displaystyle=\mathcal{L}_{s\to
t}^{-1}\left[\tilde{I}_{1}(s)\right]=\frac{1}{4r}\left(2rt+e^{-rt}\text{Erfc}\left[\frac{v\sqrt{t}}{\sqrt{2}}\right]+\frac{v}{\sqrt{v^{2}+2r}}\text{Erf}\left[\frac{\sqrt{(v^{2}+2r)t}}{\sqrt{2}}\right]-1\right),$
(56) $\displaystyle I_{2}(t)$ $\displaystyle=\mathcal{L}_{s\to
t}^{-1}\left[\tilde{I}_{1}(s)\right]=J(t)+\int_{0}^{t}~{}d\tau~{}J(\tau)~{}\frac{ve^{-\frac{v^{2}+2r}{2}(t-\tau)}}{\sqrt{2\pi(t-\tau)}},$
(57) $\displaystyle\text{where~{}~{}}J(t)$
$\displaystyle=\frac{1}{4r}\left[\gamma_{E}+\Gamma\left(0,rt\right)+\log(rt)\right].$
(58)

Inserting these inverse Laplace transforms in Eq. (53), we get

$\displaystyle\langle t_{m}(t)\rangle=I_{1}(t)+I_{2}(t).$ (59)

One can also proceed to compute the higher moments in a similar manner, but
the expressions are quite involved. Hence, we do not present them here. In
Figure 7, we have compared $\langle t_{m}(t)\rangle$ in Eq. (59) with the
numerical simulations to find a perfect agreement between them.

We now turn our attention to analyze $P_{r}(t_{m}|t)$ for the drift-diffusion
process. Similar to the simple diffusion, performing an inversion for
arbitrary values of $s$ and $k$ turns out to be difficult. Thus, we make the
approximations $r\gg s$ and $r\gg k$ to have $\phi_{d}\simeq 1$, and Eq. (52)
simplifies to

$\displaystyle\mathcal{Z}_{r}\left(k|s\right)$
$\displaystyle\simeq\frac{\sqrt{v^{2}+2(s+r)}-v}{\sqrt{v^{2}+2(s+k+r)}-v}~{}$
$\displaystyle\times\int_{0}^{1}~{}dw~{}\frac{(r+s+k)}{\left(s+rw\right)\left(s+k+rw\right)}.$
(60)

It is possible to perform the double Laplace inversion (see Appendix G) to
eventually arrive at

$\displaystyle
P_{r}(t_{m}|t)\simeq\mathbb{I}_{1}(t_{m},t)+\mathbb{I}_{2}(t_{m},t)+\mathbb{I}_{3}(t_{m},t),$
(61)

where $\mathbb{I}$-functions are given in Eqs. (132-134). Note again that this
expression is only valid in the limits of $rt\to\infty$ and $rt_{m}\to\infty$.
We verify this result in Fig 5 (bottom panel). One can again approximate the
above expression in large time (like we have shown in the case of simple
diffusion) to find the leading order behavior:
$P_{r}(t_{m}|t)\simeq\frac{1}{t}$, which is again independent of the process
details. However, one would expect that there will be some correction (sub-
leading) terms to this leading behavior. To find them, we first perform the
small $k$ expansion of $\phi_{d}$ to obtain

$\displaystyle\phi_{d}$ $\displaystyle\simeq$ $\displaystyle 1-b(r,v)k~{},$
$\displaystyle~{}\text{with}~{}b(r,v)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{v^{2}+2r}(\sqrt{v^{2}+2r}-v)}.$ (62)

Further, we approximate $w^{\phi_{d}}=e^{\phi_{d}\log w}\simeq e^{(1-bk)\log
w}\simeq w(1-bk\log w)$. Moreover, we also take the large $t$, large $t_{m}$
limit so that $s+r\simeq r$ and $s+r+k\simeq r$ and thus

$\displaystyle\mathcal{Z}_{r}\left(k|s\right)=r~{}\int_{0}^{1}~{}\frac{dw}{(s+rw-
rbwk\log w)(s+k+rw)}.$ (63)

We now first take the inverse Laplace transform with respect to $k\to t_{m}$,
and then we take the inverse Laplace transform with respect to $s\to t$ to
find

$\displaystyle P_{r}(t_{m}|t)=\int_{0}^{1}~{}dw\frac{re^{-rtw}}{1+brw\log
w}\left[1-\Theta\left(t+\frac{t_{m}}{brw\log w}\right)\right].$ (64)

Let us now do the following transform $rtw=y$ and then we have

$\displaystyle w\log w=\frac{y}{rt}\log\frac{y}{rt}\simeq-\frac{y}{rt}\log
rt~{}~{}~{}~{}\text{for}~{}~{}~{}rt\gg 1~{}.$ (65)

The argument inside $\Theta$ function in Eq. (64) becomes negative, and hence
this term does not contribute. Eventually, we are left with

$\displaystyle
P_{r}(t_{m}|t)=r~{}\int_{0}^{rt}~{}\frac{dy}{rt}~{}\frac{e^{-y}}{1-a(r,v)y}~{},$
(66)

where $a(r,v)=b(r,v)r~{}\frac{\log rt}{rt}\ll 1~{}~{}\text{when~{}~{}}rt\gg
1$. Expanding in small $a$ argument and also setting the upper limit to
$+\infty$, we get

$\displaystyle P_{r}(t_{m}|t)\simeq$
$\displaystyle\frac{1}{t}+\frac{a(r,v)}{t},$ $\displaystyle\simeq$
$\displaystyle\frac{1}{t}+b(r,v)~{}\frac{\log rt}{t^{2}},$ (67)

where $b(r,v)$ is given by Eq. (62). Comparing this expression with that of
the simple diffusion in Eq. (50), we see that even though the leading order
behaviour of $P_{r}(t_{m}|t)$ in both cases is $1/t$, the sub-leading terms
are rather different. The subleading term in Eq. (67) depends on the drift $v$
(limit for simple diffusion can be checked easily by noting $b(r,v)=1/2r$).
Finally, the limiting distribution in Eq. (67) also gives the moments at large
$t$ which read

$\displaystyle\langle
t_{m}^{n}\rangle\simeq\frac{t^{n}}{1+n}+\frac{b(r,v)t^{n-1}\log
rt}{r(1+n)}~{}~{},~{}~{}t\gg\frac{1}{r},$ (68)

where again the logarithmic correction to the universal form of the moments is
observed. So far, we have presented rigorous results for the statistics of
$t_{m}$ for two canonical models namely the simple diffusion followed by the
drift-diffusion process. The large time universal form
$P_{r}(t_{m}|t)\simeq\frac{1}{t}$ is noteworthy in both cases. In the
following section, we present general arguments to show why this is indeed a
robust and universal characteristic for the arg-max for generic stochastic
processes.

Figure 7: Comparison of $\langle t_{m}(t)\rangle$ in Eq. (59) with the
numerical simulations. We have chosen $r=1$ and $v=1$ for the simulation
purpose.

##  V Limiting distribution of $t_{m}$ for general stochastic process

In the previous sections, we observed that the density of arg-max converges to
a uniform distribution of the form $P_{r}(t_{m}|t)\simeq 1/t$ when $t,t_{m}\gg
r^{-1}$ and $t_{m}\neq t$ for both diffusion and drifted diffusion. Naturally,
the question arises what are the ramifications of resetting to other
stochastic processes. Here, we show that the $1/t$-form of $P_{r}(t_{m}|t)$ is
completely universal and independent of the underlying stochastic process as
long as the process starts afresh after every reset event. To prove this, we
analyse our main formula for $\mathcal{Z}_{r}\left(M,k|s\right)$ in Eq. (5) in
the limit of large $rt$. Note that the survival probability $S_{0}(0,M|t)$ for
any stochastic process is bounded as $0\leq S_{0}(0,M|t)\leq 1$ for all $t$.
This bound suggests that the Laplace transform $\bar{S}_{0}(0,M|s)$ can be
written as

$\displaystyle\bar{S}_{0}(0,M|s)=\frac{1}{s}-\frac{1}{s}~{}\mathcal{U}(M,s),$
(69)

where $\mathcal{U}(M,s)$ is a general function with the constraints that we
discuss in the following. Since $\bar{S}_{0}(0,M|s)\geq 0$, we have
$\mathcal{U}(M,s)\leq 1$. Furthermore $\bar{S}_{0}(0,M|s)\leq 1/s$, which in
terms of $\mathcal{U}(M,s)$ becomes $\mathcal{U}(M,s)\geq 0$. Also, we expect
that the particle will survive without getting absorbed at $x=M\to\infty$
which implies $\bar{S}_{0}(0,M\to\infty|s)\simeq 1/s$ or equivalently
$\mathcal{U}\left(M\to\infty,s\right)\simeq 0$. In what follows, we use Eq.
(69) along with these constraints to analyze
$\mathcal{Z}_{r}\left(M,k|s\right)$ given in Eq. (5).

Figure 8: Comparison of $P_{r}(t_{m}|t)$ in Eq. (75) (shown by solid green
line) with the simulation results for five different underlying stochastic
processes, namely (i) diffusion (cyan), (ii) diffusion with positive drift
(blue), (iii) diffusion with negative drift (red), (iv) random acceleration
(black) and (v) Ornstein-Uhlenbeck process (magenta). While (i)-(iii) & (v)
are Markov processes, (iv) is a non-Markov one. Simulations are conducted for
the parameters $t=1$ and $r=20$ in all the cases (see Sec. H for more details
on the model systems).

Inserting $\bar{S}(0,M|s)$ from Eq. (69) in Eq. (5), we get

$\displaystyle\mathcal{Z}_{r}\left(M,k|s\right)=\frac{(r+s)(r+s+k)\mathcal{Z}_{0}\left(M,k|r+s\right)}{\left[s+r\mathcal{U}(M,r+s)\right]\left[s+k+r\mathcal{U}(M,r+s+k)\right]}.$
(70)

For $s\to 0$ and $k\to 0$, we approximate $\mathcal{Z}_{r}\left(M,k|s\right)$
as

$\displaystyle\mathcal{Z}_{r}\left(M,k|s\right)\simeq\frac{r^{2}}{\left[s+r\mathcal{U}(M,r)\right]\left[s+k+r\mathcal{U}(M,r)\right]}~{}\frac{d\bar{S}_{0}(0,M|r)}{dM},$
(71)

where we have used
$\mathcal{Z}_{0}\left(M,k=0|r\right)=\frac{d\bar{S}_{0}(0,M|r)}{dM}$. This
stems from the fact that $S(0,M|t)$ is the cumulative distribution of $M(t)$
i.e.,

$\displaystyle S_{0}(0,M|t)=\text{Prob}\left[M(t)\leq M\right].$ (72)

We now perform the double inverse Laplace transformation of Eq. (71) with
respect to $k$ and $s$ to get the joint distribution $P_{r}(M,t_{m}|t)$ for
$t,t_{m}\gg r^{-1}$ and $t_{m}\neq t$. The expression of $P_{r}(M,t_{m}|t)$
reads

$\displaystyle P_{r}(M,t_{m}|t)\simeq
r^{2}e^{-rt~{}\mathcal{U}(M,r)}~{}\frac{d\bar{S}_{0}(0,M|r)}{dM}.$ (73)

Finally to get the distribution of $t_{m}$, we integrate $P_{r}(M,t_{m}|t)$
for all $M$ to yield

$\displaystyle
P_{r}(t_{m}|t)\simeq\frac{1}{t}\left[e^{-rt~{}\mathcal{U}(M\to\infty,r)}-e^{-rt~{}\mathcal{U}(M\to
0,r)}\right].$ (74)

Using $\mathcal{U}(M\to\infty,r)=0$ and $\mathcal{U}(M\to 0,r)\geq 0$, we get

$\displaystyle P_{r}(t_{m}|t)\simeq\frac{1}{t}.$ (75)

In Figure 8, we have compared our analytic expression for $P_{r}(t_{m}|t)$
(given in Eq. (75)) to the results of the numerical simulations for five
different stochastic processes namely diffusion, diffusion with positive and
negative drift, random acceleration and Ornstein Uhlenbeck process. It is
important to remark that random acceleration process (RAP) is a non-Markov
process while the others are Markovian in nature. We observe an excellent
agreement between our analytical prediction and the simulations in all the
cases (we have consigned the details of simulation in Appendix H). As stressed
before, this result (75) is indeed independent of the nature of the underlying
stochastic process.

We end this section by discussing the origin of this universal limiting
distribution for $t_{m}$. Consider a long trajectory $x(\tau)$ with
$0\leq\tau\leq t$ with many resetting intervals. However, these intervals are
statistically independent since the entire configuration (all the variables)
is renewed after each resetting event, and hence there are no correlations
between the intervals. Hence, we can map our problem to a simple example of a
discrete time intervals $\\{\tau_{1},\tau_{2},\cdots,\tau_{N-1}\\}$ of $N-1$
IID entries, each drawn from a PDF $p(\tau)$. As the observation time $t$
becomes large, effect of the last interval $\tau_{N}$ becomes negligible i.e.,
$t-\tau_{N}\approx t$. Note that the maximum $M$ can be in any one of these
$N$ intervals with equal probability $1/N$ for a given $N$. However, $N$ is a
random variable in a fixed time interval $t$. In fact, $N$ is a Poisson
process with $P(N)=\frac{(rt)^{N}}{N!}e^{-rt}$. Taking the average, we find
that arg-max $t_{m}$ converges to a uniform distribution at large $t$ namely
Eq. (75). Thus, this result is completely universal, i.e., independent of the
PDF $p(\tau)$ and the underlying stochastic motion at large time. The
essential key point in this derivation is that resetting makes the intervals
completely independent to each other. Thus, even though for simple process
like Brownian motion (without resetting), computation of $P_{0}(t_{m}|t)$ can
not be made using this simple argument, resetting simplifies the problem
elegantly in many folds and the resulting density pertains to a uniform
distribution as we have already shown.

##  VI Conclusion

In summary, we have extensively studied statistics of the maximum distance
$M(t)$ and the time $t_{m}(t)$ taken to reach this maximum distance (upto an
observation time $t$) by a stochastic process which is subject to a resetting
mechanism. Resetting occurs at a constant rate $r$ which reinstates the
particle back to its initial position intermittently. The process is renewed
after each resetting event, and the memory from the previous trial is erased.
Utilizing this key property, we derive a renewal formula (Eq. (5)) for the
joint distribution of $M(t)$ and $t_{m}$ in the presence of resetting ($r>0$)
in terms of the same but with $r=0$. Our derivation is quite generic and holds
for both Markov and non-Markov underlying process.

Next, we use Eq. (5) and marginalize it to study statistics of $M(t)$ and
$t_{m}(t)$ respectively. We choose simple diffusion and diffusion with drift
as the underlying process and add resetting to them. In the case of simple
diffusion, we explicitly computed all moments of $M(t)$ from which we showed
that they have logarithmic time dependence at large time. Surprisingly,
although the position density converge to time independent steady state,
$\langle M^{n}(t)\rangle$ grows with time but rather slowly with logarithmic
dependence (see also MajumdarMori2020 ). Our results are consistent to
demonstrate that the limiting distribution of $M(t)$ belongs to a Gumbel class
which was recently understood from the EVS theory of weakly interacting random
variables Restart1 ; review ; EVS-review-cor . For the drift-diffusion case,
we have also computed all moments of $M(t)$. As in the case of simple
diffusion, we again observe the logarithmic growth for $\langle
M^{n}(t)\rangle$ despite the fact that drift-diffusion process reaches a
steady state in the presence of resetting. Finally, we do a consistency check
to show that the distribution of the maximum reproduces the Gumbel law as
expected from the EVS theory.

We then turn our attention to the statistics of the arg-max $t_{m}$. We first
derived the moments generating function for $t_{m}(t)$ in the Laplace space
for simple diffusion with reset in Eq. (42). This allowed us to compute the
first two moments exactly for all $t$. We next show that the arg-max density
at large time converges to a uniform distribution which only depends on the
observation time $t$ but not on the specifics of the underlying process. The
sub-leading correction terms are shown to be process dependent. For the drift-
diffusion process, computation of the higher order moments beyond the first
one becomes quite tedious however at large time we extracted the leading
contributions with logarithmic sub-leading terms. The density at large time is
again uniform with process dependent correction terms.

Borrowing wisdom from these exact results, we next analyzed the density of
$t_{m}$ for generic stochastic processes subject to stochastic resetting with
a rate $r$. Following an asymptotic expression for the joint distribution of
$M(t)$ and $t_{m}(t)$, we show that indeed at large time
$P_{r}(t_{m}|t)\simeq\frac{1}{t}$ which is independent of the underlying
process but the sub-leading terms are naturally process dependent as
demonstrated for simple diffusion and drift-diffusion. We provide a
probabilistic interpretation of the result based on the renewal property of
the resetting phenomena. Numerical simulations covering many of the above-
mentioned processes are in perfect agreement with our analytical predictions.

Estimating the probability of extreme events is an important problem in
statistics as well as in statistical, mathematical and condensed matter
physics and in other interdisciplinary subjects. Despite its paramount
relevance, EVS poses serious computational challenges and exact results are
scarce. In this paper, we showcase one such example for a renewal process
namely stochastic resetting which has gained immense interest in recent times.
Although the subject has been very dynamic from the perspective of non-
equilibrium transport properties or the first passage estimation, obtaining
exact results and limiting distributions related to EVS has been very limited.
Finding such results can be often subtle due to the complexity inherent to the
systems. However, drawing inspirations from the canonical model systems such
as simple diffusion and diffusion with drift, we have been able to unravel
some of the universal features of extremals in stochastic resetting systems.
We conclude by stating that the scope of the current formalism is not
restricted only to instantaneous resetting but also can accommodate scenarios
when returns are spatio-temporally correlated HRS ; return-0 ; return-1 ;
return-2 ; return-3 ; return-4 .

###### Acknowledgements.

Prashant Singh acknowledges useful discussions during ICTS-TIFR programs ‘BSSP
X’ and ‘Fluctuations in Nonequilibrium Systems: Theory and Applications’.
Arnab Pal gratefully acknowledges support from the Raymond and Beverly Sackler
Post-Doctoral Scholarship and the Ratner Center for Single Molecule Science at
Tel-Aviv University.

##  Appendix A Derivation of the renewal formula in Eq. (5)

In this section, we present the derivation of the renewal formula in Eq. (5)
which was presented in the main text. For brevity, we recall that the maximum
$M$ can occur in any of the $N$-intervals. Following the main text, these
contributions are given by

$\displaystyle\mathcal{C}_{1}=$
$\displaystyle\left(\prod_{i=1}^{N}\int_{0}^{\infty}d\tau_{i}\right)\int_{0}^{\infty}~{}d\tau~{}P_{0}(M,\tau|\tau_{1})p(\tau_{1})\left[\prod_{i=2}^{N-1}~{}S_{0}(0,\tau_{i}|M)p(\tau_{i})\right]\left[e^{-r\tau_{N}}S_{0}(0,\tau_{N}|M)\right]~{}\delta(t-\sum_{i=1}^{N}\tau_{i})~{}\delta(\tau-
t_{m}),$ $\displaystyle\mathcal{C}_{2}=$
$\displaystyle\left(\prod_{i=1}^{N}\int_{0}^{\infty}d\tau_{i}\right)~{}\int_{0}^{\infty}~{}d\tau~{}P_{0}(M,\tau|\tau_{2})p(\tau_{2})\left[\prod_{i=1,i\neq
2}^{N-1}~{}S_{0}(0,\tau_{i}|M)p(\tau_{i})\right]\left[e^{-r\tau_{N}}S_{0}(0,\tau_{N}|M)\right]\delta(t-\sum_{i=1}^{N}\tau_{i})~{}\delta(\tau_{1}+\tau-
t_{m}),$ $\displaystyle\mathcal{C}_{3}=$ $\displaystyle\cdots$
$\displaystyle\mathcal{C}_{4}=$ $\displaystyle\cdots$ $\displaystyle\cdots$
$\displaystyle\mathcal{C}_{N}=$
$\displaystyle\left(\prod_{i=1}^{N}\int_{0}^{\infty}d\tau_{i}\right)~{}\int_{0}^{\infty}~{}d\tau~{}P_{0}(M,\tau|\tau_{N})e^{-r\tau_{N}}\left[\prod_{i=1}^{N-1}~{}S_{0}(0,\tau_{i}|M)p(\tau_{i})\right]\delta(t-\sum_{i=1}^{N}\tau_{i})~{}\delta(\sum_{i=1}^{N-1}\tau_{i}+\tau-
t_{m}).$ (76)

To obtain joint distribution $P_{r}\left(M,t_{m},N|t\right)$, one has to sum
all contributions $\mathcal{C}_{1},~{}\mathcal{C}_{2},...,\mathcal{C}_{N}$.
Inserting the contributions from Eqs. (76) and noting that
$e^{-r\tau_{N}}=\frac{1}{r}p(\tau_{N})$ with $p(\tau_{i})=re^{-r\tau_{i}}$,
the joint distribution $\mathcal{P}_{r}\left(M,t_{m},N|t\right)$ can be
formally written as

$\displaystyle\mathcal{P}_{r}\left(M,t_{m},N|t\right)$
$\displaystyle=\frac{1}{r}\sum_{j=1}^{N}\int_{0}^{\infty}d\tau_{j}~{}d\tau~{}P_{0}\left(M,\tau|\tau_{j}\right)p(\tau_{j})\left(\prod_{j^{\prime}=1,j^{\prime}\neq
j}^{N}\int_{0}^{\infty}d\tau_{j^{\prime}}S_{0}(0,\tau_{j^{\prime}}|M)p(\tau_{j^{\prime}})d\tau_{j^{\prime}}\right)$
$\displaystyle\times\delta\left(\sum_{i=1}^{j-1}\tau_{i}+\tau-
t_{max}\right)\delta(t-\sum_{i=1}^{N}\tau_{i}),$ (77)

where the second delta function ensures that the total observation time is
$t$. To simplify this expression, it is useful to perform Laplace
transformations with respect to $t~{}(\to s)$ and $t_{m}~{}(\to k)$. Denoting
the Laplace transformation of $\mathcal{P}_{r}\left(M,t_{m},N|t\right)$ by
$\mathcal{Z}_{r}\left(M,k,N|s\right)$, we take the Laplace transformation of
Eq. (77) to yield

$\displaystyle\mathcal{Z}_{r}\left(M,k,N|s\right)$
$\displaystyle=\sum_{m=1}^{N}\mathcal{Z}_{0}\left(M,k|r+s\right)\left[r\bar{S}_{0}(0,s+r+k|M)\right]^{m-1}\left[r\bar{S}_{0}(0,s+r|M)\right]^{N-m},$
(78)
$\displaystyle=\frac{\mathcal{Z}_{0}\left(M,k|r+s\right)}{r\left[\bar{S}_{0}(0,s+r|M)-\bar{S}_{0}(0,s+r+k|M)\right]}\left[\left(r\bar{S}_{0}(0,s+r|M)\right)^{N}-\left(r\bar{S}_{0}(0,s+r+k|M)\right)^{N}\right].$
(79)

In the first line, we have used the notation $\bar{S}_{0}(0,s,M)$ as the
Laplace transformation of $S_{0}(0,t|M)$. To get the joint distribution of $M$
and $t_{m}$ in the Laplace space of $k$ and $s$, we sum
$\mathcal{Z}_{r}\left(M,k,N|s\right)$ for all values of $N$ from 1 to $\infty$
which results in

$\displaystyle\mathcal{Z}_{r}\left(M,k|s\right)$
$\displaystyle=\sum_{N=1}^{\infty}\mathcal{Z}_{r}\left(M,k,N|s\right),$
$\displaystyle=\frac{\mathcal{Z}_{0}\left(M,k|r+s\right)}{\left[1-r\bar{S}_{0}(0,s+r|M)\right]~{}\left[1-r\bar{S}_{0}(0,s+r+k|M)\right]},$
(80)

where in going to the second line from the first line, we have substituted
$\mathcal{Z}_{r}\left(M,k,N|s\right)$ from Eq. (79). This concludes the proof
for Eq. (5) in the main text.

##  Appendix B Asymptotic forms of $\langle M^{n}(t)\rangle$ for Brownian
motion

In this appendix, we derive the asymptotic behaviour of $\langle
M^{n}(t)\rangle$ for Brownian motion whose exact expression is given in Eq.
(15). Below, we look at the behaviour of $\langle M^{n}(t)\rangle$ for large
and short times separately.

###  B.1 Case I: $\langle M^{n}(t)\rangle$ for $t\gg\frac{1}{r}$

To begin with, we consider the Laplace transformation of $P_{r}(M|t)$ from Eq.
(12) and rewrite here as

$\displaystyle\mathcal{Z}_{r}\left(M,k=0|s\right)=\frac{\sqrt{2}\left(r+s\right)^{3/2}~{}e^{-\sqrt{2(r+s)}M}}{\left(s+re^{-\sqrt{2(r+s)}M}\right)^{2}}.$
(81)

For small $r\gg s$ (which corresponds to large $r\ll t^{-1}$), we approximate
$r+s\simeq s$ in Eq. (81) to yield

$\displaystyle\mathcal{Z}_{r}\left(M,k=0|s\right)\simeq\frac{\sqrt{2}r^{3/2}~{}e^{-\sqrt{2r}M}}{\left(s+re^{-\sqrt{2r}M}\right)^{2}}.$
(82)

To get the distribution in the time domain, we perform the inverse Laplace
transform of Eq. (82) by using $\mathcal{L}_{s\to
t}^{-1}\left[\frac{1}{(s+a)^{2}}\right]=te^{-at}$ for $a\geq 0$ to yield

$\displaystyle
P_{r}(M|t)\simeq\sqrt{2r^{3}}te^{-\sqrt{2r}M}\text{exp}\left(-rte^{-\sqrt{2r}M}\right).$
(83)

We emphasise that the approximate equality in this equation indicates that it
is valid only for $t\gg\frac{1}{r}$. We use this form of the distribution to
get the moments at large $t$ as

$\displaystyle\langle M^{n}(t)\rangle$
$\displaystyle=\int_{0}^{\infty}dMM^{n}~{}P_{r}(M|t),$ (84)
$\displaystyle\simeq\frac{1}{(2r)^{n/2}}\log^{n}(rt)+O\left(\log^{n-1}(rt)\right),$
(85)

where in going to the second line, we have inserted $P_{r}(M|t)$ from Eq. (83)
and performed the integration over $M$. Comparing Eq. (85) with the scaling
form in Eq. (15), we find that the scaling function $H_{n}(z)$ at large $z$ is
given by

$\displaystyle H_{n}(z)\simeq\log^{n}z+O\left(\log^{n-1}z\right).$ (86)

This result has been quoted in Eq. (17) of the main text.

###  B.2 Case II: $\langle M^{n}(t)\rangle$ for $t\ll\frac{1}{r}$

To find the moments at times $t\ll\frac{1}{r}$, we need the form of the
distribution $P_{r}(M|t)$ at these times as indicated by Eq. (84). In the
Laplace space, this will correspond to the $r\ll s$ behaviour of
$\mathcal{Z}_{r}\left(M,k=0|s\right)$ in Eq. (81). By direct expansion of Eq.
(81) in $r$, we find

$\displaystyle\mathcal{Z}_{r}\left(M,k=0|s\right)$
$\displaystyle\simeq\sqrt{\frac{2}{s}}e^{-\sqrt{2s}M}+\frac{r}{2s^{3/2}}\left[\left(3\sqrt{2}-2M\sqrt{s}\right)\right.$
$\displaystyle\left.\times
e^{-\sqrt{2s}M}-4\sqrt{2}~{}e^{-2\sqrt{2s}M}\right].$ (87)

We next use Eq.(13) (first line) to get the moments in the Laplace space and
then perform inverse Laplace inversion to yield

$\displaystyle\langle
M^{n}(t)\rangle\simeq\frac{n!}{2^{\frac{n}{2}}\Gamma\left(1+\frac{n}{2}\right)}~{}t^{\frac{n}{2}}+\frac{\mathcal{B}_{n}r}{2^{n/2}}t^{\frac{n}{2}+1},~{}~{}\text{with}$
(88)
$\displaystyle\mathcal{B}_{n}=\frac{\Gamma(n+1)\left(3\sqrt{2}-2^{\frac{3}{2}-n}\right)-\sqrt{2}\Gamma(n+2)}{2\sqrt{2}~{}\Gamma\left(\frac{n+4}{2}\right)}.$
(89)

Finally comparing this equation with the scaling form in Eq. (15), we find
that the scaling function $H_{n}(z)$ at small $z$ is given by

$\displaystyle
H_{n}(z)\simeq\frac{n!}{\Gamma\left(1+\frac{n}{2}\right)}z^{\frac{n}{2}}+\mathcal{B}_{n}z^{\frac{n}{2}+1},$
(90)

which is Eq. (18) in the main text.

##  Appendix C Derivation of $\langle t_{m}(t)\rangle$ and $\langle
t^{2}_{m}(t)\rangle$ for Brownian motion

In this appendix, we derive the exact expressions of $\langle t_{m}(t)\rangle$
and $\langle t^{2}_{m}(t)\rangle$ which are presented in Eqs. (44) and (45)
respectively. To derive these results, we insert the expression of
$\mathcal{Z}_{r}\left(k|s\right)$ from Eq. (42) in Eq. (43) for $n=1,2$ to get

$\displaystyle\int_{0}^{\infty}dte^{-st}\langle t_{m}(t)\rangle$
$\displaystyle=\frac{2r^{2}+3rs+s^{2}}{4s^{2}(r+s)^{2}}+\frac{\log\left(\frac{r+s}{s}\right)}{4rs},$
(91) $\displaystyle\int_{0}^{\infty}dte^{-st}\langle t_{m}^{2}(t)\rangle$
$\displaystyle=\frac{16r^{2}+36rs+15s^{2}}{24s^{3}(r+s)^{2}}-\frac{\text{Li}_{2}\left(-\frac{r}{s}\right)}{12rs(r+s)}$
$\displaystyle+\frac{(4r+s)\log\left(\frac{r+s}{s}\right)}{24rs^{2}(r+s)}.$
(92)

Let us invert Eq. (91) with respect to $s$. To this aim, we use the following
inverse Laplace transformations:

$\displaystyle\mathcal{L}_{s\to
t}^{-1}\left[\frac{\log\left(\frac{r+s}{s}\right)}{s}\right]=\gamma_{E}+\Gamma(0,rt)+\log
rt,$ (93) $\displaystyle\mathcal{L}_{s\to
t}^{-1}\left[\frac{2r^{2}+3rs+s^{2}}{4s^{2}(r+s)^{2}}\right]=\frac{2rt+e^{-rt}-1}{4r}.$
(94)

Using these two inverse Laplace transforms in Eq. (91), we recover the
expression of $\langle t_{m}(t)\rangle$ as written in Eq. (44). We next look
at Eq. (92) to get $\langle t_{m}^{2}(t)\rangle$ for which we need three
inverse Laplace transformations. Inverse Laplace transform of the first term
in the RHS of Eq. (92) is given by

$\displaystyle\mathcal{L}_{s\to
t}^{-1}\left[\frac{16r^{2}+36rs+15s^{2}}{24s^{3}(r+s)^{2}}\right]=$
$\displaystyle\frac{1}{24r^{2}}\left[e^{-rt}(9+5rt)-9\right.$
$\displaystyle\left.+4rt+8r^{2}t^{2}\right].$ (95)

Next, we turn to the second term in the RHS of Eq. (92). Note that this term
is the product of $\frac{1}{r+s}$ and
$\frac{1}{s}\text{Li}_{2}\left(-\frac{r}{s}\right)$ which implies that the
convolution theorem for Laplace transforms can be directly used. Doing so, we
find

$\displaystyle\mathcal{L}_{s\to
t}^{-1}\left[\frac{\text{Li}_{2}\left(-\frac{r}{s}\right)}{s(r+s)}\right]=\int_{0}^{t}dt_{1}e^{-r(t-t_{1})}f_{1}(t_{1}),~{}~{}~{}\text{where}$
(96) $\displaystyle f_{1}(t_{1})=\mathcal{L}_{s\to
t_{1}}^{-1}\left[\frac{\text{Li}_{2}\left(-\frac{r}{s}\right)}{s}\right],$
(97)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}=-rt_{1}~{}_{3}F_{3}\Big{(}\\{1,1,1\\},\\{2,2,2\\},\\{-rt_{1}\\}\Big{)}.$
(98)

We now look at the third term in the R.H.S. of Eq. (92). Once again, we notice
that it has the product form because of which we use the convolution theorem.
The inverse Laplace transform then reads

$\displaystyle\mathcal{L}_{s\to
t}^{-1}\left[\frac{(4r+s)\log\left(\frac{r+s}{s}\right)}{24rs^{2}(r+s)}\right]=\int_{0}^{t}dt_{1}f_{2}(t-t_{1})f_{3}(t_{1}),$
(99)

where the functions $f_{2}(t)$ and $f_{3}(t)$ are given by

$\displaystyle f_{2}(t)$ $\displaystyle=\mathcal{L}_{s\to
t}^{-1}\left[\log\left(\frac{r+s}{s}\right)\right],$ (100)
$\displaystyle=\frac{1-e^{-rt}}{t},$ (101) $\displaystyle f_{3}(t)$
$\displaystyle=\mathcal{L}_{s\to
t}^{-1}\left[\frac{(4r+s)}{24rs^{2}(r+s)}\right],$ (102)
$\displaystyle=\frac{4rt-3+3e^{-rt}}{24r^{2}}.$ (103)

Finally inserting Eqs. (95), (96) and (99) in Eq. (92), we recover the form of
$\langle t_{m}^{2}(t)\rangle$ as quoted in Eq. (45).

##  Appendix D Asymptotic forms of $\langle t_{m}^{n}(t)\rangle$ for Brownian
motion

This appendix deals with the derivation of the forms of $\langle
t_{m}^{n}(t)\rangle$ for Brownian motion when $t\ll r^{-1}$ and $t\gg r^{-1}$.
These asymptotic forms are presented in Eqs. (46) and (47). To begin with, we
analyse $\mathcal{Z}_{r}\left(k|s\right)$ in Eq. (42) in various limits of
$sr^{-1}$ using which we compute the distribution $P_{r}(t_{m}|t)$ for
different regimes of $t$. We then use $P_{r}(t_{m}|t)$ to compute moments.
Below we consider the cases $t\ll r^{-1}$ and $t\gg r^{-1}$ separately.

###  D.1 Case I: $t\ll\frac{1}{r}$

In the Laplace space of $s$, the limit $t\ll\frac{1}{r}$ corresponds to
$rs^{-1}\to 0$ which for fixed $s$ means $r\to 0$. Taking the limit $r\to 0$
in $\mathcal{Z}_{r}\left(k|s\right)$ in Eq. (42), we get

$\displaystyle\mathcal{Z}_{r}\left(k|s\right)\simeq\frac{1}{\sqrt{s(s+k)}}+\frac{r}{s^{3/2}}\left[\frac{1}{2\sqrt{s+k}}-\frac{1}{\sqrt{s}+\sqrt{s+k}}\right].$
(104)

We next invert this Laplace transform to obtain the distribution
$P_{r}(t_{m}|t)$ using which we compute moments. Note that we have to perform
double inverse Laplace transformations: one from $s\to t$ and the other from
$k\to t_{m}$. To invert Eq. (104), we use the following double inverse Laplace
transformations:

$\displaystyle\mathcal{L}_{s\to t}^{-1}\mathcal{L}_{k\to
t_{m}}^{-1}\left[\frac{1}{\sqrt{s(s+k)}}\right]=\frac{1}{\pi\sqrt{t_{m}(t-t_{m})}},$
(105) $\displaystyle\mathcal{L}_{s\to t}^{-1}\mathcal{L}_{k\to
t_{m}}^{-1}\left[\frac{s^{-3/2}}{(\sqrt{s}+\sqrt{s+k})}\right]=\frac{2}{\pi}\left[\sqrt{\frac{t-t_{m}}{t_{m}}}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-\cos^{-1}\sqrt{\frac{t_{m}}{t}}\right].$
(106)

Using these two equations in $\mathcal{Z}_{r}\left(k|s\right)$ in Eq. (104),
we get

$\displaystyle
P_{r}(t_{m}|t)\simeq\frac{1}{\pi\sqrt{t_{m}(t-t_{m})}}+\frac{r}{\pi}\left(2\cos^{-1}\sqrt{\frac{t_{m}}{t}}-\sqrt{\frac{t-t_{m}}{t_{m}}}\right).$
(107)

We emphasize that this expression is valid only in the limit $rt\to 0$.
Finally, we use this form of $P_{r}(t_{m}|t)$ to compute the moments of
$t_{m}$ which then read

$\displaystyle\langle t_{m}^{n}(t)\rangle$
$\displaystyle\simeq\frac{(2n-1)!t^{n}}{n!(n-1)!2^{2n-1}}+\mathcal{A}_{n}\gamma
t^{n+1},~{}~{}~{}\text{with}$ (108) $\displaystyle\mathcal{A}_{n}$
$\displaystyle=\frac{1}{\Gamma(n+2)}\left[\frac{\Gamma\left(3/2+n\right)}{\sqrt{\pi}(n+1)}-\frac{(2n-1)!}{2^{2n}(n-1)!}\right],$
(109)

which is Eq. (46) in the main text.

###  D.2 Case II: $t\gg\frac{1}{r}$

We next look at the moments when $t\gg\frac{1}{r}$. Once again we begin with
the expression of $\mathcal{Z}_{r}\left(k|s\right)$ in Eq. (42) and analyse it
in the limit of large $r$. For large $r$, we approximate $r+s\simeq r$ and
$w^{\sqrt{\frac{r+s}{r+s+k}}}\simeq w-\frac{wk\log w}{2r}$ and insert them in
Eq. (42) to yield

$\displaystyle\mathcal{Z}_{r}\left(k|s\right)\simeq
r\int_{0}^{1}\frac{dw}{(s+k+rw)\left(s+rw-\frac{wk\log w}{2}\right)}.$ (110)

To get the distribution in the time domain from Eq. (110), we now use the
following standard inverse Laplace transformation

$\displaystyle\mathcal{L}_{s\to
t}^{-1}\left[\frac{e^{-bs}}{s+a}\right]=e^{-a(t-b)}\Theta(t-b),~{}~{}\text{with
}a,b\geq 0,$ (111)

in Eq. (110) to get

$\displaystyle P_{r}(t_{m}|t)\simeq 2r\int_{0}^{1}dw\frac{e^{-rtw}}{2+w\log
w}\left[1-\Theta\left(t+\frac{2t_{m}}{w\log w}\right)\right].$ (112)

We are now left with the integration over $w$. This can be done by making the
transformation $rtw=y$ in Eq. (112). To proceed further, we approximate $\log
w=\log y-\log rt\simeq-\log rt$ and $\frac{rt}{\log rt}\to\infty$ for
$rt\to\infty$. With these approximations, the integration in Eq. (112) can be
performed explicitly to get

$\displaystyle P_{r}(t_{m}|t)\simeq\frac{1}{t}+\frac{\log rt}{2rt^{2}}.$ (113)

Using this form of $P_{r}(t_{m}|t)$ for $t\gg r^{-1}$, it is straightforward
to show the moment is given by

$\displaystyle\langle
t_{m}^{n}(t)\rangle\simeq\frac{t^{n}}{n+1}+\frac{t^{n-1}\log(rt)}{2r(n+1)},$
(114)

which has been mentioned in Eq. (47).

##  Appendix E Derivation of $P_{r}(t_{m}|t)$ in Eq. (49) for Brownian motion

In this appendix, we perform the double inverse Laplace transformation of
$\mathcal{Z}_{r}\left(k|s\right)$ in Eq. (48) to get the distribution
$P_{r}(t_{m}|t)$ in Eq. (49). We first perform the inversion of Eq. (48) with
respect to $k$ for which we use the following:

$\displaystyle\mathcal{L}_{k\to
t_{m}}^{-1}\left[\frac{\sqrt{k+b}}{k+a}\right]=$
$\displaystyle\frac{e^{-bt_{m}}}{\sqrt{\pi t_{m}}}+\sqrt{b-a}~{}e^{-at_{m}}$
$\displaystyle\times\text{Erf}\left(\sqrt{(b-a)t_{m}}\right),$ (115)

with $a,b\geq 0$. Using this in Eq. (48) by reading appropriately $a$ and $b$,
we get

$\displaystyle\bar{P}_{r}(t_{m}|s)=\bar{Q}_{1}(t_{m}|s)+\bar{Q}_{2}(t_{m}|s),$
(116)

where $\bar{P}_{r}(t_{m}|s)$ stands for the Laplace transformation of
$P_{r}(t_{m}|t)$. Also, the functions $\bar{Q}_{1}(t_{m}|s)$ and
$\bar{Q}_{2}(t_{m}|s)$ are given by

$\displaystyle\bar{Q}_{1}(t_{m}|s)$
$\displaystyle=\int_{0}^{1}dw\frac{\sqrt{r+s}~{}e^{-(r+s)t_{m}}}{\sqrt{\pi
t_{m}}(s+rw)},$ (117) $\displaystyle\bar{Q}_{2}(t_{m}|s)$
$\displaystyle=\int_{0}^{1}dw\frac{\sqrt{r(r+s)(1-w)}~{}e^{-(s+rw)t_{m}}}{s+rw}$
$\displaystyle\times\text{Erf}\left(\sqrt{rt_{m}(1-w)}\right).$ (118)

To obtain the distribution $P_{r}(t_{m}|t)$, we have to perform inverse
Laplace transformation in Eq. (116). Let us first perform the inversion for
$\bar{Q}_{1}(t_{m}|s)$ in Eq. (117) for which we use the inverse Laplace
transformation in Eq. (115) with $k$ replaced by $s$. Denoting the inverse
Laplace transformation of $\bar{Q}_{1}(t_{m}|s)$ by $Q_{1}(t_{m}|t)$, we get

$\displaystyle Q_{1}(t_{m}|t)$
$\displaystyle=\frac{e^{-rt}}{\pi\sqrt{t_{m}(t-t_{m})}}+\frac{e^{-rt_{m}}}{\sqrt{\pi
t_{m}}}\int_{0}^{1}dw\sqrt{r(1-w)}$ $\displaystyle\times
e^{-rw(t-t_{m})}~{}\text{Erf}\left(\sqrt{r(1-w)(t-t_{m})}\right).$ (119)

To perform the integration in the second term, we make the following change in
variable: $y=r(1-w)(t-t_{m})$, which, in turn, yields

$\displaystyle Q_{1}(t_{m}|t)$
$\displaystyle=\frac{e^{-rt}}{\pi\sqrt{t_{m}(t-t_{m})}}\left[1+\frac{\sqrt{\pi}}{r(t-t_{m})}\right.$
$\displaystyle\left.\times\int_{0}^{r(t-t_{m})}dy\sqrt{y}~{}e^{y}~{}\text{Erf}(\sqrt{y})\right].$
(120)

The integration over $y$ in the second line can be explicitly performed using
Mathematica in terms of the generalized hypergeometric functions. The final
result reads

$\displaystyle
Q_{1}(t_{m}|t)=\frac{e^{-rt}~{}_{2}F_{2}\left(\\{1,1\\},\\{1/2,2\\},r(t-t_{m})\right)}{\pi\sqrt{t_{m}(t-t_{m})}}.$
(121)

We next turn to $\bar{Q}_{2}(t_{m}|s)$ in Eq. (116). For
$\bar{Q}_{2}(t_{m}|s)$, it turns out that one has to follow similar steps as
done for $\bar{Q}_{1}(t_{m}|s)$. To avoid repetition, we only present here the
final expression of $Q_{2}(t_{m}|t)$ which reads

$\displaystyle Q_{2}(t_{m}|t)$
$\displaystyle=e^{-rt}\frac{\left[~{}_{2}F_{2}\left(\\{1,1\\},\\{1/2,2\\},r(t-t_{m})\right)-1\right]}{\pi\sqrt{t_{m}(t-t_{m})}}$
$\displaystyle+re^{-rt}\int_{0}^{1}~{}dw~{}w~{}e^{rtw}~{}\text{Erf}\left(\sqrt{rwt_{m}}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\text{Erf}\left(\sqrt{rw(t-t_{m})}\right).$
(122)

Finally using Eqs. (121) and (122) in Eq. (116), we recover the result for
$P_{r}(t_{m}|t)$ in Eq. (49).

##  Appendix F Derivation of $\langle M^{n}(t)\rangle$ for drift-diffusion
process

This appendix provides the derivation of the scaling relation in Eq. (29) for
$\langle M^{n}(t)\rangle$. We begin with the Laplace transform of $\langle
M^{n}(t)\rangle$ in Eq. (28). Looking at this equation, we find that the
Laplace transform is product to two terms which leads us to use convolution
property for Laplace transforms. Using this property, we get

$\displaystyle\langle
M^{n}(t)\rangle=-\frac{n!}{2^{n/2}}\sum_{k=1}^{\infty}\frac{(-r)^{k}}{k^{n}}\int_{0}^{t}d\tau
f_{4}(t-\tau),f_{3}^{n}(\tau)$ (123)

where the functions $f_{3}(t)$ and $f_{4}(t)$ are given below.

$\displaystyle f_{3}(t)$ $\displaystyle=\mathcal{L}_{s\to
t}^{-1}\left[\frac{1}{\left(\sqrt{s-\beta}+\gamma\right)^{n}}\right],$
$\displaystyle=\frac{(-1)^{n-1}}{(n-1)!}\left[\frac{e^{\beta
t}~{}\delta_{n,1}}{\sqrt{\pi t}}-\frac{d^{n-1}}{d\gamma^{n-1}}\left\\{\gamma
e^{(\beta+\gamma^{2})t}\text{Erfc}\left(\gamma\sqrt{t}\right)\right\\}\right]$
$\displaystyle f_{4}(t)$ $\displaystyle=\mathcal{L}_{s\to
t}^{-1}\left[\frac{s+r}{s^{k+1}}\right],$
$\displaystyle=\frac{t^{k+1}}{\Gamma(k)}+\frac{rt^{k}}{\Gamma(k+1)},$ (124)

where $\beta=-\left(r+\gamma^{2}\right)$ and $\gamma=-v/\sqrt{2}$. Inserting
the forms of $f_{3}(t)$ and $f_{4}(t)$ in the expression of $\langle
M^{n}(t)\rangle$ in Eq. (123) and then changing the variable $\tau=tw$, we get
the scaling relation in Eq. (29).

##  Appendix G Derivation of $P_{r}(t_{m}|t)$ in Eq. (61) for drift-diffusion
process

In this section, we present derivation for the density $P_{r}(t_{m}|t)$ namely
Eq. (61) which was presented in the main text. We recall from the main text
(Eq. (60)) that

$\displaystyle\mathcal{Z}_{r}\left(k|s\right)$
$\displaystyle\simeq\frac{\sqrt{v^{2}+2(s+r)}-v}{\sqrt{v^{2}+2(s+k+r)}-v}~{}\int_{0}^{1}~{}dw~{}\frac{(r+s+k)}{\left(s+rw\right)\left(s+k+rw\right)}.$
(125)

To invert this expression, we do the inverse Laplace transforms with respect
to $k$ and $s$ respectively. Next, we perform the integral. Following the
first inversion, we rewrite

$\displaystyle\mathcal{Z}_{r}\left(t_{m}|s\right)\simeq\left[\sqrt{v^{2}+2(s+r)}-v\right]\int_{0}^{1}~{}\frac{dw}{\left(s+rw\right)}\mathcal{L}_{k\to
t_{m}}^{-1}\left[\underbrace{\frac{s+r+k}{s+rw+k}\frac{1}{\sqrt{v^{2}+2(s+k+r)}-v}}_{\mathcal{I}(k)}\right].$
(126)

We can now rearrange $\mathcal{I}(k)$ and then do the inversion to find

$\displaystyle\mathcal{L}_{k\to t_{m}}^{-1}\left[\mathcal{I}(k)\right]$
$\displaystyle=\mathcal{L}_{k\to
t_{m}}^{-1}\left[\frac{1}{2}\frac{v}{s+rw+k}+\frac{1}{2}\frac{\sqrt{v^{2}+2(s+k+r)}}{s+rw+k}\right]$
$\displaystyle=\frac{e^{-(\frac{v^{2}}{2}+s+r)t_{m}}}{\sqrt{2\pi
t_{m}}}+\frac{ve^{-(s+rw)t_{m}}}{2}+\frac{\sqrt{\frac{v^{2}}{2}+r(1-w)}}{\sqrt{2}}e^{-(s+rw)t_{m}}\text{Erf}\left[\sqrt{\left(\frac{v^{2}}{2}+r(1-w)\right)t_{m}}\right].$
(127)

Now, we need to perform the Laplace inversion wrt $s$ i.e.,

$P_{r}(t_{m}|t)=\mathcal{L}_{s\to
t}^{-1}\left[\mathcal{Z}_{r}\left(t_{m}|s\right)\right],$ (128)

where we have

$\displaystyle\mathcal{Z}_{r}\left(t_{m}|s\right)$
$\displaystyle=\int_{0}^{1}~{}dw~{}\frac{e^{-(\frac{v^{2}}{2}+s+r)t_{m}}}{\sqrt{2\pi
t_{m}}}\frac{\sqrt{v^{2}+2(s+r)}-v}{s+rw}$
$\displaystyle+\int_{0}^{1}~{}dw~{}\frac{ve^{-(s+rw)t_{m}}}{2}\frac{\sqrt{v^{2}+2(s+r)}-v}{s+rw}$
$\displaystyle+\int_{0}^{1}~{}dw~{}\frac{\sqrt{\frac{v^{2}}{2}+r(1-w)}}{\sqrt{2}}~{}\text{Erf}\left[\sqrt{\left(\frac{v^{2}}{2}+r(1-w)\right)t_{m}}\right]~{}e^{-(s+rw)t_{m}}\frac{\sqrt{v^{2}+2(s+r)}-v}{s+rw}.$
(129)

Substituting Eq. (129) into Eq. (128) and performing the Laplace inversions,
we obtain

$\displaystyle P_{r}(t_{m}|t)$
$\displaystyle\simeq\frac{e^{-(\frac{v^{2}}{2}+r)t_{m}}}{\sqrt{2\pi
t_{m}}}\int_{0}^{1}~{}dw~{}\mathcal{J}_{\ell}(t,w)+\frac{v}{2}\int_{0}^{1}~{}dw~{}e^{-rwt_{m}}~{}\mathcal{J}_{\ell}(t,w)$
$\displaystyle+\int_{0}^{1}~{}dw~{}\frac{\sqrt{\frac{v^{2}}{2}+r(1-w)}}{\sqrt{2}}~{}\text{Erf}\left[\sqrt{\left(\frac{v^{2}}{2}+r(1-w)\right)t_{m}}\right]~{}e^{-rwt_{m}}~{}\mathcal{J}_{\ell}(t,w),$
$\displaystyle\simeq\mathbb{I}_{1}(t_{m},t)+\mathbb{I}_{2}(t_{m},t)+\mathbb{I}_{3}(t_{m},t),$
(130)

where we have used the following inverse Laplace transform to arrive at Eq.
(130)

$\displaystyle\mathcal{J}_{\ell}(t,w)$ $\displaystyle=\mathcal{L}_{s\to
t}^{-1}\left[\frac{\sqrt{v^{2}+2(s+r)}-v}{s+rw}e^{-st_{m}}\right]=-ve^{-rw(t-t_{m})}\Theta(t-t_{m})$
$\displaystyle+\sqrt{2}\left[\frac{e^{-(t-t_{m})(\frac{v^{2}}{2}+r)}}{\sqrt{\pi(t-t_{m})}}+e^{-rw(t-t_{m})}\sqrt{\frac{v^{2}}{2}+r(1-w)}~{}\text{Erf}\left(\sqrt{(t-t_{m})(\frac{v^{2}}{2}+r-rw)}\right)\right]\Theta(t-t_{m}).$
(131)

The $\mathbb{I}$-functions introduced in Eq. (130) are formally defined in the
following. They can also be simplified occasionally. The first component in
Eq. (130) reads

$\displaystyle\mathbb{I}_{1}(t_{m},t)$
$\displaystyle=\frac{e^{-(\frac{v^{2}}{2}+r)t_{m}}}{\sqrt{2\pi
t_{m}}}\int_{0}^{1}~{}dw~{}\mathcal{J}_{\ell}(t,w),$
$\displaystyle=\frac{e^{-(\frac{v^{2}}{2}+r)t}}{\pi\sqrt{t_{m}(t-t_{m})}}-\frac{ve^{-(\frac{v^{2}}{2}+r)t_{m}}}{\sqrt{2\pi
t_{m}}}~{}\frac{1-e^{-r(t-t_{m})}}{r(t-t_{m})}+\frac{e^{-(\frac{v^{2}}{2}+r)t}}{\pi\sqrt{t_{m}(t-t_{m})}}\int_{\frac{v^{2}}{2r}}^{\frac{v^{2}}{2r}+1}~{}dz~{}e^{r(t-t_{m})z}\sqrt{rz}~{}\text{Erf}\left[\sqrt{r(t-t_{m})}\sqrt{z}\right],$
$\displaystyle=\frac{e^{-(\frac{v^{2}}{2}+r)t}}{\pi\sqrt{t_{m}(t-t_{m})}}-v\frac{e^{-(\frac{v^{2}}{2}+r)t_{m}}}{\sqrt{2\pi
t_{m}}}~{}\frac{1-e^{-r(t-t_{m})}}{r(t-t_{m})}$
$\displaystyle+\frac{e^{-(\frac{v^{2}}{2}+r)t}}{\pi(t-t_{m})\sqrt{\pi
t_{m}}}\left(-\frac{v^{2}\,_{2}F_{2}\left[1,1;\frac{1}{2},2;\frac{(t-t_{m})v^{2}}{2}\right]}{2r}+\left(\frac{v^{2}}{2r}+1\right)\,_{2}F_{2}\left[1,1;\frac{1}{2},2;r(t-t_{m})\left(\frac{v^{2}}{2r}+1\right)\right]-1\right).$
(132)

Similarly, the second component in Eq. (130) gives

$\displaystyle\mathbb{I}_{2}(t_{m},t)$
$\displaystyle=\frac{v}{2}\int_{0}^{1}~{}dw~{}e^{-rwt_{m}}~{}\mathcal{J}_{\ell}(t,w),$
$\displaystyle=\frac{ve^{-(\frac{v^{2}}{2}+r)(t-t_{m})}}{\
\sqrt{2\pi(t-t_{m})}}~{}\frac{1-e^{-rt_{m}}}{rt_{m}}-\frac{v^{2}}{2}\frac{1-e^{-rt}}{rt}+\frac{ve^{-(\frac{v^{2}}{2}+r)t}}{\sqrt{2}}\int_{\frac{v^{2}}{2r}}^{\frac{v^{2}}{2r}+1}~{}dz~{}e^{rtz}\sqrt{rz}~{}\text{Erf}\left[\sqrt{r(t-t_{m})}\sqrt{z}\right].$
(133)

Finally, simplifying the third component in Eq. (130), we get

$\displaystyle\mathbb{I}_{3}(t_{m},t)$
$\displaystyle=\int_{0}^{1}~{}dw~{}\frac{\sqrt{\frac{v^{2}}{2}+r(1-w)}}{\sqrt{2}}~{}\text{Erf}\left[\sqrt{\left(\frac{v^{2}}{2}+r(1-w)\right)t_{m}}\right]~{}e^{-rwt_{m}}~{}\mathcal{J}_{\ell}(t,w),$
$\displaystyle=\frac{e^{-(\frac{v^{2}}{2}+r)t}}{\sqrt{\pi(t-t_{m})}}\int_{\frac{v^{2}}{2r}}^{\frac{v^{2}}{2r}+1}~{}dz~{}e^{rt_{m}z}\sqrt{rz}~{}\text{Erf}\left[\sqrt{rt_{m}}\sqrt{z}\right]-\frac{ve^{-(\frac{v^{2}}{2}+r)t}}{\sqrt{2}}\int_{\frac{v^{2}}{2r}}^{\frac{v^{2}}{2r}+1}~{}dz~{}e^{rtz}\sqrt{rz}~{}\text{Erf}\left[\sqrt{rt_{m}}\sqrt{z}\right],$
$\displaystyle+re^{-(\frac{v^{2}}{2}+r)t}\int_{\frac{v^{2}}{2r}}^{\frac{v^{2}}{2r}+1}~{}dz~{}z~{}e^{rtz}~{}\text{Erf}\left[\sqrt{rt_{m}}\sqrt{z}\right]\text{Erf}\left[\sqrt{r(t-t_{m})}\sqrt{z}\right]$
$\displaystyle=\frac{e^{-(\frac{v^{2}}{2}+r)t}}{\pi
t_{m}\sqrt{\pi(t-t_{m})}}\left(-\frac{v^{2}\,_{2}F_{2}\left[1,1;\frac{1}{2},2;\frac{t_{m}v^{2}}{2}\right]}{2r}+\left(\frac{v^{2}}{2r}+1\right)\,_{2}F_{2}\left[1,1;\frac{1}{2},2;rt_{m}\left(\frac{v^{2}}{2r}+1\right)\right]-1\right)$
$\displaystyle-\frac{ve^{-(\frac{v^{2}}{2}+r)t}}{\sqrt{2}}\int_{\frac{v^{2}}{2r}}^{\frac{v^{2}}{2r}+1}~{}dz~{}e^{rtz}\sqrt{rz}~{}\text{Erf}\left[\sqrt{rt_{m}}\sqrt{z}\right]$
$\displaystyle+re^{-(\frac{v^{2}}{2}+r)t}\int_{\frac{v^{2}}{2r}}^{\frac{v^{2}}{2r}+1}~{}dz~{}z~{}e^{rtz}~{}\text{Erf}\left[\sqrt{rt_{m}}\sqrt{z}\right]\text{Erf}\left[\sqrt{r(t-t_{m})}\sqrt{z}\right].$
(134)

Joining all the $\mathbb{I}$-functions results in the expression (61) for
$P_{r}(t_{m}|t)$ which was announced in the main text.

##  Appendix H Details of the model systems used in the simulation in Sec. V

In this section, we present details of the processes used in the simulation in
Sec. V. We have used four different model systems as underlying processes.
Three of them are Markovian in nature while one is a non-Markov process. All
of them are subjected to resetting at a rate $r$ which means that the time
intervals between the resetting events were taken from an exponential
distribution namely $p(\tau)=re^{-r\tau}$. We observe trajectories governed by
these processes for a fixed observation time $t$, and compute the maximum
displacement $M$ that it undertook by this time. Moreover, we also note down
the time $t_{m}$ at which this maximum took place. Details of the model
systems are as follows:

  1. 1.

Simple diffusion: Motion of the particle for a simple diffusing particle is
given by

$\displaystyle\frac{dx}{d\tau}=\eta(\tau),$ (135)

where $\eta(\tau)$ is the Gaussian white noise with mean zero and variance
$2D\tau$. This is a Markov process.

  2. 2.

Diffusion with drift: Here, we consider the diffusing particle in the presence
of a drift velocity $v>0$ so that

$\displaystyle\frac{dx}{d\tau}=v+\eta(\tau),$ (136)

which is also a Markov process.

  3. 3.

Random acceleration process: In this case, the position $x(t)$ of the particle
evolves via

$\displaystyle\frac{dx}{d\tau}=v,~{}~{}~{}~{}\frac{dv}{d\tau}=\eta(\tau),$
(137)

so that the process becomes non-Markov in $x$-variable.

  4. 4.

Ornstein-Uhlenbeck process: Here, a diffusing particle is placed in a harmonic
trap with potential strength $\frac{1}{2}x^{2}$ so that position of the
particle evolves as

$\displaystyle\frac{dx}{d\tau}=-x+\eta(\tau).$ (138)

This is also another canonical example of a Markov process.

## References

  * (1) Fisher, R.A. and Tippett, L.H.C., 1928, April. Limiting forms of the frequency distribution of the largest or smallest member of a sample. In Mathematical proceedings of the Cambridge philosophical society (Vol. 24, No. 2, pp. 180-190). Cambridge University Press.
  * (2) Gumbel, E.J., 2004. Statistics of extremes. Courier Corporation.
  * (3) Gnedenko, B., 1943. Sur la distribution limite du terme maximum d’une serie aleatoire. Annals of mathematics, pp.423-453.
  * (4) Leadbetter, M.R., Lindgren, G. and Rootzén, H., 2012. Extremes and related properties of random sequences and processes. Springer Science & Business Media.
  * (5) Fortin, J.Y. and Clusel, M., 2015. Applications of extreme value statistics in physics. Journal of Physics A: Mathematical and Theoretical, 48(18), p.183001.
  * (6) Bouchaud, J.P. and Mézard, M., 1997. Universality classes for extreme-value statistics. Journal of Physics A: Mathematical and General, 30(23), p.7997.
  * (7) Albeverio, S., Jentsch, V. and Kantz, H. eds., 2006. Extreme events in nature and society. Springer Science & Business Media.
  * (8) Majumdar, S.N., Pal, A. and Schehr, G., 2020. Extreme value statistics of correlated random variables: a pedagogical review. Physics Reports, 840, pp.1-32.
  * (9) Fyodorov, Y.V. and Bouchaud, J.P., 2008. Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. Journal of Physics A: Mathematical and Theoretical, 41(37), p.372001.
  * (10) Fyodorov, Y.V., 2010. Multifractality and freezing phenomena in random energy landscapes: An introduction. Physica A: Statistical Mechanics and its Applications, 389(20), pp.4229-4254.
  * (11) Raychaudhuri, S., Cranston, M., Przybyla, C. and Shapir, Y., 2001. Maximal height scaling of kinetically growing surfaces. Physical review letters, 87(13), p.136101.
  * (12) Majumdar, S.N. and Comtet, A., 2004. Exact maximal height distribution of fluctuating interfaces. Physical review letters, 92(22), p.225501.
  * (13) Bar, A., Majumdar, S.N., Schehr, G. and Mukamel, D., 2016. Exact extreme-value statistics at mixed-order transitions. Physical Review E, 93(5), p.052130.
  * (14) Majumdar, S.N., 2010. Real-space condensation in stochastic mass transport models. Exact Methods in Low-dimensional Statistical Physics and Quantum Computing: Lecture Notes of the Les Houches Summer School: Volume 89, July 2008, p.407.
  * (15) Guillet, A., Roldán, E. and Jülicher, F., 2020. Extreme-value statistics of stochastic transport processes. New Journal of Physics, 22(12), p.123038.
  * (16) Dean, D.S. and Majumdar, S.N., 2006. Large deviations of extreme eigenvalues of random matrices. Physical review letters, 97(16), p.160201.
  * (17) Majumdar, S.N. and Vergassola, M., 2009. Large deviations of the maximum eigenvalue for Wishart and Gaussian random matrices. Physical review letters, 102(6), p.060601.
  * (18) Majumdar, S.N. and Schehr, G., 2014. Top eigenvalue of a random matrix: large deviations and third order phase transition. Journal of Statistical Mechanics: Theory and Experiment, 2014(1), p.P01012.
  * (19) Dumonteil, E., Majumdar, S.N., Rosso, A. and Zoia, A., 2013. Spatial extent of an outbreak in animal epidemics. Proceedings of the National Academy of Sciences, 110(11), pp.4239-4244.
  * (20) Krapivsky, P.L. and Majumdar, S.N., 2000. Traveling waves, front selection, and exact nontrivial exponents in a random fragmentation problem. Physical review letters, 85(26), p.5492.
  * (21) S. N. Majumdar, P. L. Krapivsky, Extreme value statistics and traveling fronts: Application to computer science, Phys. Rev. E 65, 036127 (2002).
  * (22) S. N. Majumdar, Traveling front solutions to directed diffusion-limited aggregation, digital search trees, and the Lempel-Ziv data compression algorithm, Phys. Rev. E 68, 026103 (2003).
  * (23) Taloni, A., Vodret, M., Costantini, G. and Zapperi, S., 2018. Size effects on the fracture of microscale and nanoscale materials. Nature Reviews Materials, 3(7), pp.211-224.
  * (24) Schehr, G. and Majumdar, S.N., 2014. Exact record and order statistics of random walks via first-passage ideas. In First-passage phenomena and their applications (pp. 226-251).
  * (25) Schehr, G., Majumdar, S.N., Comtet, A. and Randon-Furling, J., 2008. Exact distribution of the maximal height of p vicious walkers. Physical review letters, 101(15), p.150601.
  * (26) Brunet, É. and Derrida, B., 2009. Statistics at the tip of a branching random walk and the delay of traveling waves. EPL (Europhysics Letters), 87(6), p.60010.
  * (27) Höll, M., Wang, W. and Barkai, E., 2020. Extreme value theory for constrained physical systems. Physical Review E, 102(4), p.042141.
  * (28) Evans, M.R. and Majumdar, S.N., 2008. Condensation and extreme value statistics. Journal of Statistical Mechanics: Theory and Experiment, 2008(05), p.P05004.
  * (29) Frachebourg, L., Ispolatov, I. and Krapivsky, P.L., 1995. Extremal properties of random systems. Physical Review E, 52(6), p.R5727.
  * (30) Godreche, C., Majumdar, S.N. and Schehr, G., 2009. Longest excursion of stochastic processes in nonequilibrium systems. Physical review letters, 102(24), p.240602.
  * (31) Lévy, P., 1940. On certain homogeneous stochastic processes. Compositio mathematica , 7 , pp. 283-339.
  * (32) Andersen, E.S., 1954. On the fluctuations of sums of random variables. Mathematica Scandinavica, pp.263-285.
  * (33) Randon-Furling, J. and Majumdar, S.N., 2007. Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time. Journal of Statistical Mechanics: Theory and Experiment, 2007(10), p.P10008.
  * (34) Majumdar, S.N., Randon-Furling, J., Kearney, M.J. and Yor, M., 2008. On the time to reach maximum for a variety of constrained Brownian motions. Journal of Physics A: Mathematical and Theoretical, 41(36), p.365005.
  * (35) Majumdar, S.N. and Bouchaud, J.P., 2008. Optimal time to sell a stock in the Black–Scholes model: comment on ‘Thou Shalt Buy and Hold’, by A. Shiryaev, Z. Xu and XY Zhou. Quantitative Finance, 8(8), pp.753-760.
  * (36) Schehr, G. and Le Doussal, P., 2010. Extreme value statistics from the real space renormalization group: Brownian motion, Bessel processes and continuous time random walks. Journal of Statistical Mechanics: Theory and Experiment, 2010(01), p.P01009.
  * (37) Majumdar, S.N., Rosso, A. and Zoia, A., 2010. Time at which the maximum of a random acceleration process is reached. Journal of Physics A: Mathematical and Theoretical, 43(11), p.115001.
  * (38) Singh, P. and Kundu, A., 2019. Generalised ‘Arcsine’laws for run-and-tumble particle in one dimension. Journal of Statistical Mechanics: Theory and Experiment, 2019(8), p.083205.
  * (39) Mori, F., Le Doussal, P., Majumdar, S.N. and Schehr, G., 2020. Universal survival probability for a d-dimensional run-and-tumble particle. Physical review letters, 124(9), p.090603.
  * (40) Wiese, K.J., Majumdar, S.N. and Rosso, A., 2011. Perturbation theory for fractional Brownian motion in presence of absorbing boundaries. Physical Review E, 83(6), p.061141.
  * (41) Delorme, M. and Wiese, K.J., 2016. Extreme-value statistics of fractional Brownian motion bridges. Physical Review E, 94(5), p.052105.
  * (42) Sadhu, T., Delorme, M. and Wiese, K.J., 2018. Generalized arcsine laws for fractional Brownian motion. Physical review letters, 120(4), p.040603.
  * (43) Majumdar, S.N., Rosso, A. and Zoia, A., 2010. Hitting probability for anomalous diffusion processes. Physical review letters, 104(2), p.020602.
  * (44) Rambeau, J. and Schehr, G., 2010. Extremal statistics of curved growing interfaces in 1+ 1 dimensions. EPL (Europhysics Letters), 91(6), p.60006.
  * (45) Evans, M.R., Majumdar, S.N. and Schehr, G., 2020. Stochastic resetting and applications. Journal of Physics A: Mathematical and Theoretical, 53(19), p.193001.
  * (46) Evans, M.R. and Majumdar, S.N., 2011. Diffusion with stochastic resetting. Physical review letters, 106(16), p.160601.
  * (47) Evans, M.R. and Majumdar, S.N., 2011. Diffusion with optimal resetting. Journal of Physics A: Mathematical and Theoretical, 44(43), p.435001.
  * (48) Majumdar, S.N., Sabhapandit, S. and Schehr, G., 2015. Dynamical transition in the temporal relaxation of stochastic processes under resetting. Physical Review E, 91(5), p.052131.
  * (49) Pal, A., 2015. Diffusion in a potential landscape with stochastic resetting. Physical Review E, 91(1), p.012113.
  * (50) Eule, S. and Metzger, J.J., 2016. Non-equilibrium steady states of stochastic processes with intermittent resetting. New Journal of Physics, 18(3), p.033006.
  * (51) Nagar, A. and Gupta, S., 2016. Diffusion with stochastic resetting at power-law times. Physical Review E, 93(6), p.060102.
  * (52) Pal, A., Kundu, A. and Evans, M.R., 2016. Diffusion under time-dependent resetting. Journal of Physics A: Mathematical and Theoretical, 49(22), p.225001.
  * (53) Singh, R.K., Metzler, R. and Sandev, T., 2020. Resetting dynamics in a confining potential. Journal of Physics A: Mathematical and Theoretical, 53(50), p.505003.
  * (54) Kusmierz, L., Majumdar, S.N., Sabhapandit, S. and Schehr, G., 2014. First order transition for the optimal search time of Lévy flights with resetting. Physical review letters, 113(22), p.220602.
  * (55) Meylahn, J.M., Sabhapandit, S. and Touchette, H., 2015. Large deviations for Markov processes with resetting. Physical Review E, 92(6), p.062148.
  * (56) Den Hollander, F., Majumdar, S.N., Meylahn, J.M. and Touchette, H., 2019. Properties of additive functionals of Brownian motion with resetting. Journal of Physics A: Mathematical and Theoretical, 52(17), p.175001.
  * (57) Pal, A., Chatterjee, R., Reuveni, S. and Kundu, A., 2019. Local time of diffusion with stochastic resetting. Journal of Physics A: Mathematical and Theoretical, 52(26), p.264002.
  * (58) Domazetoski, V., Masó-Puigdellosas, A., Sandev, T., Méndez, V., Iomin, A. and Kocarev, L., 2020. Stochastic resetting on comblike structures. Physical Review Research, 2(3), p.033027.
  * (59) Reuveni, S., Urbakh, M. and Klafter, J., 2014. Role of substrate unbinding in Michaelis–Menten enzymatic reactions. Proceedings of the National Academy of Sciences, 111(12), pp.4391-4396.
  * (60) Roldan, E., Lisica, A., Sanchez-Taltavull, D. and Grill, S.W., 2016. Stochastic resetting in backtrack recovery by RNA polymerases. Physical Review E, 93(6), p.062411.
  * (61) Bressloff, P.C., 2020. Modeling active cellular transport as a directed search process with stochastic resetting and delays. Journal of Physics A: Mathematical and Theoretical, 53(35), p.355001.
  * (62) Ramoso, Angelo Marco, et al. Stochastic resetting antiviral therapies prevent drug resistance development. EPL (Europhysics Letters) 132.5 (2020): 50003.
  * (63) Luby, M., Sinclair, A. and Zuckerman, D., 1993. Optimal speedup of Las Vegas algorithms. Information Processing Letters, 47(4), pp.173-180.
  * (64) Hamlin, P., Thrasher, W.J., Keyrouz, W. and Mascagni, M., 2019. Geometry entrapment in Walk-on-Subdomains. Monte Carlo Methods and Applications, 25(4), pp.329-340.
  * (65) Pal, A., Kuśmierz, Ł and Reuveni, S., 2020. Search with home returns provides advantage under high uncertainty. Physical Review Research, 2(4), p.043174.
  * (66) Montanari, A. and Zecchina, R., 2002. Optimizing searches via rare events. Physical review letters, 88(17), p.178701.
  * (67) Bressloff, P.C., 2020. Directed intermittent search with stochastic resetting. Journal of Physics A: Mathematical and Theoretical, 53(10), p.105001.
  * (68) Pal, A. and Reuveni, S., 2017. First Passage under Restart. Physical review letters, 118(3), p.030603.
  * (69) Reuveni, S., 2016. Optimal stochastic restart renders fluctuations in first passage times universal. Physical review letters, 116(17), p.170601.
  * (70) Pal, A., Eliazar, I. and Reuveni, S., 2019. First passage under restart with branching. Physical review letters, 122(2), p.020602.
  * (71) Belan, S., 2018. Restart could optimize the probability of success in a Bernoulli trial. Physical review letters, 120(8), p.080601.
  * (72) Chechkin, A. and Sokolov, I.M., 2018. Random search with resetting: a unified renewal approach. Physical review letters, 121(5), p.050601.
  * (73) Pal, A. and Prasad, V.V., 2019. First passage under stochastic resetting in an interval. Physical Review E, 99(3), p.032123.
  * (74) Ray, S., Mondal, D. and Reuveni, S., 2019. Peclet number governs transition to acceleratory restart in drift-diffusion. Journal of Physics A: Mathematical and Theoretical, 52(25), p.255002.
  * (75) Ray, S., 2020. Space-dependent diffusion with stochastic resetting: A first-passage study. The Journal of Chemical Physics, 153(23), p.234904.
  * (76) Pal, A. and Prasad, V.V., 2019. Landau-like expansion for phase transitions in stochastic resetting. Physical Review Research, 1(3), p.032001.
  * (77) Gupta, D., 2019. Stochastic resetting in underdamped Brownian motion. Journal of Statistical Mechanics: Theory and Experiment, 2019(3), p.033212.
  * (78) Bodrova, A.S., Chechkin, A.V. and Sokolov, I.M., 2019. Scaled Brownian motion with renewal resetting. Physical Review E, 100(1), p.012120.
  * (79) Singh, P., 2020. Random acceleration process under stochastic resetting. Journal of Physics A: Mathematical and Theoretical, 53(40), p.405005.
  * (80) Evans, M.R. and Majumdar, S.N., 2018. Run and tumble particle under resetting: a renewal approach. Journal of Physics A: Mathematical and Theoretical, 51(47), p.475003.
  * (81) Santra, I., Basu, U. and Sabhapandit, S., 2020. Run-and-tumble particles in two dimensions under stochastic resetting conditions. Journal of Statistical Mechanics: Theory and Experiment, 2020(11), p.113206.
  * (82) Kumar, V., Sadekar, O. and Basu, U., 2020. Active Brownian motion in two dimensions under stochastic resetting. Physical Review E, 102(5), p.052129.
  * (83) Fuchs, J., Goldt, S. and Seifert, U., 2016. Stochastic thermodynamics of resetting. EPL (Europhysics Letters), 113(6), p.60009.
  * (84) Pal, A. and Rahav, S., 2017. Integral fluctuation theorems for stochastic resetting systems. Physical Review E, 96(6), p.062135.
  * (85) Gupta, D., Plata, C.A. and Pal, A., 2020. Work fluctuations and Jarzynski equality in stochastic resetting. Physical review letters, 124(11), p.110608.
  * (86) Mukherjee, B., Sengupta, K. and Majumdar, S.N., 2018. Quantum dynamics with stochastic reset. Physical Review B, 98(10), p.104309.
  * (87) Basu, U., Kundu, A. and Pal, A., 2019. Symmetric exclusion process under stochastic resetting. Physical Review E, 100(3), p.032136.
  * (88) Karthika, S. and Nagar, A., 2020. Totally asymmetric simple exclusion process with resetting. Journal of Physics A: Mathematical and Theoretical, 53(11), p.115003.
  * (89) Magoni, M., Majumdar, S.N. and Schehr, G., 2020. Ising model with stochastic resetting. Physical Review Research, 2(3), p.033182.
  * (90) Ray, A., Pal, A., Ghosh, D., Dana, S.K. and Hens, C., 2021. Mitigating long transient time in deterministic systems by resetting. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31(1), p.011103.
  * (91) Tal-Friedman, O., Pal, A., Sekhon, A., Reuveni, S. and Roichman, Y., 2020. Experimental realization of diffusion with stochastic resetting. J. Phys. Chem. Lett. 2020, 11, 17, 7350–7355
  * (92) Besga, B., Bovon, A., Petrosyan, A., Majumdar, S.N. and Ciliberto, S., 2020. Optimal mean first-passage time for a Brownian searcher subjected to resetting: experimental and theoretical results. Physical Review Research, 2(3), p.032029.
  * (93) Majumdar, S.N., Mori, F., Schawe, H. and Schehr, G., 2020. Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting. arXiv preprint arXiv:2011.06668.
  * (94) Villarroel, J. and Montero, M., 2018. Continuous-time ballistic process with random resets. Journal of Statistical Mechanics: Theory and Experiment, 2018(12), p.123204.
  * (95) Gradshteyn, I.S. and Ryzhik, I.M., 2014. Table of integrals, series, and products. Academic press.
  * (96) Pal, A., Kuśmierz, Ł. and Reuveni, S., 2019. Time-dependent density of diffusion with stochastic resetting is invariant to return speed. Physical Review E, 100(4), p.040101.
  * (97) Pal, A., Kuśmierz, Ł. and Reuveni, S., 2019. Invariants of motion with stochastic resetting and space-time coupled returns. New Journal of Physics, 21(11), p.113024.
  * (98) Gupta, D., Plata, C.A., Kundu, A. and Pal, A., 2020. Stochastic resetting with stochastic returns using external trap. Journal of Physics A: Mathematical and Theoretical, 54(2), p.025003.
  * (99) Bodrova, A.S. and Sokolov, I.M., 2020. Resetting processes with noninstantaneous return. Physical Review E, 101(5), p.052130.
  * (100) Masó-Puigdellosas, A., Campos, D. and Méndez, V., 2019. Transport properties of random walks under stochastic noninstantaneous resetting. Physical Review E, 100(4), p.042104.