# First-passage functionals for Ornstein Uhlenbeck process with stochastic
resetting

Ashutosh Dubey and Arnab Pal  The Institute of Mathematical Sciences, CIT
Campus, Taramani, Chennai 600113, India & Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar, Mumbai 400094, India
ashutoshrd<EMAIL_ADDRESS>
###### Abstract

We study the statistical properties of first-passage Brownian functionals
(FPBFs) of an Ornstein-Uhlenbeck (OU) process in the presence of stochastic
resetting. We consider a one dimensional set-up where the diffusing particle
sets off from $x_{0}$ and resets to $x_{R}$ at a certain rate $r$. The
particle diffuses in a harmonic potential (with strength $k$) which is
centered around the origin. The center also serves as an absorbing boundary
for the particle and we denote the first passage time of the particle to the
center as $t_{f}$. In this set-up, we investigate the following functionals:
(i) local time $T_{loc}=\int_{0}^{t_{f}}d\tau~{}\delta(x-x_{R})$ i.e., the
time a particle spends around $x_{R}$ until the first passage, (ii) occupation
or residence time $T_{res}=\int_{0}^{t_{f}}d\tau~{}\theta(x-x_{R})$ i.e., the
time a particle typically spends above $x_{R}$ until the first passage and
(iii) the first passage time $t_{f}$ to the origin. We employ the Feynman-Kac
formalism for renewal process to derive the analytical expression for the
first moment of all the three FPBFs mentioned above. In particular, we find
that resetting can either prolong or shorten the mean residence and first
passage time depending on the system parameters. The transition between these
two behaviors or phases can be characterized precisely in terms of optimal
resetting rates, which interestingly undergo a continuous transition as we
vary the trap stiffness $k$. We characterize this transition and identify the
critical -parameter & -coefficient for both the cases. We also showcase other
interesting interplay between the resetting rate and potential strength on the
statistics of these observables. Our analytical results are in excellent
agreement with the numerical simulations.

##  1 Introduction

Brownian functionals, in their various different forms, appear ubiquitously in
many different fields such as statistical physics [1, 2, 3, 4], stochastic
processes [5, 6, 7], chemistry [8, 9], finance [10], and computer science [1].
The Brownian functional over fixed time interval $[0,t]$ is defined as
$V=\int_{0}^{t}U[x(\tau)]d\tau$, where $x(\tau)$ is a Brownian path and $U[x]$
is some specified function of the path, which depends on the quantity one
wants to calculate [1]. Often, we are also interested in an another class of
Brownian functional namely the first passage Brownian functional (FPBF), which
is defined over random interval of time $[0,t_{f}]$ so that

$\displaystyle V=\int_{0}^{t_{f}}U[x(\tau)]d\tau~{},$ (1)

where $t_{f}$ is the first-passage time which is random and will be defined
more precisely later. In here, we consider an Ornstein-Uhlenbeck (OU) process
i.e., a Brownian particle in a harmonic potential $\frac{1}{2}kx^{2}$, where
$k$ is the strength or stiffness of the potential. The potential is centered
at the origin. The equation of motion for the position $x(\tau)$ of the
particle can be written in terms of the overdamped Langevin equation

$\displaystyle\frac{dx(\tau)}{d\tau}=-kx+\xi(\tau),$ (2)

where $\xi(\tau)$ is the Gaussian white noise of zero mean and correlations
$\langle\xi(\tau)\xi(\tau^{\prime})\rangle=2D\delta(\tau-\tau^{\prime})$.
Here, $D$ is the diffusion constant and for simplicity, we have set $D=1$
unless specified otherwise. In addition, we will assume that there is an
absorbing boundary at the origin and the process ends as soon as the particle
hits the origin, and so, $t_{f}$ encodes the statistics of this time. Due to
the absorption, naturally there is no equilibration for the particle in this
confining potential.

In literature, Brownian functionals appear in a wide variety of problems
across different disciplines. For instance, an important studied Brownian
functional is the local time which quantifies the amount of time spent by the
Brownian particle in the neighborhood of a desired location [1, 17, 12, 11,
13, 14, 15, 16]. Another important example of such Brownian functional is
occupation or residence time, which specifies the total time spent by a
diffusing particle in a certain domain. The celebrated Lévy arc-sine law is an
estimation of residence time for simple 1D diffusion process when considering
the total time spent above the origin [1, 18, 19]. Over the years, residence
time statistics has been computed for simple diffusion [8, 9] diffusion in
random and porous landscape [16, 20], heterogeneous medium [21] or in a
potential [17], in confinements[22], in active particle systems [23], in
molecular biology [24] and in experiments [25]. The list is not exhaustive but
most of these Brownian functionals have been calculated for a fixed interval
of time.

Another important class of functional is the first passage Brownian
functionals (FPBFs) which are measured upto a random interval of time $t_{f}$,
which keeps a record of time when a process satisfies certain criterion e.g. a
particle reaching a desired location or marking turnover of an enzymatic
reaction. The field of first passage time itself is remarkably broad and there
are a myriad of applications that span over many different topics. We refer to
[3, 26, 27, 28] for an extensive read of the subject. The other important
FPBFs studied in the literature are area and extreme maximum which have
applications in statistical physics and queuing theory [29, 30, 31]. The FPBF
has also been calculated in the context of snowmelt dynamics [32], biopolymer
translocation dynamics [33], DNA breathing dynamics [34] and barrierless
reactions [35]. There has been a renewed interest for studying functionals in
the presence of stochastic resetting. Brownian functionals such as the simple
first passage time [36, 37, 38, 39, 40, 41]; local time [42] & occupation time
[43, 44] till fixed interval, and the same along with the area functional till
first passage event [45] have been calculated in the presence of stochastic
resetting.

In this work, we focus on various first passage functionals for OU process in
the presence of stochastic resetting. To the best of our knowledge, treatment
for OU process with resetting has been limited to non-equilibrium properties
such as steady states and relaxation [46, 47, 48], large deviations of time-
additive functions [49], estimating work fluctuations [50], and the first
passage time [51], but a comprehensive study for the local and
occupation/residence time functionals appears to be missing. This work tries
to fill some of this void and while doing so, it also reveals an optimization
and resetting transition for the residence time similar to the first passage
time. For completeness, we mention a few works along this direction but in the
absence of resetting. The first passage time [7], area [52, 53, 54] and local
time [55] have been calculated in the absence of stochastic resetting. Such
computation for OU functionals has been normally invoked in biological systems
particularly in the modeling of neuronal activity as mentioned in the
reference [7].

The central goal here is to study the statistical properties of FPBF-s for OU
process in presence of resetting. We consider that the particle is reset to a
location $x_{R}>0$ at a rate $r$. Under this assumption, motion of the
particle in microscopic time step $\Delta t$ can be updated in the following
way

$\displaystyle x(t+\Delta t)=\begin{cases}&x(t)-kx(t)\Delta t+\xi(t)\Delta
t,~{}~{}~{}~{}~{}~{}\text{with prob }(1-r\Delta t),\\\
&x_{R},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{with
prob }r\Delta t.\end{cases}$ (3)

We consider the following observables in this study:

  1. 1.

Local time: Local time $T_{loc}$ refers to the amount of time spent by the
particle in the neighbourhood of some desired position (say $x_{\ell}$) till
the first passage time. For this quantity, $U(x)=\delta(x-x_{\ell})$ and hence
$T_{loc}(x_{\ell})=\int_{0}^{t_{f}}~{}d\tau~{}\delta[x(\tau)-x_{\ell}]$. The
$T_{loc}$ has explicitly been calculated near the resetting position in which
case $x_{\ell}=x_{R}$.

  2. 2.

Residence time: The second functional is residence/occupation time, which
estimates the cumulative time spent by the Brownian particle in a certain
domain till the first passage time. In this case, $U(x)=\theta(x-x_{R})$,
where $\theta(x)$ is the Heaviside step function and the residence time takes
the form $T_{res}(x_{R})=\int_{0}^{t_{f}}~{}d\tau~{}\theta(x(\tau)-x_{R})$.

  3. 3.

First-passage time: Finally, we consider the Brownian functional of the first-
passage time itself. In this case $U(x)=1$ and hence the relevant functional
$V=\int_{0}^{t_{f}}1~{}d\tau=t_{f}$ is the first passage time.

In this manuscript, we have illustrated, in detail, the effect of resetting
and potential strength on the first moment of the above-mentioned FPBFs for
the OU process. We find that the mean local time monotonically increases as a
function of the resetting rate $r$. Furthermore, the mean residence time and
the first passage time display both monotonic and non-monotonic behavior as a
function of the resetting rate $r$. For non-monotonic case, it is generically
found that resetting renders these times minima. The optimal resetting rates
are shown to undergo a continuous phase transition as a function of the
potential stiffness. We identify the critical strengths and characterize them
using the generic framework of first passage under resetting wherever
applicable. The manuscript is organized as follows. In section 2, we revisit
Feynman-Kac formalism and derive a backward differential equation for the
moment generating function of an arbitrary functional. Utilizing this backward
differential equation, we investigate the statistics of the local time in Sec.
3, residence time in Sec. 4, and the first passage time in Sec. 5. We conclude
our manuscript with a brief summary and future outlook in Sec. 6.

##  2 General formulation

This section will sketch out the essential steps to compute the first passage
OU functionals in the presence of stochastic resetting. To this end, we will
follow the Feynman-Kac formalism [1, 45]. Since we are focusing on the first
passage functionals, the first step will be to derive a backward master
equation for the moment generating function which is defined as

$\displaystyle Q(p,x_{0})$
$\displaystyle=\int_{0}^{\infty}dV~{}e^{-pV}~{}P_{R}(V,x_{0})$ (4)
$\displaystyle=\langle e^{-p\int_{0}^{t_{f}}U[x(\tau)]d\tau}\rangle,$ (5)

where the average $\langle..\rangle$ in Eq. 5 represents the averaging over
all such paths which start at some point $x_{0}$ at time $\tau=0$ and
terminate at the origin at time $\tau=t_{f}$ in the presence of repeated
resetting to $x_{R}$. There are three components of stochasticity in this
problem (i) intrinsic thermal noise in the OU process, (ii) temporal
randomness due to resetting, and (iii) randomness incurred to the final
observation time $\tau=t_{f}$ due to the absorbing boundary. The moments of
first passage Brownian functional $V$ can be derived directly from the moment
generating function i.e.,

$\displaystyle\langle
V^{m}\rangle=(-1)^{m}\left(\frac{\partial^{m}Q(p,x_{0})}{\partial
p^{m}}\right)\bigg{|}_{p\to 0}.$ (6)

To proceed further, consider one realization of a stochastic path
$\\{x(\tau);0\leq\tau\leq t_{f}\\}$ and split that into two parts: (i) the
first interval $[0,\Delta t]$ and (ii) the rest interval $[\Delta t,t_{f}]$
with $\Delta t\to 0$. In the first part, the Brownian path propagates from
$x_{0}$ at $\tau=0$ to $x_{0}^{\prime}=x_{0}+\Delta x$ in time $\Delta t$. In
the rest interval, the process continues until the particle, which is at
$x_{0}^{\prime}$ at time $\Delta t$, is absorbed at the origin at a later time
$t_{f}$, possibly undergoing many resetting events. This path-decomposition
leads us to break the integration in Eq. (5) as
$\int_{0}^{t_{f}}=\int_{0}^{\Delta t}+\int_{\Delta t}^{t_{f}}$ and in the
limit $\Delta t\to 0$, we get

$\displaystyle Q(p,x_{0})$ $\displaystyle=\langle e^{-pU(x_{0})\Delta
t}~{}e^{-p\int_{\Delta t}^{t_{f}}U[x(\tau)]d\tau}\rangle,$
$\displaystyle=\langle e^{-pU(x_{0})\Delta t}Q(p,x_{0}^{\prime})\rangle.$ (7)

To evaluate $x_{0}^{\prime}$, note that the Brownian particle can reset to
$x_{R}$ with probability $r\Delta t$ in an infinitesimal time $\Delta t$ or it
can keep evolving according to Eq. (3). Putting these two contributions
together, using the noise properties $\langle\xi(0)\rangle$ and
$\langle\xi^{2}(0)\rangle$, and keeping only the leading order terms in
$\Delta t$, we obtain the following backward equation in the presence of
resetting

$\displaystyle\frac{\partial^{2}Q(p,x_{0})}{\partial
x_{0}^{2}}-kx_{0}\frac{\partial Q(p,x_{0})}{\partial
x_{0}}-pU(x_{0})Q(p,x_{0})-rQ(p,x_{0})+rQ(p,x_{R})=0.$ (8)

Eq. (8), valid in the domain $x_{0}\in[0,\infty]$, is the key equation of this
work. This is moreover supplemented with the following boundary conditions

$\displaystyle Q(p,x_{0}\to 0^{+})=1,~{}$ (9) $\displaystyle
Q(p,x_{0}\to\infty)<\infty.$ (10)

The first boundary condition is obvious since the particle, starting close to
the origin i.e., $x_{0}\rightarrow 0$, will immediately be absorbed yielding
FPT $t_{f}\rightarrow 0$. Therefore, substituting
$V=\int_{0}^{t_{f}}~{}d\tau~{}U[x(\tau)]\to 0$ in Eq. (5) results in the first
boundary condition (9). The second boundary condition can be explained in the
following way. Consider a situation when the particle had started from
$x_{0}\rightarrow\infty$. Nonetheless, the next resetting move will bring it
back to $x_{R}$ in a finite time span. Therefore, the FPT $t_{f}$ should be
generally finite for any non-zero and finite value of the resetting rate $r$.
Hence, the functional $V$ should remain finite for the fixed value of
resetting position $x_{R}$ even when the initial position
$x_{0}\rightarrow\infty$. This leads to the second boundary condition (10).
Moving forward, the steps are as follows. Depending on the functional
$V[x(\tau)]$ of interest, one first solves the corresponding backward
differential Eq. (8) with the appropriate boundary conditions. This gives us
the moment generating function $Q(p,x_{0})$ which then leads to the moments of
the functional using Eq. 6. We study each of the aforementioned functionals in
the subsequent sections.

##  3 Local time

The local time density that a particle spents at the position $x_{\ell}$ until
the first-passage time $t_{f}$ is given by

$\displaystyle
T_{loc}(x_{\ell})=\int_{0}^{t_{f}}d\tau~{}\delta[x(\tau)-x_{\ell}]~{}.$ (11)

In Eq. (12) the delta function is defined within the limiting sense as below:

$\displaystyle T_{loc}(x_{\ell})=\lim_{\epsilon\to
0}\frac{T_{2\epsilon}(x_{\ell})}{2\epsilon},~{}~{}~{}\text{where}~{}~{}T_{2\epsilon}(x_{\ell})=\int_{0}^{t_{f}}d\tau~{}[\theta(x(\tau)-x_{\ell}-\epsilon)-\theta(x(\tau)-x_{\ell}+\epsilon)].$
(12)

Here, $T_{2\epsilon}(x_{\ell})$ measure of the total time spent by the
particle inside the box $[x_{\ell}-\epsilon,x_{\ell}+\epsilon]$ till the first
passage event. Thus, the normalization condition can be understood as
$\int_{0}^{\infty}T_{loc}(x_{\ell})dx_{\ell}=t_{f}$. Since we are interested
in estimating the local time density near the resetting location $x_{R}$, we
substitute $U(x)=\delta(x-x_{R})$ into the backward Eq. (8) to obtain

$\displaystyle\frac{\partial^{2}Q(p,x_{0})}{\partial
x_{0}^{2}}-kx_{0}\frac{\partial Q(p,x_{0})}{\partial
x_{0}}-p~{}\delta(x_{0}-x_{R})Q(p,x_{0})-rQ(p,x_{0})+rQ(p,x_{R})=0.$ (13)

For $x_{0}\neq x_{R}$, the $\delta(x_{0}-x_{R})$ term is eliminated and Eq.
(13) can be written as

$\displaystyle\frac{\partial^{2}Q(p,x_{0})}{\partial
x_{0}^{2}}-kx_{0}\frac{\partial Q(p,x_{0})}{\partial
x_{0}}-rQ(p,x_{0})+rQ(p,x_{R})=0.$ (14)

The homogeneous part of the Eq. (14)

$\displaystyle\frac{\partial^{2}Q(p,x_{0})}{\partial
x_{0}^{2}}-kx_{0}\frac{\partial Q(p,x_{0})}{\partial x_{0}}-rQ(p,x_{0})=0$
(15)

can be solved by changing the variable from $Q(p,x_{0})$ to
$\tilde{Q}(p,x_{0})$ using the following transformation

$\displaystyle\tilde{Q}(p,x_{0})=\exp\bigg{(}-\frac{1}{2}\int^{x_{0}}kx^{\prime}dx^{\prime}\bigg{)}Q(p,x_{0})=\exp\bigg{(}-\frac{kx_{0}^{2}}{4}\bigg{)}Q(p,x_{0}).$
(16)

Using (16) in (15), we obtain the following differential equation for
$\tilde{Q}(p,x_{0})$

$\displaystyle\frac{\partial^{2}\tilde{Q}(p,x_{0})}{\partial
x_{0}^{2}}+\bigg{(}-r+\frac{k}{2}-\frac{k^{2}x^{2}}{4}\bigg{)}\tilde{Q}(p,x_{0})=0,$
(17)

which has a standard solution [56, 57]

$\displaystyle\tilde{Q}(p,x_{0})=\mathcal{A}D_{-\frac{r}{k}}(\sqrt{k}x_{0})+\mathcal{B}D_{\frac{r}{k}-1}(i\sqrt{k}x_{0}),$
(18)

where $D_{\lambda}(z)$ is a parabolic cylinder or Weber function and
$i=\sqrt{-1}$ is the imaginary number. Transforming back to the original
variable, we have the following solutions

$\displaystyle
Q(p,x_{0})=\begin{cases}&e^{kx_{0}^{2}/4}[\mathcal{A}D_{-\frac{r}{k}}(\sqrt{k}x_{0})+\mathcal{B}D_{\frac{r}{k}-1}(i\sqrt{k}x_{0})]+Q(p,x_{R}),~{}~{}~{}~{}\text{for
}x_{0}<x_{R},\\\
&e^{kx_{0}^{2}/4}[\mathcal{C}D_{-\frac{r}{k}}(\sqrt{k}x_{0})+\mathcal{D}D_{\frac{r}{k}-1}(i\sqrt{k}x_{0})]+Q(p,x_{R}),~{}~{}~{}~{}\text{for
}x_{0}>x_{R}.\end{cases}$ (19)

There are four constants namely $\mathcal{A},~{}\mathcal{B},~{}\mathcal{C}$
and $\mathcal{D}$ in the equation above. To solve them, four boundary
conditions on $Q(p,x_{0})$ need to be specified. Out of these, two boundary
conditions originate from the behaviour of $Q(p,x_{0})$ as $x_{0}\to 0^{+}$
and $x_{0}\to\infty$ as mentioned in Eqs. (9) and (10) respectively. The other
two conditions are essentially matching conditions – one of them being the
continuity of $Q(p,x_{0})$ across the point $x_{0}=x_{R}$ i.e.,

$\displaystyle~{}~{}Q(p,x_{R}^{+})=Q(p,x_{R}^{-}).$ (20)

The other one is a discontinuity condition in the derivative of $Q(p,x_{0})$
across the point $x_{0}=x_{R}$ which can be obtained by integrating Eq. (13)
from $x_{0}=x_{R}-\delta$ to $x_{0}=x_{R}+\delta$ and taking $\delta\to 0^{+}$
so that

$\displaystyle\left(\frac{\partial Q}{\partial
x_{0}}\right)_{x_{R}^{+}}-\left(\frac{\partial Q}{\partial
x_{0}}\right)_{x_{R}^{-}}=pQ(p,x_{R}).$ (21)

The constant $\mathcal{D}$ becomes zero owing to the boundary condition (10).
Setting $x_{0}=x_{R}$, and after some manipulations, it can be seen from Eq.
(19) that $Q(p,x_{R})$ is completely characterized by $\mathcal{C}$ which we
find to be

Figure 1: Comparison of the first moment of $T_{loc}(x_{R})$ for OU process
with numerical simulation for different values of (a) stiffness constant $k$
and (b) resetting rate $r$. Here, $x_{0}=x_{R}=2.5$. The lines and symbols are
analytical and numerical results respectively.

$\displaystyle\mathcal{C}=\frac{\displaystyle\begin{multlined}pQ(p,x_{R})e^{-kx_{R}^{2}/4}[D_{-\frac{r}{p}}(0)D_{\frac{r}{p}-1}(i\sqrt{k}x_{R})-D_{\frac{r}{p}-1}(0)D_{-\frac{r}{p}}(\sqrt{k}x_{R})]+[1-Q(p,x_{R})]\\\
\times[i\sqrt{k}D_{-\frac{r}{p}}(\sqrt{k}x_{R})D_{\frac{r}{p}}(i\sqrt{k}x_{R})-\frac{r}{\sqrt{k}}D_{-(\frac{r}{p}+1)}(\sqrt{k}x_{R})D_{\frac{r}{p}-1}(\sqrt{k}x_{R})]\end{multlined}pQ(p,x_{R})e^{-kx_{R}^{2}/4}[D_{-\frac{r}{p}}(0)D_{\frac{r}{p}-1}(i\sqrt{k}x_{R})-D_{\frac{r}{p}-1}(0)D_{-\frac{r}{p}}(\sqrt{k}x_{R})]+[1-Q(p,x_{R})]\\\
\times[i\sqrt{k}D_{-\frac{r}{p}}(\sqrt{k}x_{R})D_{\frac{r}{p}}(i\sqrt{k}x_{R})-\frac{r}{\sqrt{k}}D_{-(\frac{r}{p}+1)}(\sqrt{k}x_{R})D_{\frac{r}{p}-1}(\sqrt{k}x_{R})]}{D_{-\frac{r}{p}}(0)[i\sqrt{k}D_{-\frac{r}{p}}(\sqrt{k}x_{R})D_{\frac{r}{p}}(i\sqrt{k}x_{R})-\frac{r}{\sqrt{k}}D_{-(\frac{r}{p}+1)}(\sqrt{k}x_{R})D_{\frac{r}{p}-1}(i\sqrt{k}x_{R})]},$
(24)

where we have used the following relations [56, 57]

$\displaystyle\frac{d^{n}}{dz^{n}}\bigg{[}e^{z^{2}/4}D_{\nu}(z)\bigg{]}=(-1)^{n}(-\nu)_{n}e^{z^{2}/4}D_{\nu-n}(z)$
(25)
$\displaystyle\frac{d^{n}}{dz^{n}}\bigg{[}e^{-z^{2}/4}D_{\nu}(z)\bigg{]}=(-1)^{n}e^{-z^{2}/4}D_{\nu+n}(z)$
(26)

Substituting Eq. (24) into the second line of Eq. (19) yields the complete
solution for the local time moment generating function

$\displaystyle
Q(p,x_{R})=\left[1+\frac{pe^{-kx_{R}^{2}/4}[D_{-\frac{r}{k}}(0)D_{\frac{r}{k}-1}(i\sqrt{k}x_{R})-D_{\frac{r}{k}-1}(0)D_{-\frac{r}{k}}(\sqrt{k}x_{R})]}{\frac{r}{\sqrt{k}}D_{-(\frac{r}{k}+1)}(\sqrt{k}x_{R})D_{\frac{r}{k}-1}(i\sqrt{k}x_{R})-i\sqrt{k}D_{-\frac{r}{k}}(\sqrt{k}x_{R})D_{\frac{r}{k}}(i\sqrt{k}x_{R})}\right]^{-1},$
(27)

The first moment of $T_{loc}$ can be calculated by inserting $Q(p,x_{R})$ from
the above into Eq. (6). Skipping details, we find

$\displaystyle\langle
T_{loc}(x_{R})\rangle=\frac{2^{-(\frac{1}{2}+\frac{r}{2k})}e^{-kx_{R}^{2}/4}\sqrt{\pi
k}[2^{\frac{r}{k}}\Gamma(\frac{1}{2}+\frac{r}{2k})D_{-\frac{r}{k}}(\sqrt{k}x_{R})-\sqrt{2}\Gamma(1-\frac{r}{2k})D_{\frac{r}{k}-1}(i\sqrt{k}x_{R})]}{\Gamma(1-\frac{r}{2k})\Gamma(\frac{1}{2}+\frac{r}{2k})[ikD_{-\frac{r}{k}}(\sqrt{k}x_{R})D_{\frac{r}{k}}(i\sqrt{k}x_{R})-rD_{-\frac{r}{k}-1}(\sqrt{k}x_{R})D_{\frac{r}{p}-1}(i\sqrt{k}x_{R})]},$
(28)

where $\Gamma(.)$ is the Euler gamma function. In the $r=0$ limit, the above
Eq. (28) takes the following form

$\displaystyle\langle
T_{loc}\rangle\big{|}_{r=0}=\frac{2^{-1/2}e^{-kx_{0}^{2}/4}\sqrt{\pi
k}[\Gamma(1/2)D_{0}(\sqrt{k}x_{0})-\Gamma(1)D_{-1}(i\sqrt{k}x_{0})]}{i\Gamma(1)\Gamma(1/2)kD_{0}(\sqrt{k}x_{0})D_{0}(i\sqrt{k}x_{0})},$
(29)

which can be further simplified by noting $D_{0}(z)=e^{-z^{2}/4}$ and
$D_{-1}(z)=\sqrt{\frac{\pi}{2}}e^{z^{2}/4}[1-\text{erf}(z/\sqrt{2})]$ to find

$\displaystyle\langle
T_{loc}\rangle\big{|}_{r=0}=\sqrt{\frac{\pi}{2k}}\exp\bigg{[}-\frac{kx_{0}^{2}}{2}\bigg{]}\text{erfi
}\Big{(}\sqrt{\frac{k}{2}}x_{0}\Big{)},$ (30)

where $\text{erfi }(\cdot)$ is the imaginary error function defined by
$\text{erfi }(z)=-i\text{erf}(iz)$ [55]. In Fig. 1 (a), we have plotted the
first moment of local time density $\langle T_{loc}(x_{R})\rangle$ as a
function of resetting $r$ for different values of potential strength $k$. We
found a monotonic increase in the mean local time as the resetting rate is
gradually increased. This is due to the fact that the particle will spend more
time in the vicinity of $x_{R}$ if the frequency of resetting is higher. Fig.
1 (b) shows the monotonic decrease in $\langle T_{loc}(x_{R})\rangle$ with
$k$. As $k$ increases, the particle will have more drift towards the minimum
of the potential, which is also the location of the absorbing boundary, and it
gets absorbed more easily. As a result, the particle will spend lesser time
around the resetting location $x_{R}$. At a sufficiently large value of $k$,
$\langle T_{loc}(x_{R})\rangle$ is almost in the same order for different $r$
since the trap is so strong that the particle is not able to diffuse much
through the space and gets absorbed almost momentarily despite a possible
number of resetting events. For completeness, we also verify the mean local
time $\langle T_{loc}\rangle\big{|}_{r=0}$ in Fig. 2a. We compared our
analytical results with the numerical simulations and found an excellent
agreement.

Figure 2: Comparison of the theoretical expressions for the first moment of
(a) local time density $\langle T_{loc}\rangle|_{r=0}$ and (b) residence time
$\langle T_{res}\rangle|_{r=0}$ as a function of stiffness constant $k$ in the
absence of resetting with numerical simulation. The lines and symbols are
analytical and numerical simulations results respectively. We have set:
$x_{0}=2.5$.

##  4 Residence time

In this section, we discuss the effect of resetting on the first moment of
occupation or residence time
$T_{res}=\int_{0}^{t_{f}}~{}d\tau~{}\theta(x(\tau)-x_{R})$. As sketched in
Sec. 2, we need to solve the backward differential equation (8) with
$U(x)=\theta(x-x_{R})$ which gives

$\displaystyle\frac{\partial^{2}Q(p,x_{0})}{\partial
x_{0}^{2}}-kx_{0}\frac{\partial Q(p,x_{0})}{\partial
x_{0}}-p\theta(x_{0}-x_{R})Q(p,x_{0})-rQ(p,x_{0})+rQ(p,x_{R})=0.$ (31)

Solving this equation separately for two regions $x_{0}\geq x_{R}$ and
$x_{0}<x_{R}$, we get

$\displaystyle
Q(p,x_{0})=\begin{cases}&e^{kx_{0}^{2}/4}[\mathcal{C}_{1}D_{-\frac{r}{k}}(\sqrt{k}x_{0})+\mathcal{C}_{2}D_{\frac{r}{k}-1}(i\sqrt{k}x_{0})]+Q(p,x_{R}),~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}x_{0}<x_{R},\\\
&e^{kx_{0}^{2}/4}[\mathcal{C}_{3}D_{-\frac{(r+p)}{k}}(\sqrt{k}x_{0})+\mathcal{C}_{4}D_{\frac{(r+p)}{k}-1}(i\sqrt{k}x_{0})]+\frac{r}{r+p}Q(p,x_{R}),~{}~{}\text{for
}x_{0}>x_{R}.\end{cases}$ (32)

As before, the four constants
$\mathcal{C}_{1},~{}\mathcal{C}_{2},~{}\mathcal{C}_{3}$ and $\mathcal{C}_{4}$
(which are functions of $x_{R}$ and $p$, but are independent of $x_{0}$) can
be evaluated using the boundary conditions mentioned in Eqs. (9) and (10)
along with the continuity conditions for $Q(p,x_{0})$ and its derivative
across $x_{0}=x_{R}$ namely

$\displaystyle Q(p,x_{R}^{+})=Q(p,x_{R}^{-}),$ (33)
$\displaystyle\left(\frac{\partial Q}{\partial
x_{0}}\right)_{x_{R}^{+}}=\left(\frac{\partial Q}{\partial
x_{0}}\right)_{x_{R}^{-}}.$ (34)

With these conditions, all the four constants can be calculated directly from
Eq. (32). As we are interested in $x_{0}=x_{R}$, we need only
$\mathcal{C}_{3}$ and $\mathcal{C}_{4}$. However, the boundary condition (10)
demands $\mathcal{C}_{4}=0$ and we are left with

$\displaystyle\mathcal{C}_{3}=\frac{\begin{multlined}[1-Q(p,x_{R})][rD_{-(\frac{r}{k}+1)}(\sqrt{k}x_{R})D_{(\frac{r}{k}-1)}(i\sqrt{k}x_{R})-ikD_{-\frac{r}{k}}(\sqrt{k}x_{R})D_{\frac{r}{k}}(i\sqrt{k}x_{R})]\\\
+\frac{p}{r+p}e^{-kx_{R}^{2}/4}Q(p,x_{R})[ikD_{-\frac{r}{k}}(0)D_{\frac{r}{k}}(i\sqrt{k}x_{R})-rD_{\frac{r}{k}-1}(0)D_{-(\frac{r}{k}+1)}(\sqrt{k}x_{R})]\end{multlined}[1-Q(p,x_{R})][rD_{-(\frac{r}{k}+1)}(\sqrt{k}x_{R})D_{(\frac{r}{k}-1)}(i\sqrt{k}x_{R})-ikD_{-\frac{r}{k}}(\sqrt{k}x_{R})D_{\frac{r}{k}}(i\sqrt{k}x_{R})]\\\
+\frac{p}{r+p}e^{-kx_{R}^{2}/4}Q(p,x_{R})[ikD_{-\frac{r}{k}}(0)D_{\frac{r}{k}}(i\sqrt{k}x_{R})-rD_{\frac{r}{k}-1}(0)D_{-(\frac{r}{k}+1)}(\sqrt{k}x_{R})]}{\begin{multlined}(r+p)D_{-(\frac{r+p}{k}+1)}(\sqrt{k}x_{R})[D_{-\frac{r}{k}}(0)D_{\frac{r}{k}-1}(i\sqrt{k}x_{R})-D_{\frac{r}{k}-1}(0)D_{-\frac{r}{k}}(\sqrt{k}x_{R})]\\\
+D_{-\frac{r+p}{k}}(\sqrt{k}x_{R})[ikD_{\frac{r}{k}}(i\sqrt{k}x_{R})D_{-\frac{r}{k}}(0)-rD_{\frac{r}{k}-1}(0)D_{-(\frac{r}{k}+1)}(\sqrt{k}x_{R})]\end{multlined}(r+p)D_{-(\frac{r+p}{k}+1)}(\sqrt{k}x_{R})[D_{-\frac{r}{k}}(0)D_{\frac{r}{k}-1}(i\sqrt{k}x_{R})-D_{\frac{r}{k}-1}(0)D_{-\frac{r}{k}}(\sqrt{k}x_{R})]\\\
+D_{-\frac{r+p}{k}}(\sqrt{k}x_{R})[ikD_{\frac{r}{k}}(i\sqrt{k}x_{R})D_{-\frac{r}{k}}(0)-rD_{\frac{r}{k}-1}(0)D_{-(\frac{r}{k}+1)}(\sqrt{k}x_{R})]}.$
(39)

On substituting (39) in the second line of Eq. (32), we finally arrive at the
following expression for the moment generating function of the residence time

$\displaystyle
Q(p,x_{R})=\left[1+\frac{pe^{-kx_{R}^{2}/4}D_{-\frac{r+p}{k}-1}(\sqrt{k}x_{R})[D_{-\frac{r}{k}}(0)D_{\frac{r}{k}-1}(i\sqrt{k}x_{R})-D_{\frac{r}{k}-1}(0)D_{-\frac{r}{k}}(\sqrt{k}x_{R})]}{D_{-\frac{r+p}{k}}(\sqrt{k}x_{R})[rD_{-\frac{r}{k}-1}(\sqrt{k}x_{R})D_{\frac{r}{k}-1}(i\sqrt{k}x_{R})-ikD_{-\frac{r}{k}}(\sqrt{k}x_{R})D_{\frac{r}{k}}(i\sqrt{k}x_{R})]}\right]^{-1}.$
(40)

Figure 3: Comparison of the first moment of residence time $\langle
T_{res}(x_{R})\rangle$ for OU process with numerical simulation as a function
of (a) stiffness constant $k$ for different values of $r$ and, (b) resetting
rate $r$ for different values of $k$. The lines and symbols are analytical and
numerical simulations results respectively. (c) Continuous transition of
optimal resetting rate $r^{c}$, which from a finite value transits to zero at
the critical point $k^{c}=0.22624$. The locus of $r^{c}$ (solid line) is
obtained by numerically minimizing $\langle T_{res}(x_{R})\rangle$ for
different values of $k$. In particular, the diamond and square markers depict
$r^{c}>0$ and $r^{c}=0$ respectively obtained by minimizing $\langle
T_{res}(x_{R})\rangle$ from 3(b) for two different values of $k$. Inset:
$r^{c}$ near the critical point $k^{c}$ shows a power law behavior. In the
simulation, we have set: $x_{0}=x_{R}=2.5$ and $D=1$.

The mean residence time can be calculated using Eq. (6) and reads

$\displaystyle\langle
T_{res}(x_{R})\rangle=\frac{i\sqrt{\pi}2^{-1-\frac{r}{2k}}e^{-kx_{R}^{2}/4}D_{-\frac{r}{k}-1}(\sqrt{k}x_{R})[2\Gamma(1-\frac{r}{2k})D_{\frac{r}{k}-1}(i\sqrt{k}x_{R})-2^{\frac{1}{2}+\frac{r}{k}}\Gamma(\frac{1}{2}+\frac{r}{2k})D_{-\frac{r}{k}}(\sqrt{k}x_{R})]}{\Gamma(1-\frac{r}{2k})\Gamma(\frac{1}{2}+\frac{r}{2k})D_{-\frac{r}{k}}(\sqrt{k}x_{R})[kD_{-\frac{r}{k}}(\sqrt{k}x_{R})D_{\frac{r}{k}}(i\sqrt{k}x_{R})+irD_{-\frac{r}{k}-1}(\sqrt{k}x_{R})D_{\frac{r}{k}-1}(i\sqrt{k}x_{R})]}.$
(41)

The above Eq. (41) takes the following form in the absence of resetting

$\displaystyle\langle
T_{res}\rangle\big{|}_{r=0}=\frac{\pi}{2k}\bigg{[}1-\text{erf}{\bigg{(}\sqrt{\frac{k}{2}}x_{0}\bigg{)}\bigg{]}}\text{erfi
}\bigg{(}\sqrt{\frac{k}{2}}x_{0}\bigg{)}.$ (42)

Fig. 3 (a) shows the monotonic decrease of $\langle T_{res}(x_{R})\rangle$ as
a response to the strength of potential $k$. This is attributed to the fact
that the Brownian particle can easily reach the target with an increase in
$k$. For sufficiently larger values of $k$, $\langle T_{res}(x_{R})\rangle$
saturates for different resetting rates. We have compared our analytical
results (both for $r=0$ and $r>0$ case) with the numerical simulations and
found an excellent agreement.

Fig. 3 (b) showcases various interesting features of the mean residence time
as a function of resetting rate. For large values of $k$, the mean residence
time increases monotonically as a function of resetting rate since the
potential is strong enough to bring the particle to the origin faster and thus
with an increasing resetting rate the completion can only be delayed. As we
keep decreasing the stiffness, an interesting regime appears where we see a a
non-monotonic behaviour of $\langle T_{res}(x_{R})\rangle$ with resetting rate
$r$. We find that resetting first reduces $\langle T_{res}(x_{R})\rangle$ and
there is an optimal rate $r^{c}$ when $\langle T_{res}(x_{R})\rangle$ is
minimum. Following that, the mean residence time keeps increasing
monotonically. A qualitative way of understanding this behavior is as follows:
For the low values of $k$, the particle can easily diffuse through space and
can easily spend more time above $x_{R}$. With resetting, the particle is
brought back to $x_{R}$ and consequently it spends less time above $x_{R}$ and
$\langle T_{res}(x_{R})\rangle$ starts to decrease up to a critical resetting
rate. However, as $r$ is further increased, the particle is brought back to
$x_{R}$ more frequently which increases the first passage time of the process,
and in turn, the particle also gets to spend more time above $x_{R}$.

It is evident from the above discussion that there is a critical strength
$k^{c}$ above (and below) of which resetting has different impact on $\langle
T_{res}(x_{R})\rangle$. To capture this behavior, we have looked into the
optimal rate $r^{c}$ as a function of $k$. For the large values of $k$, the
mean residence time increases monotonically. Thus, the optimal resetting rate
$r^{c}$ determined from $\frac{d\langle
T_{res}(x_{R})\rangle}{dr}|_{r=r^{c}}=0$ is always at zero in this case.
However, as we decrease $k$, a non-monotonic behavior emerges and we start
observing a finite optimal rate $r^{c}>0$. Henceforth, there exists a critical
potential strength $k^{c}$ where this transition occurs. In Fig. 3 (c), we
have shown the phase diagram in $(k,r^{c})$ plane from which one can easily
extract the value of critical strength $k^{c}$. For fixed values of
$x_{0}=2.5$ and $D=1$, we find $k^{c}\approx 0.22624$. The optimal rate
$r^{c}$ seems to exhibit a continuous phase transition alike behavior from
$r^{c}>0$ to $r^{c}=0$. Any non-zero value of $k<k^{c}$ always renders in
$r^{c}>0$ which essentially states that resetting will always minimize the
mean residence time. However, for $k>k^{c}$, no benefits can be gained by the
introduction of resetting. In other words, $\langle T_{res}(x_{R})\rangle$
will increase monotonically as a function of resetting rate. At the proximity
of the critical point $k^{c}$, one would expect a power law behavior
$r^{c}\sim|k-k^{c}|^{\beta}$ similar to the thermodynamical second order phase
transition. In the inset of Fig. 3 (c), we fit the numerical data with this
ansatz to find the following critical coefficient $\beta=1$. This scaling
behavior is similar to the one that we have found also for the MFPT $\langle
t_{f}(r)\rangle$ as will be shown in the next section.

##  5 First passage time

In this section, we investigate the first moment of the first passage time
$t_{f}$ of an OU particle, starting from $x_{0}$, to an absorbing target
placed at the origin in the presence of stochastic resetting at $x_{R}>0$. To
this end, we simply substitute $U(x)=1$ in Eq. (8) to find

$\displaystyle\frac{\partial^{2}Q(p,x_{0})}{\partial
x_{0}^{2}}-kx_{0}\frac{\partial Q(p,x_{0})}{\partial
x_{0}}-(p+r)Q(p,x_{0})+rQ(p,x_{R})=0,$ (43)

with the boundary conditions 9 and 10. The solution of above Eq. 43 is

$\displaystyle
Q(p,x_{0})=e^{kx_{0}^{2}/4}[\mathcal{A}D_{-\frac{p+r}{k}}(\sqrt{k}x_{0})+\mathcal{B}D_{\frac{p+r}{k}-1}(i\sqrt{k}x_{0})]+\frac{r}{r+p}Q(p,x_{R}).$
(44)

Using boundary conditions 9 and 10 in Eq. 44, we obtain $\mathcal{B}=0$ and

$\displaystyle\mathcal{A}=\frac{1-\frac{r}{(r+p)}Q(p,x_{R})}{D_{-\frac{p+r}{k}}(0)}.$
(45)

Figure 4: Comparison of the MFPT $\langle t_{f}(r)\rangle$ for OU process with
numerical simulations for different values of (a) resetting rate $r$ and (b)
spring constant $k$. The lines and symbols are analytical and numerical
simulation results respectively. (c) Second order like phase transition of the
optimal resetting rate $r^{*}$ which continuously reaches to zero from a
finite value at the critical point $k^{*}=0.1749$. The locus of $r^{*}$ (solid
line) is obtained by numerically minimizing $\langle t_{f}(r)\rangle$ for
different values of $k$. In particular, the diamond and square markers depict
$r^{*}>0$ and $r^{*}=0$ respectively obtained by minimizing $\langle
t_{f}(r)\rangle$ from 4(b) for two different values of $k$. Inset: Optimal
resetting rate $r^{*}$ close to the critical point $k^{*}$ shows a power law
behavior $r^{*}\sim|k-k^{*}|^{\alpha}$ with $\alpha=1$. (d) Phase diagram for
resetting transition and verification of the critical point $k^{*}$ from the
$CV>1$ criterion. The $CV$ as a function of $k$ intersects unity (dashed
horizontal line) exactly at the value $k^{*}=0.174895$. For $k>k^{*}$ i.e.,
when $CV<1$, resetting is detrimental while for $k<k^{*}$ i.e., when $CV>1$,
resetting always remains beneficial. This is in accordance with Fig. 4(c). In
the simulation, we have set: $x_{0}=x_{R}=2.5$ and $D=1$.

Substituting the $\mathcal{A}$ and $\mathcal{B}$ in 44 and setting
$x_{0}=x_{R}$, we finally obtain

$\displaystyle
Q(p,x_{R})=\frac{(r+p)e^{kx_{R}^{2}/4}D_{-\frac{r+p}{k}}(\sqrt{k}x_{R})}{D_{-\frac{r+p}{k}}(0)p+re^{kx_{R}^{2}/4}D_{-\frac{r+p}{k}}(\sqrt{k}x_{R})}.$
(46)

Inserting $Q(p,x_{R})$ from Eq. (46) into Eq. (6), we arrive at the following
expression for the mean first passage time

$\displaystyle\langle
t_{f}(r)\rangle=\frac{\sqrt{\pi}2^{-\frac{r}{2k}}e^{-kx_{R}^{2}/4}}{r\Gamma(\frac{r+k}{2k})D_{-\frac{r}{k}}(\sqrt{k}x_{R})}-\frac{1}{r},$
(47)

which was also obtained in [51]. In the limit $k\rightarrow 0$, one can use
the following identity [56, 57]

$\displaystyle
D_{\nu}(z)=2^{\nu/2}\sqrt{\pi}e^{-z^{2}/4}\bigg{[}\frac{1}{\Gamma(\frac{1-\nu}{2})}F\bigg{(}-\frac{\nu}{2},\frac{1}{2};\frac{z^{2}}{2}\bigg{)}-\frac{\sqrt{2}z}{\Gamma(-\frac{\nu}{2})}F\bigg{(}\frac{1-\nu}{2},\frac{3}{2};\frac{z^{2}}{2}\bigg{)}\bigg{]},$
(48)

where
$F(\alpha,\gamma;z)=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{(\gamma)_{n}}\frac{z^{n}}{n!}$
is the Kummer function with $(\delta)_{n}=\beta(\beta+1)....(\beta+n-1)$ in
(47), to find

$\displaystyle\langle t_{f}(r)\rangle_{k\rightarrow
0}=\frac{1}{r}(e^{\sqrt{r}x_{R}}-1),$ (49)

which was first obtained by Evans and Majumdar in the case of simple diffusion
[36].

In Figs. 4 (a) and (b), we have plotted $\langle t_{f}(r)\rangle$ as a
function of the strength of potential $k$ for different values of resetting
rates $r$ and as a function of $r$ for different values of $k$ respectively.
The solid/dashed lines represent the analytical results while the markers are
the points from numerical simulations. An excellent agreement between them is
found. The qualitative behaviour of $\langle t_{f}(r)\rangle$ is similar as
$\langle T_{res}(x_{R})\rangle$. At low values of $k$, $\langle
t_{f}(r)\rangle$ increases with an increase in $r$ and becomes almost
independent of the resetting rates when the potential is much steeper with
higher $k$ [see Fig. 4 (a)]. On the other hand, Fig 4 (b) shows that at low
values of $k$, where the drift is not sufficiently strong, $\langle
t_{f}(r)\rangle$ decreases with an increase in $r$ and reaches some minimum
value at the optimal resetting rate $r^{*}>0$ and then increases again. For
sufficiently larger values of $k$, resetting always delays the completion
rendering in $r^{*}=0$. Henceforth, there exists a critical value of potential
strength namely $k^{*}$ below (above) which resetting is helpful
(detrimental). To better understand this behavior, we have made a phase
diagram, as done in the case of residence time, in the $(k,r^{*})$ plane – see
Fig. 4 (c) for fixed values of $D=1$ and $x_{0}=x_{R}=2.5$. For this
configuration, the optimal resetting rate $r^{*}$ undergoes a continuous
transition from the $r^{*}>0$ to $r^{*}=0$ phase and there is a sharp critical
point in the potential strength at $k^{*}=0.1749$ where this behavioral
transition occurs [see also [51] where this transition was analyzed
numerically]. In a nutshell, we have

$\displaystyle\begin{array}[]{l}r^{*}\left\\{\begin{array}[]{lll}=0&&\text{if
~{}~{}}k>k^{*}\text{ }\\\ &\text{ \ \ }\\\ >0&&\text{if~{}~{} }k<k^{*}\text{
.}\end{array}\right.\text{ }\end{array}$ (54)

The continuous transition of the optimal resetting rate compels us to believe
that at the close proximity of the critical point $k^{*}$, the optimal
resetting rate has a power law behavior $r^{*}\sim|k-k^{*}|^{\alpha}$, where
$\alpha$ is the critical coefficient. Fitting with the numerical data, we find
$\alpha=1$ – see inset of Fig. 4(c) and also in [51]. This is in accordance
with the Landau theory of resetting transition where one can show that indeed
$\alpha=1$ is a universal critical coefficient [61] in a very general
scenario. This is quite striking since the power law behavior is reminiscent
of the same observed, e.g., in liquid-gas or ferromagnetic systems where the
critical coefficient $\alpha$ is found to be universal.

Eq. (54) can also be understood from a more general viewpoint laid out by the
framework of first passage under resetting [58, 59] and a Landau like phase
transition theory in stochastic resetting [61]. To see this, one can express
the MFPT $\langle t_{f}(r)\rangle$ as a power series in terms of the restart
rate $r$ near the transition and show how by utilizing the relations between
the coefficients, it is possible to predict the emergence of second-order like
continuous transitions [61, 51]. Expanding $\langle t_{f}(r)\rangle$ around
$r=0$ results in

$\displaystyle\langle t_{f}(r)\rangle=b_{0}+b_{1}r+b_{2}r^{2}+\cdots$ (55)

where $b_{i}-$s are the expansion coefficients. To identify the coefficients,
we make use of the generic expression for MFPT under stochastic resetting for
an arbitrary stochastic process given by [58, 59]

$\displaystyle\langle
t_{f}(r)\rangle=\frac{1-\tilde{t}_{f}(r)}{r\tilde{t}_{f}(r)},$ (56)

where $\tilde{t}_{f}(r)=\langle e^{-rt_{f}}\rangle$ is the moment generating
function for the FPT $t_{f}$. Doing a Taylor series expansion of $\langle
t_{f}(r)\rangle$ and comparing term by term with Eq. (55), we can easily find
the structure of the coefficients namely $b_{0}=\langle t_{f}\rangle|_{r=0}$,
$b_{1}=[-\frac{\langle t_{f}^{2}\rangle}{2}+\langle
t_{f}\rangle^{2}]\big{|}_{r=0}$, $b_{2}=[\frac{\langle
t_{f}^{3}\rangle}{6}+\langle t_{f}\rangle^{3}-\langle t_{f}\rangle\langle
t_{f}^{2}\rangle]\big{|}_{r=0}$ and so on, notably $\langle
t_{f}^{n}\rangle|_{r=0}$ is the $n$-th moment of FPT distribution for the
resetting free process. For resetting to be beneficial in the small $r$ limit,
one should have $b_{1}<0$ which after some rearrangement leads to the
following universal criterion [61, 58, 59, 60]

$\displaystyle CV=\frac{\sigma(t_{f})}{\langle t_{f}\rangle}\bigg{|}_{r=0}>1,$
(57)

where $\sigma(t_{f})$ stands for the standard deviation of $t_{f}$ and $CV$ is
called the coefficient of variation. The $CV$ condition provides a phase
diagram which allows us to scan the parameter space and identify the desired
regions where in resetting can be beneficial. Within our set-up, we find

$\displaystyle
b_{0}=\frac{\exp\Big{(}-\frac{kx_{R}^{2}}{4}\Big{)}\bigg{(}2D^{(1,0)}_{0}(\sqrt{k}x_{R})-log(2)D_{0}(\sqrt{k}x_{R})-D_{0}(\sqrt{k}x_{R})\psi^{(0)}(1/2)\bigg{)}}{2k[D_{0}(\sqrt{k}x_{R})]^{2}},$
(58)

and

$\displaystyle
b_{1}=\frac{\begin{multlined}\exp\Big{(}-\frac{kx_{R}^{2}}{4}\Big{)}\bigg{[}[D_{0}(\sqrt{k}x_{R})]^{2}\Big{(}2[log(2)]^{2}+4log(2)\psi^{(0)}(1/2)+2[\psi^{(0)}(1/2)]^{2}-\pi^{2}\Big{)}\\\
-8D_{0}(\sqrt{k}x_{R})D^{(1,0)}_{0}(\sqrt{k}x_{R})\Big{(}log(2)+\psi_{0}(1/2)\Big{)}+16[D^{(1,0)}_{0}(\sqrt{k}x_{R})]^{2}-8D_{0}(\sqrt{k}x_{R})D^{(2,0)}_{0}(\sqrt{k}x_{R})\bigg{]}\end{multlined}\exp\Big{(}-\frac{kx_{R}^{2}}{4}\Big{)}\bigg{[}[D_{0}(\sqrt{k}x_{R})]^{2}\Big{(}2[log(2)]^{2}+4log(2)\psi^{(0)}(1/2)+2[\psi^{(0)}(1/2)]^{2}-\pi^{2}\Big{)}\\\
-8D_{0}(\sqrt{k}x_{R})D^{(1,0)}_{0}(\sqrt{k}x_{R})\Big{(}log(2)+\psi_{0}(1/2)\Big{)}+16[D^{(1,0)}_{0}(\sqrt{k}x_{R})]^{2}-8D_{0}(\sqrt{k}x_{R})D^{(2,0)}_{0}(\sqrt{k}x_{R})\bigg{]}}{16k^{2}[D_{0}(\sqrt{k}x_{R})]^{3}},$
(61)

where $D^{n,0}(\nu,z)$ is the $n^{th}$ partial derivative of parabolic
cylinder function with respect to $\nu$ and $\psi^{0}(z)$ is the digamma
function defined in terms of the gamma function $\Gamma(z)$ as
$\psi^{(0)}(z)=\frac{\Gamma^{\prime}(z)}{\Gamma(z)}$ [56, 57]. Now
substituting $b_{0}$ and $b_{1}$ from Eq. (58) and (61) into
$CV=\sqrt{1-\frac{2b_{1}}{b_{0}^{2}}}$ allows us to examine the span of the
parameters e.g., the potential strength in our case by setting the condition
$CV>1$. In Fig. 4 (d), we do this task by plotting CV as a function of $k$ for
fixed $x_{R}$. From the intersection of $CV$ with unity, we can immediately
identify the critical strength $k^{*}=0.174895$. This is in perfect synchrony
with the same obtained from the analysis of the optimization of $\langle
t_{f}(r)\rangle$ as done around Eq. (54). It is also evident that for
$k>k^{*}$, $CV$ becomes lesser than unity and thus resetting is only
detrimental. On the other hand, $CV$ is greater than unity for $k<k^{*}$, and
thus resetting remains beneficial. Thus, in contrast to simple diffusion,
where resetting always turns out to be a useful strategy [36], a trade-off
exists in the case of OU process [51]. The discussion made above exactly
pinpoints how the potential strength $k$ plays a key role in this interplay.

##  6 Conclusion

In recent times, study of stochastic processes that are further subject to
resetting has gained immense interest in statistical physics [39]. Resetting
is a natural way to break detailed balance and thus resetting induced dynamics
showcases a plethora of exciting non-equilibrium phenomena [36, 46, 62, 63,
64, 65, 66, 67, 68, 69]. In the context of search or completion of a process,
resetting has been instrumental to curtail long waiting times and improve the
efficiency [36, 39, 58, 60]. In this paper, we have analyzed various first
passage functionals for an OU process subjected to stochastic resetting.
Employing the celebrated Feynman-Kac formalism, we have computed the
analytical forms for first moments of local time $T_{loc}$, residence time
$T_{res}$ and first passage time $t_{f}$. While mean local time behaves
monotonically as a function of the resetting rate, an intriguing feature is
observed for the residence time. We have found that resetting can either
shorten or lengthen the mean residence time and there exists an optimal
resetting rate for which mean residence time can further be minimized.
Furthermore, we can quantify the transition in terms of the optimal resetting
rate which can be interpreted as an order parameter in this problem. We show
that the optimal resetting rate for the residence time undergoes a continuous
transition as a function of the critical stiffness which we could exactly
quantify. Moreover, we extracted the critical coefficient from the numerical
analysis.

Finally, we study the mean first passage time of the OU process under
resetting. Here too, resetting can either accelerate or hinder the completion
of the first passage process. The transition between these two states or
phases is characterized by the behavioral change in the order parameter of the
system namely the optimal restart rate which undergoes a second-order like
continuous transition depending on the details of the system parameters. We
quantify this transition via direct optimization of the mean first passage
time and then provide a cross-verification of the same using the restart
transition theory of optimal resetting rate [61]. Such resetting transition
has been observed in various other systems such as diffusion in confinement
[70] or channel [71], diffusion in potential [72, 51, 73, 74, 75, 76, 77, 78],
random walks [79, 80] and the universal applicability of the general condition
[58, 60] that dictates such transition is quite remarkable.

Concluding, we have shown resetting induced optimization of the mean residence
time and the mean first passage time in OU process that depends crucially on
the potential strength. Various other intricate interplay between the
potential strength and resetting has been showcased. It will be interesting to
measure these observables using widely applicable optical trap experiments
[81, 82], calibrated with harmonic potential. Another interesting direction
would be to understand the effect of various other non-exponential resetting
protocols [40, 58, 83, 84] on these observable statistics. In particular,
resetting at a fixed interval of time or sharp resetting has turned out to be
quite useful in global optimization of mean first passage time among all the
strategies [58, 85] with intriguing applications to operation research [86],
and nonlinear dynamics [87]. These frontiers remain open as a prospective
future direction.

##  7 Acknowledgment

AD acknowledges support from the IMSc Post-Doctoral Scholarship. AP gratefully
acknowledges research support from the Department of Science and Technology,
India, SERB Start-up Research Grant Number SRG/2022/000080 and Department of
Atomic Energy, India.

##  Appendix A Details of numerical simulations

The statistical properties of FPBFs studied here in this work can be
numerically computed by integrating the overdamped Langevin equation in
presence of resetting. The numerical scheme is as follows. Starting from an
initial location $x_{0}$, position of the Brownian particle is updated using
the following simple rules

$\displaystyle x(t+\Delta t)=\begin{cases}&x(t)-kx(t)\Delta
t+\Gamma_{0},~{}~{}~{}~{}~{}~{}\text{with prob }(1-r\Delta t),\\\
&x_{R},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{with
prob }r\Delta t.\end{cases}$ (A.1)

Where, $\Gamma_{0}$ is a random number sampled from a Gaussian distribution
with zero mean and width is given by $\langle\Gamma_{0}^{2}\rangle=2Dt$ (here,
$D=1$). The protocol (A.1) continues till the particle reaches very close to
the origin within a prescribed tolerance level. We consider that to be a first
passage event of the process. Computation of the three quantities discussed in
the Sec. 1 is done as follows:

  1. 1.

Local time: Since we are interested in the local time in the vicinity of
initial (or resetting) location $x_{R}$, we compute the time
$T_{2\epsilon}(x_{R})$ spent by the particle inside the box of length
$[x_{R}-\epsilon,x_{R}+\epsilon]$. This process is observed till the particle
reaches the origin and gets absorbed. We have chosen $\epsilon=0.1$. Finally,
to compute the Local time density we devide the time $T_{2\epsilon}(x_{R})$ by
the box length as defined in the Eq. (12).

  2. 2.

Residence time: We numerically evaluate the total time $T_{res}(x_{R})$ spent
by the Brownian particle in the region $x>x_{R}$ till the first passage event.

  3. 3.

First passage time (FPT): To compute the FPT, we let the position evolve
according to (A.1). As soon as the particle is close to the origin within a
tolerance level, we stop the process and record the total time spent so far as
the FPT.

For averaging purpose, we repeat the same process for $10^{5}$ number of
realizations.

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# Extending a result on the minimality of the complement manifold of toric
arrangements

Elia Saini  (Elia Saini) Independent researcher, Via San Rocco 40/A, IT-23036,
Teglio (SO<EMAIL_ADDRESS>
###### Abstract.

The homotopy type of the complement manifold of a complexified toric
arrangement has been investigated by d’Antonio and Delucchi in a paper that
shows the minimality of such topological space. In this work we associate to a
given toric arrangement a matrix that represents the arrangement over the
integers. Then, we consider the family of toric arrangements for which this
matrix has maximal rank. Our goal is to prove, by means of basic linear
algebra arguments, that the complement manifold of the toric arrangements that
belong to this family is diffeomorphic to that of centered toric arrangements
and thus it is a minimal topological space, too.

## Introduction

Toric arrangements are a well studied class of hypersurface arrangements that
appear in many branches of mathematics such as algebraic geometry and topology
as well as combinatorics. Following the seminal works of Lefschetz [Lef24] and
Deligne [Del71], toric arrangements provide a generalization of hyperplane
arrangements in the context of the study of the complements of normal crossing
divisors in smooth projective varieties. A very rich area of research focuses
on the description of the combinatorial and homotopic invariants of these
arrangements.

The Betti numbers of the complement manifold of a toric arrangement were first
computed by Looijenga in [Loo93] using several facts from sheaf theory.
Afterwards, a presentation of the complex cohomology ring has been given in
[DCP05]. To get this result, De Concini and Procesi developed techniques from
algebra and combinatorics that allowed them to find a surprising connection
between partition functions, box splines and toric arrangements. These topics
are summarized in [DCP11].

Several authors such as Brändén, D’Adderio, Lenz and Moci explored in the
series of papers [BM14], [DM13], [Len17] and [Moc12] the very rich
combinatorial structure of toric arrangements and their deep relationship with
arithmetic matroids. The central question of describing the homotopy type of
the complement manifold of a toric arrangements was first teackled by Moci and
Settepanella in [MS11]. Later, d’Antonio and Delucchi in the works [dD12] and
[dD15] exploited the toric complex introduced in [MS11] to describe the
fundamental group and to prove a minimality result for those toric
arrangements that restricts to a real arrangement. A direct consequence of
this minimality result is the torsion-freeness of the integer cohomology of
this class of arrangements.

Finally, improving on the study of the toric complex and the poset of layers,
Callegaro and Delucchi exhibited in [CD17] a presentation of the integer
cohomology of the complement manifold of centered toric arrangements. In order
to describe those algebraic and topological invariants that are
combinatorially determined, Pagaria provided in [Pag19] an example of two
toric arrangements with the same poset of layers but with different integer
cohomologies, while De Concini and Gaiffi showed in [DCG21] that the rational
cohomology ring of toric arrangements is combinatorial.

In this work we provide a description of a wider class of toric arrangements
for which the minimality results previously described hold. This class
consists of those toric arrangements that have an associated matrix of maximal
rank. Using techniques from basic linear algebra, we are going to build a
sequence of diffeomorphic changes of coordinates that transform these
arrengements into centered ones, enabling us to apply the results on
complexified arrangements developed by d’Antonio and Delucchi in [dD15]. In
particular, this implies that the integer cohomology of any of these toric
arrangements is torsion-free.

Overview. Section 1 provides some basic definitions and results on toric
arrangements as well as complexified and centered arrangements, while Section
2 is devoted to the proof of our result on toric arrangements that have an
associated matrix of maximal rank.

##  1\. Basics

The purpose of this section is to recall some basic definitions and results
about toric arrangements and complexified arrangements, in order to fix
notations and set up the frame for the subsequent part of this work. For a
general theory of toric arrangements we refer to the paper [DCP05] and the
survey [DCP11], while for a detailed study of the homotopy type of the
complement manifold of complexified toric arrangements we point to the article
[dD15].

###  1.1. Toric arrangements

A toric arrangement in $(\mathbb{C}^{\ast})^{n}$ is a finite collection
$\mathcal{A}=\\{H_{1},\ldots,H_{m}\\}$ of subspaces of
$(\mathbb{C}^{\ast})^{n}$, called subtori, of the form

$H_{i}=\left\\{(z_{1},\ldots,z_{n})\in(\mathbb{C}^{\ast})^{n}\left|z_{1}^{p_{i,1}}\cdots
z_{n}^{p_{i,n}}=\alpha_{i}\right.\right\\}$

where $p_{i,j}\in\mathbb{Z}$ and $\alpha_{i}\in\mathbb{C}^{\ast}$ as $1\leq
i\leq m$ and $1\leq j\leq n$. The complement manifold $M(\mathcal{A})$ is the
complement of the union of the subtori $H_{i}$ in $(\mathbb{C}^{\ast})^{n}$,
that is, the topological space

$M(\mathcal{A})=(\mathbb{C}^{\ast})^{n}\setminus\bigcup_{i=1}^{m}H_{i}$

The matrix associated to the toric arrangement $\mathcal{A}$ is the integer
matrix of $m$ rows and $n$ columns $\tilde{M}_{\mathcal{A}}\in
M_{m,n}(\mathbb{Z})$ defined by setting

$\tilde{M}_{\mathcal{A}}=\left(\begin{array}[]{ccc}p_{1,1}&\cdots&p_{1,n}\\\
\vdots&\vdots&\vdots\\\ p_{m,1}&\cdots&p_{m,n}\\\ \end{array}\right)$

######  Note 1.1.

A square integer matrix is unimodular if its determinant equals $1$ or $-1$.
In this case, a basic result in linear algebra ensures us that the inverse
matrix still has integer entries.

###  1.2. Complexified arrangements

A toric arrangement $\mathcal{A}=\\{H_{1},\ldots,H_{m}\\}$ in
$(\mathbb{C}^{\ast})^{n}$ is complexified if it restricts to a real
arrangement, that is, if $\alpha_{i}\in S^{1}$ for all $i=1,\ldots,m$ while it
is centered if all the $\alpha_{i}\text{'s}$ equal $1$. Clearly a centered
toric arrangement is complexified, too. A topological space $X$ is minimal if
it is homotopically equivalent to a CW complex with exactly $b_{k}$ cells in
dimension $k$, where $b_{k}$ denotes the $k\text{-th}$ Betti number of $X$.
The minimality of the complement manifold of a complexified toric arrangement
has been proved by d’Antonio and Delucchi in the following theorem, that is
the basement on which our work is funded.

######  Theorem 1.1 ([dD15], Corollay 93).

Let $\mathcal{A}=\\{H_{1},\ldots,H_{m}\\}$ be a toric arrangement in
$(\mathbb{C}^{\ast})^{n}$. If $\mathcal{A}$ is complexified, then the
complement manifold $M(\mathcal{A})$ is a minimal topological space.

##  2\. Results

The aim of this section is to prove the minimality of the complement manifold
of a toric arrangement with associated matrix that has maximal rank. Our goal
is to extend to this wider class of toric arrangements the ideas described in
[CD17, Section 7.2] also presented by Pagaria in a series of conference talks.
To do this we are going to perform a sequence of changes of coordinates that
transform the given toric arrangement into a centered one (compare Section
1.2) so that we are allowed to apply Theorem 1.1.

######  Theorem 2.1.

Let $n\geq m\geq 1$ and let $\mathcal{A}=\\{H_{1},\ldots,H_{m}\\}$ be a toric
arrangement in $(\mathbb{C}^{\ast})^{n}$ with associated matrix
$\tilde{M}_{\mathcal{A}}$. If $\tilde{M}_{\mathcal{A}}$ has rank $m$, then the
complement manifold $M(\mathcal{A})$ is a minimal topological space.

With the same arguments of [dD15, Corollary 14] this implies that the
complement manifold of these toric arrangements is torsion-free, too.

######  Corollary 2.1.

Let $n\geq m\geq 1$ and let $\mathcal{A}=\\{H_{1},\ldots,H_{m}\\}$ be a toric
arrangement in $(\mathbb{C}^{\ast})^{n}$ with associated matrix
$\tilde{M}_{\mathcal{A}}$. If $\tilde{M}_{\mathcal{A}}$ has rank $m$, then the
cohomology groups $H^{k}(M(\mathcal{A}),\mathbb{Z})$ are torsion-free.

###### Proof of Theorem 2.1.

Step 1. Up to permuting coordinates in $(\mathbb{C}^{\ast})^{n}$ we can assume
that the first $m$ columns of the matrix $\tilde{M}_{\mathcal{A}}$ form a
$m\times m$ minor $M_{\mathcal{A}}$ of maximal rank. Combining Gaussian
elimination and Euclidean integer division, [New72, Theorem II.2] ensures us
that there exist a unimodular matrix $H_{\mathcal{A}}$ such that
$M_{\mathcal{A}}H_{\mathcal{A}}$ is an upper triangular matrix. Since
$H_{\mathcal{A}}\in M_{m,m}(\mathbb{Z})$ is unimodular, from Note 1.1 we know
that its inverse $K_{\mathcal{A}}$ still has integer entries. Let us denote
the entries of $H_{\mathcal{A}}$ and $K_{\mathcal{A}}$ by $h_{i,j}$ and
$k_{i,j}$, respectively. Here, $i$ stands for the row index and $j$ for the
column index.

Step 2. Let $z_{1},\ldots,z_{n}$ be coordinates in $(\mathbb{C}^{\ast})^{n}$.
For $1\leq i\leq n$ set

(1) $t_{i}=\left\\{\begin{array}[]{lll}z_{1}^{k_{i,1}}\cdots
z_{m}^{k_{i,m}}&\text{if}&1\leq i\leq m\\\ z_{i}&\text{if}&m+1\leq i\leq n\\\
\end{array}\right.$

This defines a change of coordinates
$(z_{1},\ldots,z_{n})\mapsto(t_{1},\ldots,t_{n})$ in $(\mathbb{C}^{\ast})^{n}$
with inverse given componentwise by

$z_{i}=\left\\{\begin{array}[]{lll}t_{1}^{h_{i,1}}\cdots
t_{m}^{h_{i,m}}&\text{if}&1\leq i\leq m\\\ t_{i}&\text{if}&m+1\leq i\leq n\\\
\end{array}\right.$

as the following check shows. Indeed, since the matrices $H_{\mathcal{A}}$ and
$K_{\mathcal{A}}$ are one the inverse of the other, for $1\leq i\leq m$ we
have

$\displaystyle z_{i}$ $\displaystyle=t_{1}^{h_{i,1}}\cdots t_{m}^{h_{i,m}}=$
$\displaystyle=\left(z_{1}^{k_{1,1}}\cdots
z_{m}^{k_{1,m}}\right)^{h_{i,1}}\cdots\left(z_{1}^{k_{m,1}}\cdots
z_{m}^{k_{m,m}}\right)^{h_{i,m}}=$
$\displaystyle=z_{1}^{\left(h_{i,1}k_{1,1}+\ldots+h_{i,m}k_{m,1}\right)}\cdots
z_{m}^{\left(h_{i,1}k_{1,m}+\ldots+h_{i,m}k_{m,m}\right)}=$
$\displaystyle=z_{i}$

as well as

$\displaystyle t_{i}$ $\displaystyle=z_{1}^{k_{i,1}}\cdots z_{m}^{k_{i,m}}=$
$\displaystyle=\left(t_{1}^{h_{1,1}}\cdots
t_{m}^{h_{1,m}}\right)^{k_{i,1}}\cdots\left(t_{1}^{h_{m,1}}\cdots
t_{m}^{h_{m,m}}\right)^{k_{i,m}}=$
$\displaystyle=t_{1}^{\left(k_{i,1}h_{1,1}+\ldots+k_{i,m}h_{m,1}\right)}\cdots
t_{m}^{\left(k_{i,1}h_{1,m}+\ldots+k_{i,m}h_{m,m}\right)}=$
$\displaystyle=t_{i}$

while by construction $z_{i}=t_{i}$ as $m+1\leq i\leq n$. The change of
coordinates defined in (1) transforms the toric arrangement
$\mathcal{A}=\\{H_{1},\ldots,H_{m}\\}$ with subtori

$H_{i}=\left\\{(z_{1},\ldots,z_{n})\in(\mathbb{C}^{\ast})^{n}\left|z_{1}^{p_{i,1}}\cdots
z_{n}^{p_{i,n}}=\alpha_{i}\right.\right\\}$

into the toric arrangement $\mathcal{B}=\\{K_{1},\ldots,K_{m}\\}$ with subtori

$K_{i}=\left\\{(t_{1},\ldots,t_{n})\in(\mathbb{C}^{\ast})^{n}\left|t_{i}^{d_{i}}t_{i+1}^{a_{i,i+1}}\cdots
t_{m}^{a_{i,m}}t_{m+1}^{p_{i,m+1}}\cdots
t_{n}^{p_{i,n}}=\alpha_{i}\right.\right\\}$

To see this, let $(z_{1},\ldots,z_{n})$ be a point of
$(\mathbb{C}^{\ast})^{n}$ that belongs to the subtorus $H_{i}$. Therefore, its
coordinates fulfill condition

$z_{1}^{p_{i,1}}\cdots z_{n}^{p_{i,n}}=\alpha_{i}$

Exploiting the change of coordinates given by (1) the previous condition is
then equivalent to

$t_{i}^{d_{i}}t_{i+1}^{a_{i,i+1}}\cdots
t_{m}^{a_{i,m}}t_{m+1}^{p_{i,m+1}}\cdots t_{n}^{p_{i,n}}=\alpha_{i}$

where $a_{i,s}=p_{i,1}k_{1,s}+\ldots+p_{i,m}k_{m,s}$ for $1\leq i\leq m$ and
$i+1\leq s\leq m$. Indeed, since from the first step of our proof

$M_{\mathcal{A}}H_{\mathcal{A}}=\left(\begin{array}[]{ccccc}d_{1}&a_{1,2}&\cdots&\cdots&a_{1,m}\\\
&d_{2}&a_{2,3}&\cdots&a_{2,m}\\\ &&\ddots&&\vdots\\\ &&&d_{m-1}&a_{m-1,m}\\\
&&&&d_{m}\\\ \end{array}\right)$

is an upper triangular matrix, we have

$\begin{array}[]{ll}z_{1}^{p_{i,1}}\cdots
z_{n}^{p_{i,n}}=\alpha_{i}&\Longleftrightarrow\\\ \left(t_{1}^{h_{1,1}}\cdots
t_{m}^{h_{1,m}}\right)^{p_{i,1}}\cdots\left(t_{1}^{h_{m,1}}\cdots
t_{m}^{h_{m,m}}\right)^{p_{i,m}}t_{m+1}^{p_{i,m+1}}\cdots
t_{n}^{p_{i,n}}=\alpha_{i}&\Longleftrightarrow\\\
t_{1}^{p_{i,1}h_{1,1}+\ldots+p_{i,m}h_{m,1}}\cdots
t_{m}^{p_{i,1}h_{1,m}+\ldots+p_{i,m}h_{m,m}}t_{m+1}^{p_{i,m+1}}\cdots
t_{n}^{p_{i,n}}=\alpha_{i}&\Longleftrightarrow\\\
t_{1}^{(M_{\mathcal{A}})_{i}(H_{\mathcal{A}})^{1}}\cdots
t_{m}^{(M_{\mathcal{A}})_{i}(H_{\mathcal{A}})^{m}}t_{m+1}^{p_{i,m+1}}\cdots
t_{n}^{p_{i,n}}=\alpha_{i}&\Longleftrightarrow\\\
t_{i}^{d_{i}}t_{i+1}^{a_{i,i+1}}\cdots
t_{m}^{a_{i,m}}t_{m+1}^{p_{i,m+1}}\cdots t_{n}^{p_{i,n}}=\alpha_{i}&\\\
\end{array}$

Here $(M_{\mathcal{A}})_{i}$ and $(H_{\mathcal{A}})^{l}$ stand for the
$i\text{-th}$ row and the $l\text{-th}$ column of the matrices
$M_{\mathcal{A}}$ and $H_{\mathcal{A}}$, respectively.

Step3. Let us consider the toric arrangement
$\mathcal{B}=\\{K_{1},\ldots,K_{m}\\}$ and let $t_{1},\ldots,t_{m}$ be
coordinates in $(\mathbb{C}^{\ast})^{n}$. For $1\leq i\leq n$ set

(2) $t_{(1)i}=\left\\{\begin{array}[]{lll}t_{i}&\text{if}&i\neq m\\\
t_{m}\beta_{m}^{-1}&\text{if}&i=m\\\ \end{array}\right.$

where $\beta_{m}$ is a $d_{m}\text{-th}$ root of $\alpha_{m}$. Thus,
$(t_{1},\ldots,t_{n})\mapsto(t_{(1)1},\ldots,t_{(1)n})$ defines a change of
coordinates in $(\mathbb{C}^{\ast})^{n}$ with inverse given componentwise by

$t_{i}=\left\\{\begin{array}[]{lll}t_{(1)i}&\text{if}&i\neq m\\\
t_{(1)m}\beta_{m}&\text{if}&i=m\\\ \end{array}\right.$

that transforms the toric arrangement $\mathcal{B}=\\{K_{1},\ldots,K_{m}\\}$
into the toric arrangement $\mathcal{B}_{(1)}=\\{K_{(1)1},\ldots,K_{(1)m}\\}$
where $K_{(1)i}$ are subtori of the form

$K_{(1)i}=\left\\{t_{(1)i}^{d_{i}}t_{(1)i+1}^{a_{i,i+1}}\cdots
t_{(1)m}^{a_{i,m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=\alpha_{(1)i}\right\\}$

with

$\alpha_{(1)i}=\left\\{\begin{array}[]{lll}\alpha_{i}\beta_{m}^{-a_{i,m}}&\text{if}&1\leq
i\leq m-1\\\ 1&\text{if}&i=m\\\ \end{array}\right.$

In order to prove this, let $(t_{1},\ldots,t_{n})$ be a point of
$(\mathbb{C}^{\ast})^{n}$ that belongs to the subtorus $K_{i}$. Then, its
coordinates satisfy condition

$t_{i}^{d_{i}}t_{i+1}^{a_{i,i+1}}\cdots
t_{m}^{a_{i,m}}t_{m+1}^{p_{i,m+1}}\cdots t_{n}^{p_{i,n}}=\alpha_{i}$

Thanks to the change of coordinates defined by (2) this is equivalent to

$t_{(1)i}^{d_{i}}t_{(1)i+1}^{a_{i,i+1}}\cdots
t_{(1)m}^{a_{i,m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=\alpha_{(1)i}$

To see this we have to distinguish between two cases. If $1\leq i\leq m-1$ we
have

$\begin{array}[]{ll}t_{i}^{d_{i}}t_{i+1}^{a_{i,i+1}}\cdots
t_{m}^{a_{i,m}}t_{m+1}^{p_{i,m+1}}\cdots
t_{n}^{p_{i,n}}=\alpha_{i}&\Longleftrightarrow\\\
t_{(1)i}^{d_{i}}t_{(1)i+1}^{a_{i,i+1}}\cdots
t_{(1)m-1}^{a_{i,m-1}}(t_{(1)m}\beta_{m})^{a_{i,m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=\alpha_{i}&\Longleftrightarrow\\\
t_{(1)i}^{d_{i}}t_{(1)i+1}^{a_{i,i+1}}\cdots
t_{(1)m}^{a_{i,m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=\alpha_{i}\beta_{m}^{-a_{i,m}}&\Longleftrightarrow\\\
t_{(1)i}^{d_{i}}t_{(1)i+1}^{a_{i,i+1}}\cdots
t_{(1)m}^{a_{i,m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=\alpha_{(1)i}&\\\ \end{array}$

and similarly, if $i=m$ we have

$\begin{array}[]{ll}t_{m}^{d_{m}}t_{m+1}^{p_{i,m+1}}\cdots
t_{n}^{p_{i,n}}=\alpha_{m}&\Longleftrightarrow\\\
(t_{(1)m}\beta_{m})^{d_{m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=\alpha_{m}&\Longleftrightarrow\\\
t_{(1)m}^{d_{m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=\frac{\alpha_{m}}{\alpha_{m}}&\Longleftrightarrow\\\
t_{(1)m}^{d_{m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=1&\Longleftrightarrow\\\
t_{(1)m}^{d_{m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=\alpha_{(1)m}&\\\ \end{array}$

Now, set

(3) $t_{(2)i}=\left\\{\begin{array}[]{lll}t_{(1)i}&\text{if}&i\neq m-1\\\
t_{(1)m-1}\beta_{m-1}^{-1}&\text{if}&i=m-1\\\ \end{array}\right.$

where $\beta_{m-1}$ is a $d_{m-1}\text{-th}$ root of $\alpha_{(1)m-1}$. This
defines a change of coordinates
$(t_{(1)1},\ldots,t_{(1)n})\mapsto(t_{(2)1},\ldots,t_{(2)n})$ in
$(\mathbb{C}^{\ast})^{n}$ with inverse defined componentwise by

$t_{(1)i}=\left\\{\begin{array}[]{lll}t_{(2)i}&\text{if}&i\neq m-1\\\
t_{(2)m-1}\beta_{m-1}&\text{if}&i=m-1\\\ \end{array}\right.$

that transforms the toric arrangement
$\mathcal{B}_{(1)}=\\{K_{(1)1},\ldots,K_{(1)m}\\}$ into the toric arrangement
$\mathcal{B}_{(2)}=\\{K_{(2)1},\ldots,K_{(2)m}\\}$ where $K_{(2)i}$ are
subtori of the form

$K_{(2)i}=\left\\{t_{(2)i}^{d_{i}}t_{(2)i+1}^{a_{i,i+1}}\cdots
t_{(2)m}^{a_{i,m}}t_{(2)m+1}^{p_{i,m+1}}\cdots
t_{(2)n}^{p_{i,n}}=\alpha_{(2)i}\right\\}$

with

$\alpha_{(2)i}=\left\\{\begin{array}[]{lll}\alpha_{(1)i}\beta_{m-1}^{-a_{i,m-1}}&\text{if}&1\leq
i\leq m-2\\\ 1&\text{if}&m-1\leq i\leq m\\\ \end{array}\right.$

Again, to see this let $(t_{(1)1},\ldots,t_{(1)n})$ be a point in
$(\mathbb{C}^{\ast})^{n}$ that belongs to the subtorus $K_{(1)i}$. Then, its
coordinates fulfill

$t_{(1)i}^{d_{i}}t_{(1)i+1}^{a_{i,i+1}}\cdots
t_{(1)m}^{a_{i,m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=\alpha_{(1)i}.$

Applying the change of coordinates defined by (3), for $1\leq i\leq m-2$ we
have

$\begin{array}[]{l}t_{(1)i}^{d_{i}}t_{(1)i+1}^{a_{i,i+1}}\cdots
t_{(1)m}^{a_{i,m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=\alpha_{(1)i}\\\ \Longleftrightarrow\\\
t_{(2)i}^{d_{i}}t_{(2)i+1}^{a_{i,i+1}}\cdots
t_{(2)m-2}^{a_{i,m-2}}(t_{(2)m-1}\beta_{m-1})^{a_{i,m-1}}t_{(2)m}^{a_{i,m}}t_{(2)m+1}^{p_{i,m+1}}\cdots
t_{(2)n}^{p_{i,n}}=\alpha_{(1)i}\\\ \Longleftrightarrow\\\
t_{(2)i}^{d_{i}}t_{(2)i+1}^{a_{i,i+1}}\cdots
t_{(2)m-2}^{a_{i,m-2}}t_{(2)m-1}^{a_{i,m-1}}t_{(2)m}^{a_{i,m}}t_{(2)m+1}^{p_{i,m+1}}\cdots
t_{(2)n}^{p_{i,n}}=\alpha_{(1)i}\beta_{m-1}^{-a_{i,m-1}}\\\
\Longleftrightarrow\\\ t_{(2)i}^{d_{i}}t_{(2)i+1}^{a_{i,i+1}}\cdots
t_{(2)m-2}^{a_{i,m-2}}t_{(2)m-1}^{a_{i,m-1}}t_{(2)m}^{a_{i,m}}t_{(2)m+1}^{p_{i,m+1}}\cdots
t_{(2)n}^{p_{i,n}}=\alpha_{(2)i}\\\ \end{array}$

Recalling that $\alpha_{(1)m}=1$, if $i=m$ we can see that

$\begin{array}[]{ll}t_{(1)m}^{d_{m}}t_{(1)m+1}^{a_{i,m+1}}\cdots
t_{(1)n}^{a_{i,m}}=\alpha_{(1)m}&\Longleftrightarrow\\\
t_{(1)m}^{d_{m}}t_{(1)m+1}^{a_{i,m+1}}\cdots
t_{(1)n}^{a_{i,m}}=1&\Longleftrightarrow\\\
t_{(2)m}^{d_{m}}t_{(2)m+1}^{a_{i,m+1}}\cdots
t_{(2)n}^{a_{i,m}}=1&\Longleftrightarrow\\\
t_{(2)m}^{d_{m}}t_{(2)m+1}^{a_{i,m+1}}\cdots
t_{(2)n}^{a_{i,m}}=\alpha_{(2)m}&\\\ \end{array}$

while if $i=m-1$ we find that

$\begin{array}[]{ll}t_{(1)m-1}^{d_{m-1}}t_{(1)m}^{a_{i,m}}t_{(1)m+1}^{p_{i,m+1}}\cdots
t_{(1)n}^{p_{i,n}}=\alpha_{(1)i}&\Longleftrightarrow\\\
(t_{(2)m-1}\beta_{m-1})^{d_{m-1}}t_{(2)m}^{a_{i,m}}t_{(2)m+1}^{p_{i,m+1}}\cdots
t_{(2)n}^{p_{i,n}}=\alpha_{(1)i}&\Longleftrightarrow\\\
t_{(2)m-1}^{d_{m-1}}t_{(2)m}^{a_{i,m}}t_{(2)m+1}^{p_{i,m+1}}\cdots
t_{(2)n}^{p_{i,n}}=\frac{\alpha_{(1)m-1}}{\alpha_{(1)m-1}}&\Longleftrightarrow\\\
t_{(2)m-1}^{d_{m-1}}t_{(2)m}^{a_{i,m}}t_{(2)m+1}^{p_{i,m+1}}\cdots
t_{(2)n}^{p_{i,n}}=1&\Longleftrightarrow\\\
t_{(2)m-1}^{d_{m-1}}t_{(2)m}^{a_{i,m}}t_{(2)m+1}^{p_{i,m+1}}\cdots
t_{(2)n}^{p_{i,n}}=\alpha_{(2)m-1}&\\\ \end{array}$

proving that $\alpha_{(2)m}$ and $\alpha_{(2)m-1}$ equal $1$. The pattern is
now clear. Exploiting the triangular structure of the matrix
$M_{\mathcal{A}}H_{\mathcal{A}}$, for $k=3,\ldots,m$ we can recursively build
a sequence of changes of coordinates
$(t_{(k-1)1},\ldots,t_{(k-1)n})\mapsto(t_{(k)1},\ldots,t_{(k)n})$ in
$(\mathbb{C}^{\ast})^{n}$ defined componentwise by

$t_{(k)i}=\left\\{\begin{array}[]{lll}t_{(k-1)i}&\text{if}&i\neq m-k+1\\\
t_{(k-1)m-k+1}\beta_{m-k+1}^{-1}&\text{if}&i=m-k+1\\\ \end{array}\right.$

that transform tha toric arrangement
$\mathcal{B}_{(k-1)}=\\{K_{(k-1)1},\ldots,K_{(k-1)m}\\}$ is transformed into
the toric arrangement $\mathcal{B}_{(k)}=\\{K_{(k)1},\ldots,K_{(k)m}\\}$ where
$K_{(k)i}$ are subtori of the form

$K_{(k)i}=\left\\{t_{(k)i}^{d_{i}}t_{(k)i+1}^{a_{i,i+1}}\cdots
t_{(k)m}^{a_{i,m}}t_{(k)m+1}^{p_{i,m+1}}\cdots
t_{(k)n}^{p_{i,n}}=\alpha_{(k)i}\right\\}$

with

$\alpha_{(k)i}=\left\\{\begin{array}[]{llll}\alpha_{(k-1)i}\beta_{m-k+1}^{-a_{i,m-k+1}}&\text{if}&1\leq
i\leq m-k+1\\\ 1&\text{if}&m-k+1\leq i\leq m\\\ \end{array}\right.$

Again, here $\beta_{m-k+1}$ is a $d_{m-k+1}\text{-th}$ root of
$\alpha_{(k-1)m-k+1}$.

Step 4. The toric arrangement $\mathcal{B}_{(m)}$ is then centered (compare
Section 1.2). Thus, Theorem 1.1 implies that $M(\mathcal{B}_{(m)})$ is a
minimal topological space. The sequence of diffeomorphic changes of
coordinates in $(\mathbb{C}^{\ast})^{n}$ that map the toric arrangement
$\mathcal{A}$ into the toric arrangement $\mathcal{B}_{(m)}$ induces by
restriction a diffeomorphism between the complement manifolds $M(\mathcal{A})$
and $M(\mathcal{B}_{(m)})$. Hence, we can conclude that $M(\mathcal{A})$ is a
minimal topological space, too.

######  Note 2.1.

The diffeomorphisms
$(t_{(k-1)1},\ldots,t_{(k-1)n})\mapsto(t_{(k)1},\ldots,t_{(k)1})$ defined in
the third step of the proof of Theorem 2.1 are not canonical. Indeed, they
depend on the recursive choice of the complex roots
$\beta_{m},\ldots,\beta_{1}$. However, the homotopy type of the complement
manifold $M(\mathcal{A})$ is uniquely determined.

∎

Acknowledgments. The author would like to mention Farhad Babaee for suggesting
the idea behind the third step of Theorem 2.1 during a conversation that took
place at the University of Fribourg (CH) in January, 2016. Moreover, the
author would express his gratitude to William C. Jagy for pointing out the
bibliographical reference [New72] as well as to Alberto Cavallo for the very
useful discussions throughout all the time this project has been developed.

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