# Mean first passage time of active fluctuating membrane with stochastic
resetting

Tapas Singha<EMAIL_ADDRESS>Institut Curie, Université PSL, Sorbonne
Université, CNRS UMR168, Laboratoire Physico Chimie Curie, 75005 Paris, France

###### Abstract

We study the mean first passage time of a one-dimensional active fluctuating
membrane that is stochastically returned to the same flat initial condition at
a finite rate. We start with a Fokker-Planck equation to describe the
evolution of the membrane coupled with an Ornstein-Uhlenbeck type of active
noise. Using the method of characteristics, we solve the equation and obtain
the joint distribution of the membrane height and active noise. In order to
obtain the mean first-passage time (MFPT), we further obtain a relation
between the MFPT and a propagator that includes stochastic resetting. The
derived relation is then used to calculate it analytically. Our studies show
that the MFPT increases with a larger resetting rate and decreases with a
smaller rate, i.e., there is an optimal resetting rate. We compare the results
in terms of MFPT of the membrane with active and thermal noises for different
membrane properties. The optimal resetting rate is much smaller with active
noise compared to thermal. When the resetting rate is much lower than the
optimal rate, we demonstrate how the MFPT scales with resetting rates,
distance to the target, and the properties of the membranes.

##  I Introduction

A biological membrane is comprised of a variety of proteins and ion channels
that interact with the intra- and extra-cellular environments. The proteins
and ion channels consume energy via the hydrolysis of adenosine triphosphate
(ATP) and exert force on the membrane[1]. On average, the exerted active force
could be either zero or nonzero, depending on the exact problem. The interplay
of the force and the mechanical properties of the membrane, such as bending
rigidity and tension, determines the shape of the membrane and the movement of
a cell.

In order to perform a specific function, a cell often needs to reach a certain
target. When it moves with an average velocity, calculating the time it will
take to reach a target seems straightforward. We consider a scenario in which
a cell finds its static single target only through membrane fluctuations. In
this case, it is interesting to understand the time a membrane takes to reach
its target for the first time. As the membrane fluctuates either towards or
away from the target, the reaching time is a stochastic variable, and in a
statistical sense, the mean first-passage time may be a sensible measure for
the process.

The mean first-passage time (MFPT) is relevant in a variety of physical,
chemical, and biological processes ranging from species extinctions in
ecology[2, 3] to molecular processes. A simple example of MFPT is the average
time a forager takes to find food or other resources for the first time[4]. In
biology, the immune cells search for cells with antigens. Therefore, it is
relevant in a range of length scales and becomes more important as the need to
understand complex systems increases.

Because the underlying stochastic dynamics of a fluctuating system can cause
MFPT to be infinitely long, depending on the system details, several search
strategies have been proposed over the last two decades[5, 6, 7]. One of the
most important strategies is stochastic resetting, in which the searcher is
returned to its initial state at a fixed rate. This may enhance the likelihood
of finding the target because the searcher may explore the surroundings, that
may reduce the likelihood of wandering away from the target. Stochastic
resetting was first proposed for a single Brownian particle[8, 9, 10],
subsequently, numerous problems have also been studied in other fields, such
as population expansion in fluctuating environments[11] and search algorithms
in computer science[12]. Several variations of resetting[13] have also been
investigated, including resetting with finite velocity[14], which may be
useful in real life.

Figure 1: Schematic diagram of fluctuating active (curvy, solid black line)
and passive (curvy, dashed blue line) membranes and their target (shown by a
solid green circle).The membrane is reset to its initial height $h=h_{0}$
(shown by a straight-dashed red line) at a finite rate.

A passive membrane driven by thermal noise satisfies the fluctuation-
dissipation theorem. A biological membrane, on the other hand, transports ions
through its embedded ion pumps, which consume ATP and cause local force (or
active noise) on the membrane. Together with thermal noise, this active noise
enhances the amplitude of membrane fluctuations, which can be experimentally
quantified by the effective temperature[1, 15]. Unlike the passive membrane,
the membrane, which actively participates in biological processes, does not
satisfy the fluctuation-dissipation theorem.

Though the passive surface growth with stochastic resetting has been
studied[16], the dynamics of a membrane with active fluctuations have yet to
be investigated. Importantly, the MFPT, which is fascinating in the context of
searching, remains elusive. In this work, we answer the question: how long
does it take for a membrane to reach its single point target when a membrane
is reset to its flat initial height with a finite rate? We illustrate the
scenario in the schematic diagram shown in Fig. (1).

To this end, we first derive the propagator for the coupled dynamical
equations for an active membrane. Further, we derive a general relation
between the MFPT and a propagator under resetting. Using the relation and the
derived propagator for the height, we obtain the MFPT. Our studies reveal that
there exists an optimal resetting rate at which MFPT becomes minimum. We
demonstrate how, for different membrane properties, MFPT scales with resetting
rates and target heights, and how an optimal resetting rate arises. Finally,
we compare the MFPT for active and passive membranes.

The resetting of a membrane may be related to the type of motility in which a
part of the cell membrane gets detached from its cortex and fluctuates, known
as a bleb. After that, the cortical actin filaments reach the detached
membrane and bring the membrane back to its initial position[17, 18]. The
membrane may grow again and find its target; however, the underlying dynamics
remain unclear.

This work is structured as follows: The model is presented in Section II. The
derivation of the single-height distribution is described in Section III.
Section IV depicts the connection between the mean first-passage time and the
propagator. In the following section, we calculate the height-height
correlations for different cases. Section VI presents the MFPT for a variety
of cases in one spatial dimension. Finally, Section VII presents discussion
and conclusions.

##  II Model

We are interested in understanding how the symmetric membrane fluctuations
eventually drive the membrane to reach a target. We consider a membrane with
the symmetric distribution of all membrane components (such as lipids and
proteins) and the same environment on both sides of the membrane that may
ensure symmetric membrane fluctuations and, thereby zero spontaneous
curvature. Here we first consider the free energy for a symmetric membrane
[19] expressed as

$\displaystyle\mathcal{F}[h]=\int dx\left\\{\frac{\nu}{2}(\nabla
h)^{2}+\frac{\kappa}{2}(\nabla^{2}h)^{2}\right\\}$ (1)

where $\nu$, and $\kappa$ are the tension and the bending rigidity of the
membrane, respectively. We study a simple dynamical equation for height field
$h(x,t)$ in one spatial dimension written as $\Gamma\partial h/\partial
t=-\delta\mathcal{F}[h]/\delta h+\eta(x,t)+av(x,t),$ where $\eta(x,t)$ is the
thermal noise with zero average and
$\langle\eta(x,t)\eta(x^{\prime},t^{\prime})\rangle=2D\delta(x-x^{\prime})\delta(t-t^{\prime}).$
The coefficient $D$ is the strength of the thermal noise. From here onwards,
we scale time $t$ with $\Gamma$, and keep it as $t$. The last term $v(x,t)$
arises from the fluctuating force of ion pumps/channels and the proteins which
affect the membrane dynamics. Considering $v(x,t)$ as an Ornstein-Uhlenbeck
type of field, we write

$\frac{\partial v(x,t)}{\partial t}=-\frac{v}{\tau_{a}}+\mu(x,t)$ (2)

where $\mu(x,t)$ is the Gaussian noise with zero average and
$\langle\mu(x,t)\mu(x^{\prime},t^{\prime})\rangle=2D_{a}\delta(x-x^{\prime})\delta(t-t^{\prime})$,
and $\tau_{a}$ is the relaxation time of an active noise field $v$. The
strength of the active fluctuations $D_{a}$ depends on the available ATP
concentration, and the density of the protein pumps [20] which are taken to be
constant in space and time. As there are no spatial derivatives or other
functions of space in Eq. (2), $v_{i}$ acts locally on the membrane $h_{i}$.
In contrast to thermal noise, active noise $v_{i}$ has a temporal correlation
with a correlation time $\tau_{a}$ resulting in membrane height $h_{i}$ being
kicked for a much longer period of time than that of thermal noise. To put it
another way, the active noise correlation effectively takes the form of the
thermal noise correlation in the limit $\tau_{a}\rightarrow 0$.

When a membrane is surrounded by fluid, the motion at one point of the
membrane affects the motion at the other point via the fluid medium, which is
known as hydrodynamic interaction. However, in one spatial dimension, the
effect of the hydrodynamic interactions on the relaxation dynamics is
marginal[21, 22]. We, therefore, neglect the hydrodynamic interactions and
write the dynamical equation for membrane height as

$\frac{\partial h(x,t)}{\partial t}=\nu\,\frac{\partial^{2}h}{\partial
x^{2}}-\kappa\,\frac{\partial^{4}h}{\partial x^{4}}+a\,v(x,t)+\eta(x,t).$ (3)

In the above equation, bending rigidity $\kappa$ and tension $\nu$ determine a
length scale $\ell_{c}=\sqrt{\kappa/\nu}$. As a result, when $\ell>\ell_{c}$,
$\nu$ dominates membrane dynamics; otherwise, $\kappa$ dominates. We study the
two limits of the membrane properties, namely tension-less active membrane
(TLAM) ($\nu=0$) in which dynamics are governed by bending rigidity $\kappa$,
and tension-dominated active membrane (TDAM) ($\kappa=0$) in which dynamics
are governed by tension $\nu$. To study active dynamics, we neglect the
thermal noise $\eta$ compared with active noise $v$, and we study the membrane
dynamics separately with thermal noise.

The growth of the height fluctuations may vary over time. For instance, the
initial height fluctuation dynamics differ from the late-time dynamics. The
mean-squared width, which is a statistical measure of height fluctuations,
grows with $t^{\beta}$ for $t\ll L^{z}$, where $\beta$ and $z$ are the growth
and the dynamic exponents, respectively, and $L$ represents the size of the
system. For $t\gg L^{z}$, the system reaches its steady state, and mean-
squared width no longer depends on time $t$, rather on $L^{\chi}$ where $\chi$
is the roughness exponent of the membrane [23]. The dynamic exponent $z$ is
determined by the properties of the membrane; for example, when tension
dominates membrane relaxation ($\nu\neq 0$ and $\kappa=0$) the dynamic
exponent $z=2$ and when bending rigidity dominates membrane relaxation
($\kappa\neq 0$ and $\nu=0$), $z=4$. For the active system, the timescale of
interest is expressed as $t\ll\tau_{a}\ll L^{z}$ for $L\rightarrow\infty.$

##  III Height distribution

We first aim to derive the propagator for the coupled equations given in Eqs.
(2), and (3). The discretized versions of Eqs. (2) and (3) are written as

$\frac{\partial}{\partial t}\begin{bmatrix}h_{i}(t)\\\
v_{i}(t)\end{bmatrix}=\begin{bmatrix}-\sum_{j}\Lambda_{ij}&a\\\
0&-\Lambda^{\prime}\end{bmatrix}\begin{bmatrix}h_{j}(t)\\\
v_{i}(t)\end{bmatrix}+\begin{bmatrix}\eta_{i}(t)\\\ \mu_{i}(t)\end{bmatrix}$
(4)

where $\Lambda_{ij}=-(\nu\Delta_{ij}-\kappa\Delta_{ij}^{2})$,
$\Lambda^{\prime}=1/\tau_{a}$, and $\sum_{j}$ in the above equation includes
the nearest neighbors. Further, we write the Fokker-Planck equation for the
joint distribution $\mathbf{W}(\\{h\\},\\{v\\},t|h^{0},v^{0},t_{0})$ as

$\displaystyle\frac{\partial\mathbf{W}}{\partial
t}=[-\sum^{N}_{i,j=1}\frac{\partial}{\partial
h_{i}}(-\Lambda_{ij}h_{j}+a\,\delta_{ij}\,v_{j})+\sum^{N}_{i,j=1}\frac{\partial}{\partial
v_{i}}(\Lambda_{ij}^{\prime}v_{j})$
$\displaystyle+\sum^{N}_{i,j=1}\frac{\partial^{2}}{\partial h_{i}\partial
h_{j}}D_{ij}+\sum^{N}_{i,j=1}\frac{\partial^{2}}{\partial v_{i}\partial
v_{j}}D^{a}_{ij}]\,\,\mathbf{W}(\tilde{h},\tilde{v},t)$ (5)

where $N$ is the total number of sites. The initial conditions read
$\mathbf{W}(\tilde{h},\tilde{v},t_{0}|h^{0},v^{0},t_{0})=\delta(\tilde{h}-\tilde{h}^{0})\,\,\delta(\tilde{v}-\tilde{v}^{0})$
where $\tilde{h}\equiv\\{h_{i}\\}$ and $\tilde{v}\equiv\\{v_{i}\\}$. Using the
method of characteristics, we solve the above equation (as detailed in
_Appendix-A_). Further, we integrate the distribution of initial noise field
$\tilde{v}^{0}$ as $\int
d\tilde{v}^{0}\,\delta(\tilde{v}^{0})\,\mathbf{W}(\tilde{h},t|\tilde{\psi}^{0})$.
We also integrate out the active noise field $\tilde{v}$ and then obtain a
marginal distribution for $\tilde{h}$ which reads as

$\displaystyle W(\tilde{h},t|0,0)$
$\displaystyle=\frac{\exp{[-\frac{1}{2}\tilde{h}^{T}\mathbf{M}^{-1}_{1}\tilde{h}]}}{(2\pi)^{L/2}\sqrt{\det{\mathbf{M}}_{1}}}$
(6)

where covariance matrix $\mathbf{M}_{1}$ is expressed as

$\displaystyle\mathbf{M}_{1}=\left[2D_{\text{tot}}f(2\Lambda)+a^{\prime
2}(f(2\Lambda^{\prime})-2f(\Lambda+\Lambda^{\prime}))\right]\delta_{ij}$ (7)

in which $f(\theta)=(1-e^{-\theta\,t})/\theta$, $D_{\text{tot}}=D+a^{\prime
2}/2$, and $a^{\prime 2}=2a^{2}D_{a}/(\Lambda^{\prime}-\Lambda)^{2}$. The
above matrix is obtained with the non-stationary active noise. Considering
$a=0$ in Eq. (6), we obtain the propagator for the passive system expressed as
$W_{\text{passive}}(\tilde{h},t|0,0)=\sqrt{\frac{1}{(2\pi)^{L}}\det{\left(\frac{\Lambda}{D\,(1-e^{-2\Lambda
t})}\right)}}\,\exp{\left[-\frac{1}{2}\,\tilde{h}^{T}\frac{\Lambda}{D(1-e^{-2\Lambda
t})}\tilde{h}\,\right]},$ which is consistent with the propagator for the
Ornstein-Uhlenbeck type of particle in a harmonic potential [24].

As we define the position of the target as being at $0$, we integrate out
heights from all the sites except $x=0$ because a single height at that point
may only reach the target as overhangs are not considered. For a homogeneous
system, we obtain the marginal distribution

$\displaystyle W(\mathbf{0},t|h_{0},0)$
$\displaystyle=\frac{1}{\sqrt{2\pi\langle
h^{2}(t)\rangle}}\exp{\left[-\frac{1}{2}\frac{h_{0}^{2}}{\langle
h^{2}(t)\rangle}\right]}.$

The obtained propagator in the above equation describes how a single height
distribution evolves with time starting from a reference height $h_{0}$.

##  IV Mean First Passage time with resetting

We next present a relation between the first-passage time and survival
probability, and then investigate how the mean first-passage time is related
to a propagator.

###  IV.1 Survival probability

Let us consider a particle driven by stochastic noise starts at position
$h=h_{0}$, and reaches the target ($h=0$) for the first time at time $T$,
known as first-passage time. The survival probability, $S(h_{0},T)$, can be
related to the first-passage time distribution, $F(h_{0},T)$, which is defined
as the likelihood that the searcher has not reached the target up to time $T$.
With this, $F(h_{0},T)$ can be expressed by the difference between the
survival probabilities at two consecutive discrete times as
$F(h_{0},T)=S(h_{0},T-1)-S(h_{0},T)$ [25]. In the continuum limit $\Delta
T\rightarrow 0$, $F(h_{0},T)\Delta T=S(h_{0},T-\Delta T)-S(h_{0},T)$ yields

$F(h_{0},T)=-\frac{\partial S(h_{0},T)}{\partial T}.$ (9)

We next study the MFPT of a stochastic variable in one spatial dimension under
stochastic resetting.

###  IV.2 MFPT with Resetting

We consider that the initial separation between a point on the membrane and a
target is $h_{0}$, and that the membrane takes time $T$ to reach the target
for the first time (see fig.1). Resetting is the immediate return of the
membrane to its starting flat initial height $h_{0}$ at a rate $r$. The
survival probability with resetting is denoted as $S_{r}(h_{0},T)$ and the
corresponding first-passage distribution as $F_{r}(h_{0},T)$. The first-
passage distribution and the survival probability with resetting are related
in the following way [26]:

$F_{r}(h_{0},T)=-\frac{\partial S_{r}(h_{0},T)}{\partial T}.$ (10)

Therefore, the mean first-passage time with resetting is defined as $\langle
T\rangle=\int^{\infty}_{0}dT\,TF_{r}(h_{0},T).$ Using Eq. (10) in the
definition of the MFPT, we get

$\langle T\rangle=\int^{\infty}_{0}dT\,S_{r}(h_{0},T).$ (11)

Now, we consider the Laplace transform (LT) of survival probability, which
reads as

$\widetilde{S}_{r}(h_{0},s)=\int_{0}^{\infty}dT\,e^{-s\,T}\,S_{r}(h_{0},T).$
(12)

When $T\rightarrow\infty$, we consider that the searcher finds the target with
probability $1$, which results in $S_{r}(h_{0},\infty)=0$. Substituting $s=0$
in Eq. (12) and equating with Eq. (11), we obtain

$\langle T\rangle=\widetilde{S}_{r}(h_{0},0),$ (13)

where $S_{r}(h_{0},\infty)=0$ is considered. Following the backward Kolmogorov
equation for first-passage time distribution, we write

$\displaystyle F_{r}(h_{0},T)=e^{-rT}F(h_{0},T)+\int_{0}^{T}$ $\displaystyle
dt^{\prime}S_{r}(h_{0},T-t^{\prime})\,r$ (14)
$\displaystyle\times\,e^{-rt^{\prime}}F(h_{0},t^{\prime}),$

where $F_{r}(h_{0},T)$ is the first-passage time distribution with resetting.
The first term on the right-hand side of the above equation indicates that
there has been no resetting in time $T$, while the second term denotes that
the last resetting took place at a time $t^{\prime}$.

Using Eq. (10), we multiply $e^{-sT}$ on both sides of the above equation and
finally integrate over $T$. This leads to

$\widetilde{S}_{r}(h_{0},s)=\frac{1-\widetilde{F}(h_{0},r+s)}{s+r\widetilde{F}(h_{0},r+s)}$
(15)

where $\widetilde{F}(h_{0},r+s)$ is the LT of the first-passage distribution
time _without_ resetting. In the above equation, we use $S_{r}(h_{0},0)=1$,
i.e., initially ($t=0$) the survival probability is $1$. Substituting $s=0$ in
the above equation, and using Eq. (13), we obtain

$\displaystyle\langle
T\rangle=\frac{1}{r}\left(\frac{1}{\widetilde{F}(h_{0},r)}-1\right).$ (16)

Following the renewal equation, we write the relation between the first-
passage time distribution and the propagator as [3, 27]

$\displaystyle
W(\mathbf{0},T|h_{0})=\int^{T}_{0}dt^{\prime}F(h_{0},t^{\prime})W$
$\displaystyle(\mathbf{0},T-t^{\prime}|\mathbf{0})dt^{\prime}$ (17)
$\displaystyle+\delta_{(0,h_{0})}\delta(T)$

where $W(\mathbf{0},T|h_{0})$ is the propagator at $h=0$ at time $T$ given
that the initial coordinate was at $h=h_{0}$. The Laplace transform of the
above equation leads to

$\widetilde{F}(h_{0},s)=\frac{\widetilde{W}(\mathbf{0},s|h_{0})}{\widetilde{W}(\mathbf{0},s|\mathbf{0})}$
(18)

where $h_{0}\neq 0$, and $s$ is the Laplace conjugate to time $T$.
Substituting the above equation in Eq. (16), we obtain

$\displaystyle\langle
T\rangle=\frac{1}{r}\left(\frac{\widetilde{W}(\mathbf{0},r|\mathbf{0})}{\widetilde{W}(\mathbf{0},r|h_{0})}-1\right).$
(19)

Despite the fact that we discuss the case of a point on the membrane, the
obtained relation is valid for any stochastic variable.

##  V Height-height correlation

The auto-correlation of active proteins from Eq.(2) can be written as

$\displaystyle\langle v_{i}(t)v_{j}(t^{\prime})\rangle$
$\displaystyle=e^{-\Lambda^{\prime}(t+t^{\prime})}\int^{t}_{t_{0}}ds\int^{t^{\prime}}_{t_{0}}ds^{\prime}e^{\Lambda^{\prime}(s+s^{\prime})}\langle\mu_{i}(s)\mu_{j}(s^{\prime})\rangle$

where $\langle v_{i}(t)\rangle=0$, and $t_{0}$ is the initial time. We
consider that the active proteins have already reached their steady state as
they relax faster than the membrane, which is achieved by setting
$t_{0}\rightarrow-\infty$. Carrying out the integration in the above equation,
we obtain

$\displaystyle\langle
v_{i}(t)v_{j}(t^{\prime})\rangle=\,D_{a}\,\frac{e^{-\Lambda^{\prime}|t-t^{\prime}|}}{\Lambda^{\prime}}\delta_{ij}.$
(21)

The derivation of $\mathbf{M}_{1}$ (given in Eq. (7)) from the Fokker-Planck
equation and $\langle h_{i}(t)h_{j}(t)\rangle$ from the coupled Langevin
equations (given in Eqs. (2),(3)) are found to be the same

$\langle h_{i}(t)h_{j}(t)\rangle=M^{ij}_{1}\,\,\delta_{ij}.$ (22)

The above relation is obtained with the non-stationary and stationary state
active noise. With non-stationary state active noise, the relation is shown in
Appendix-A, and with stationary state, the height correlation is obtained as

$\displaystyle\langle
h^{2}_{i}(t)\rangle=2\left(D+D^{\prime}_{a}\right)f(2\Lambda)$
$\displaystyle-2D^{\prime}_{a}\,f(\Lambda+\Lambda^{\prime}),$ (23)

where
$D^{\prime}_{a}=\frac{a^{2}D_{a}}{\Lambda^{\prime}(\Lambda^{\prime}-\Lambda)}.$
When $\Lambda$ is considered as the spring constant of a harmonic well, the
above equation is consistent with the displacement correlation of a particle
in a harmonic well with an Ornstein-Uhlenbeck type of active noise [28]. For
$\Lambda\rightarrow 0$, Eq. (23) becomes equivalent to the displacement
correlation of the active Ornstein-Uhlenbeck type of noise. Further,
considering $\Lambda^{\prime}=1/\tau_{a}$, the above correlation can be
obtained as follows: $\lim_{\Lambda\rightarrow 0}\langle
h^{2}_{i}(t)\rangle=2t\left(D+(a\tau_{a})^{2}D_{a}\left[1-\frac{\tau_{a}}{t}(1-e^{-t/\tau_{a}})\right]\right)$
which agrees with ref.[29]. In the passive case ($a=0$), the above correlation
becomes $2Dt$, which is the mean-square displacement of a Brownian particle in
one dimension.

In order to investigate the MFPT of fluctuating membranes, we first employ the
relation between the MFPT and the propagator given in Eq. (19). We next aim to
obtain the propagator with active or thermal noise (as given in Eq.
(LABEL:EqPropagator)), which requires the single point height-height
correlation. To this end, we begin with the Fourier transform of Eq.(3) which
reads as

$\displaystyle\frac{\partial h(q,t)}{\partial
t}=-\frac{1}{\tau_{q}}h(q,t)+\eta(q,t)+a\,v(q,t)$ (24)

where $\,\tau_{q}=(\nu q^{2}+\kappa q^{4})^{-1}$ is the membrane relaxation.
Using the flat initial condition, we get

$\displaystyle
h(q,t)=e^{-t/\tau_{q}}\int^{t}_{0}dt^{\prime}\,e^{t^{\prime}/\tau_{q}}\,\,[\eta(q,t^{\prime})+a\,v(q,t^{\prime})].$
(25)

We also obtain the auto-correlation of the stationary state active noises as

$\langle
v(q,t)v(q^{\prime},t^{\prime})\rangle=D_{a}\tau_{a}\,(2\pi)\delta(q+q^{\prime})\,e^{-|t^{\prime}-t|/\tau_{a}}.$
(26)

The inverse Fourier transform of the height-height correlation can be written
as

$\displaystyle\langle h(x,t)^{2}\rangle$
$\displaystyle=\int^{\infty}_{-\infty}\frac{dq}{(2\pi)}\int^{\infty}_{-\infty}\frac{dq^{\prime}}{(2\pi)}\,\,e^{i(q+q^{\prime})x}\,\langle
h(q,t)h(q^{\prime},t)\rangle$

in which the correlation can be calculated either with active or thermal
noises.

###  V.1 Tension-dominated membrane ($\kappa=0$)

####  V.1.1 Active

Using the active noise correlation given in Eq. (26) and considering the
Fourier transform of the height given in Eq. (25), we write the height-height
correlation as

$\displaystyle\langle
h(x,t)^{2}\rangle_{\text{active}}=a^{2}\,D_{a}\,\tau_{a}\int^{\infty}_{-\infty}$
$\displaystyle\frac{dq}{(2\pi)}e^{-2\nu
tq^{2}}\int^{t}_{0}du\int^{t}_{0}du^{\prime}$ (28) $\displaystyle\times
e^{\nu(u+u^{\prime})q^{2}}e^{-|u-u^{\prime}|/\tau_{a}}.$

We perform integration over $q$ and obtain

$\displaystyle\langle h(x,t)^{2}\rangle_{\text{active}}$
$\displaystyle=\frac{a^{2}\,D_{a}\,\tau_{a}}{2\sqrt{\pi\nu}}\,\int^{t}_{0}du\int^{t}_{0}du^{\prime}\,\frac{e^{-|u-u^{\prime}|/\tau_{a}}}{\sqrt{2t-(u+u^{\prime})}}.$

Next, we carry out the integration over $u$ and $u^{\prime}$ which results in

$\displaystyle\langle h(x,t)^{2}\rangle_{\text{active}}$
$\displaystyle=\frac{a^{2}D_{a}\tau^{3/2}_{a}}{\sqrt{\nu}}(\,[\sqrt{\frac{2t\tau_{a}}{\pi}}-\frac{\tau_{a}}{2}\,\text{Erf}\left(\sqrt{\frac{t}{\tau_{a}}}\right)]$
$\displaystyle+$
$\displaystyle\frac{\tau_{a}}{2}e^{-\frac{2t}{\tau_{a}}}[\text{Erfi}\left(\sqrt{\frac{t}{\tau_{a}}}\right)-\text{Erfi}\left(\sqrt{\frac{2t}{\tau_{a}}}\right)]).$

In this study, we are interested in the regime in which active noise drives
the membrane. As we see above, depending on time $t$ and $\tau_{a}$, the
active noise correlation given in Eq. (LABEL:TDAM_Ht-HT) has two distinct
regimes. For $t\ll\tau_{a}$, the above equation is simplified to

$\displaystyle\langle
h(x,t)^{2}\rangle_{\text{active}}\simeq\frac{4}{3}(\sqrt{2}-1)\,\frac{a^{2}\,D_{a}\tau_{a}}{\sqrt{\pi\nu}}\,t^{3/2}$
(31)

and for $t\gg\tau_{a}$

$\displaystyle\langle h(x,t)^{2}\rangle_{\text{active}}\simeq
a^{2}\,D_{a}\tau^{2}_{a}\,\sqrt{\frac{2t}{\pi\nu}}.$ (32)

####  V.1.2 Passive

Similarly, we next derive the auto-correlation with thermal noise (see
Appendix B) which reads as

$\displaystyle\langle
h(x,t)^{2}\rangle_{\text{passive}}=D\sqrt{\frac{2t}{\pi\nu}}.$ (33)

In the limit $t\gg\tau_{a}$, the auto-correlation of active noise exhibits
passive-like behavior. Because the active noise becomes uncorrelated when the
interval between two successive observations is larger than the correlation
time $\tau_{a}$. As a result, the effect of several changes in the
uncorrelated noise $v_{i}$ changes the membrane height $h_{i}$ in a random
manner. Thus, in the long time limit, this uncorrelated active noise becomes
additive to the thermal noises, and according to the central limit theorem,
the distribution of these uncorrelated noises leads to a Gaussian distribution
with increased variance.

Since we study the passive case separately, for active noise, we limit
ourselves only when $t\ll\tau_{a}$.

###  V.2 Tension-less membrane ($\nu=0$)

####  V.2.1 Active

When the membrane tension is very low compared to the bending rigidity, the
dynamics of membrane relaxation are dominated by $\kappa$. We investigate how
the height-height correlations scale in time for tension-less active membrane
(TLAM). We begin with the height-height correlation expressed as

$\displaystyle\langle
h(x,t)^{2}\rangle_{\text{active}}=a^{2}\,D_{a}\,\tau_{a}\int^{\infty}_{-\infty}\frac{dq}{(2\pi)}$
$\displaystyle\int^{t}_{0}du\int^{t}_{0}du^{\prime}e^{\kappa(u+u^{\prime}-2t)q^{4}}$
(34) $\displaystyle\times\,e^{-|u-u^{\prime}|/\tau_{a}}.$

Integrating over $q$, we obtain

$\displaystyle\langle
h(x,t)^{2}\rangle_{\text{active}}=\Gamma\left(\frac{5}{4}\right)\frac{a^{2}D_{a}\,\tau_{a}}{\pi\kappa^{1/4}}\,\int^{t}_{0}du\int^{t}_{0}du^{\prime}\,\frac{e^{-|u-u^{\prime}|/\tau_{a}}}{[2t-(u+u^{\prime})]^{\frac{1}{4}}}.$

Next, integrating over $u$ and $u^{\prime}$ and considering $t\ll\tau_{a}$, we
obtain

$\displaystyle\langle h(x,t)^{2}\rangle_{\text{active}}$
$\displaystyle\,\simeq(2^{3/4}-1)\frac{8\,\Gamma(\frac{1}{4})}{21}\frac{a^{2}\,D_{a}\,\tau_{a}}{\pi\kappa^{1/4}}\,\,t^{7/4}.$
(36)

We now carry out numerical integrations given in Eqs. (28) and (34), and
compare the numerical results with the obtained analytical results given in
Eqs. (31) and (36), respectively for $t\ll\tau_{a}$ (shown in Fig. (2))

Figure 2:  (a) The integration given in Eq.(28) for tension-dominated membrane
fluctuations is numerically solved for $\tau_{a}=50\,\,\text{and}\,\,500$
(shown by the dotted-black and dashed-green lines, respectively), and the
results are compared to the analytical findings given in Eq. (31) (shown by
the solid-red line). (b) Similarly, the integration given in Eq.(34) for
rigidity-dominated membrane fluctuations is numerically solved (shown by the
dotted-black and dashed-green lines, respectively), and the results are then
compared with Eq. (36) (shown by the dashed-red line).

####  V.2.2 Passive

We also obtain the height-height correlation for the passive membrane as

$\displaystyle\langle h(x,t)^{2}\rangle_{\text{passive}}\,$
$\displaystyle=\frac{D}{3\pi^{3/4}}\,\left[\frac{(2t)^{3}}{\pi\kappa}\right]^{1/4}.$
(37)

With active noise, the height-height correlation ($\sim t^{7/4}$) grows much
faster than the thermal noise ($\sim t^{3/4}$), on the other hand, the scaling
of $\kappa$ remains the same ($\sim\kappa^{-1/4}$) for both the noises. We
obtain similar results for tension-dominant membranes in which the correlation
$\langle h^{2}\rangle\sim\nu^{-1/2}$ for both the noises, whereas the time
dependence of the correlation gets significantly affected by the active noise
as given in Eqs. (31) and (33).

##  VI Mean First-Passage Time

In this section, we derive the MFPT for TDAM via the LT of the propagator.

###  VI.1 MFPT: Tension-dominated active membrane

As the diagonal matrix $\mathbf{M}_{1}$ is the exactly same as $\langle
h^{2}_{i}(t)\rangle$, we substitute Eq. (31) in Eq. (LABEL:EqPropagator), and
obtain

$W_{\text{active}}(h_{0},t)=\frac{1}{\sqrt{\pi}}\frac{1}{\alpha_{t}\,}\exp{\left(-\frac{h_{0}^{2}}{\alpha_{t}^{2}}\right)}$
(38)

where
$\alpha_{t}^{-1}=\sqrt{3\sqrt{\pi\nu}\,}/\left(\sqrt{8a^{2}D_{a}\tau_{a}(\sqrt{2}-1)}\,\,t^{3/4}\right).$
The Laplace transform of Eq. (38) yields

$\widetilde{W}_{\text{active}}(u_{a},r)=\frac{1}{\sqrt{\pi}\alpha}\frac{1}{r^{1/4}}\,\int^{\infty}_{0}\,\frac{d\bar{t}}{\bar{t}^{3/4}}\exp{\left(-\bar{t}-\frac{u_{a}^{2}}{\bar{t}^{\frac{3}{2}}}\right)}$
(39)

where $\alpha=\alpha_{t}/t^{3/4}$, $\bar{t}=r\,t$ and
$u_{a}=r^{3/4}\,h_{0}/\alpha$. We substitute the above equation in Eq. (19),
and obtain

$\langle
T_{\nu}\rangle_{\text{active}}=\frac{1}{r}\left(\frac{I_{a}(0)}{I_{a}(u_{a})}-1\right)$
(40)

Figure 3: The variation of mean first-passage time $\langle T\rangle$ with
resetting rate $r$ for a tension-dominated membrane ($\nu\neq 0$ and
$\kappa=0$) and fixed target height $h_{0}$. (a) For an active membrane, we
plot the MFPT, which is analytically obtained in Eq.(40), where
$u_{a}=r^{3/4}h_{0}\,\sqrt{(3\sqrt{\pi\nu})/(8a^{2}\,D_{a}\tau_{a}(\sqrt{2}-1))}.$
(b) For passive membrane, we plot MFPT obtained in Eq.(92) where
$u_{p}=r^{1/4}h_{0}/(2D\sqrt{2/(\pi\nu)})^{1/2}$ as the effective coupling
variables for active (shown in left figure (a)) and passive (shown in right
figure (b)) systems, respectively. In contrast to the passive membrane, where
the effective coupling rate $u_{p}>1$, the active membrane has an optimal
resetting rate for $u_{a}<1$.

Figure 4: Variation of mean first-passage time $\langle T\rangle$ with
resetting rate $r$ for a tension-less membrane ($\kappa\neq 0$ and $\nu=0$)
and fixed target height $h_{0}$. (a) For active membrane, we plot the MFPT via
numerical integration which is analytically obtained in Eq.(46) where
$y_{a}=r^{7/8}h_{0}/\sqrt{(16/21)(2^{3/4}-1)\Gamma(1/4)a^{2}\,D_{a}\tau_{a}/\pi\kappa^{1/4}}$.
(b) For passive membrane, we plot MFPT obtained in Eq.(100) where
$y_{p}=r^{3/8}h_{0}\sqrt{3\pi\kappa^{1/4}/(2^{7/4}D)}$ as the effective
coupling variables for active (shown in left figure (a)) and passive (shown in
right figure (b)) systems, respectively. The optimal resetting rate is
obtained for $y_{a}<1$ for active system whereas for passive membrane
$y_{p}>1$.

where

$I_{a}(u_{a})=\int^{\infty}_{0}\frac{d\bar{t}}{\bar{t}^{3/4}}\exp{\left(-\bar{t}-\frac{u_{a}^{2}}{\bar{t}^{3/2}}\right)}.$
(41)

For $u_{a}\rightarrow 0$,

$I_{a}(u_{a})\simeq
I_{a}(0)(1-b_{0}|u_{a}|^{1/3}+b_{1}|u_{a}|^{5/3}+b_{2}|u_{a}|^{2})$ (42)

where $I_{a}(0)=\Gamma(1/4)$, and $b_{0}=4\,\Gamma(5/6)/\Gamma(1/4)$,
$b_{1}=\frac{2^{2/3}}{\sqrt{3\pi}}\Gamma(-\frac{5}{3})$, and
$b_{2}=\frac{2}{\pi\,3^{3/4}}\Gamma(-\frac{5}{12})\Gamma(\frac{11}{12}).$

Correlation and Mean first-passage time
---
Membranes |  $\langle h^{2}(x,t)\rangle\sim$ |  Eqs. |  $\lim_{r\to 0}\langle T\rangle\sim$ |  Eqs.
Tension-dominated active |  $D_{a}\,\,\nu^{-1/2}\,\,t^{3/2}$ |  (31) |  $r^{-3/4}\,h_{0}^{1/3}\,\,\nu^{1/12}\,D_{a}^{-1/6}$ |  (43)
Tension-less active |  $D_{a}\,\,\kappa^{-1/4}\,t^{7/4}$ |  (36) |  $r^{-7/8}\,h_{0}^{1/7}\,\,\kappa^{1/56}\,D_{a}^{-1/14}$ |  (49)
Tension-dominated passive |  $D\,\nu^{-1/2}\,\,t^{1/2}$ |  (33) |  $r^{-1/2}\,h_{0}^{2}\,\nu^{1/2}\,\,D^{-1}$ |  (95)
Tension-less passive |  $\,D\,\kappa^{-1/4}\,t^{3/4}$ |  (37) |  $r^{-3/8}\,h_{0}^{5/3}\,\,\kappa^{5/24}\,D^{-5/6}$ |  (101)
Table 1: Single point height-height correlation and mean first-passage times
with active and passive noises. For tension-dominated passive membrane
($\nu\neq 0$), MFPT has the largest power $h_{0}^{2}$, whereas tension-less
active membrane has the lowest power $h_{0}^{1/7}$. We show the dependence of
MFPT on the membrane properties $\nu$, $\kappa$, target height $h_{0}$, and
resetting rate $r$ for $r\rightarrow 0.$ Our study reveals that active noise,
compared with the thermal, significantly reduces the dependence on the
membrane properties.

We are interested in reducing the MFPT which indeed occurs for a smaller
resetting rate $r$. Thus, the MFPT is simplified as

$\langle
T_{\nu}\rangle_{\text{active}}\simeq\frac{b_{0}}{r^{3/4}}\left(\frac{h_{0}}{\alpha}\right)^{1/3}\,\,\,\text{for}\,\,\,r\ll\left(\frac{\alpha}{h_{0}}\right)^{4/3}$
(43)

The above scaling suggests that $\langle
T_{\nu}\rangle_{\text{active}}\sim\left(a^{2}D_{a}\tau_{a}/\sqrt{\nu}\right)^{-1/6}.$

###  VI.2 MFPT: Tension-less active membrane

We next investigate the MFPT in which the membrane relaxation is dominated by
$\kappa$. We first write the propagator as

$W_{\text{active}}(h_{0},t)=\frac{1}{\sqrt{\pi}\beta_{t}}\exp{\left(-\frac{h_{0}^{2}}{\beta_{t}^{2}}\right)}$
(44)

where $\beta_{t}=\beta\,t^{7/8}$ and
$\beta=\sqrt{\frac{16}{21}(2^{3/4}-1)\,\Gamma(1/4)\frac{a^{2}D_{a}\tau_{a}}{\pi\kappa^{1/4}}}.$
We consider LT of the above equation, which is expressed as

$\widetilde{W}_{\text{active}}(y_{a},r)=\frac{1}{\sqrt{\pi}\beta}\frac{1}{r^{1/8}}\,\int^{\infty}_{0}\,\frac{d\bar{t}}{\bar{t}^{7/8}}\exp{\left(-\bar{t}-\frac{y_{a}^{2}}{\bar{t}^{\frac{7}{4}}}\right)}$
(45)

where $y_{a}=r^{7/8}\,h_{0}/\beta$ and
$\widetilde{W}_{\text{active}}(0,r)=\frac{1}{\sqrt{\pi}\beta}\Gamma{(1/8)}/r^{1/8}.$
Substituting the above equation in Eq. (19), we obtain

$\langle
T_{\kappa}\rangle_{\text{active}}=\frac{1}{r}\left(\frac{J_{a}(0)}{J_{a}(y_{a})}-1\right)$
(46)

where

$J_{a}(y_{a})=\int^{\infty}_{0}\frac{d\bar{t}}{\bar{t}^{7/8}}\exp{\left(-\bar{t}-\frac{y_{a}^{2}}{\bar{t}^{7/4}}\right)}$
(47)

where $J_{a}(0)=\Gamma(1/8)$. For $x\ll 1$, $J_{a}(y_{a})$ can be expressed as

$J_{a}(y_{a})\simeq
J_{a}(0)(1-c_{0}|y_{a}|^{1/7}+c_{1}|y_{a}|^{9/7}-c_{2}|y_{a}|^{2})$ (48)

where $c_{0}=1.111$, $c_{1}=0.294$, and $c_{2}=0.310.$ Keeping only the
leading order, we obtain

$\langle
T_{\kappa}\rangle_{\text{active}}\simeq\frac{c_{0}}{r^{7/8}}\left(\frac{h_{0}}{\beta}\right)^{1/7}\,\,\,\text{for}\,\,\,r\ll\left(\frac{\beta}{h_{0}}\right)^{8/7}$
(49)

Our study suggests that the MFPT is lowest at an intermediate value of $r$ for
all cases as shown in Figs. (3) and (4). As $r$ deviates from this value, the
MFPT increases. Because, when $r$ is very large, the membrane moves too slowly
to reach the target due to frequent resetting, and when $r$ is very small,
stochastic fluctuations may cause the membrane to move far away from the
target, resulting in a significantly longer MFPT.

The MFPT scales as $\langle
T_{\kappa}\rangle_{\text{active}}\sim\kappa^{1/56}$ for a rigidity-dominated
active membrane, whereas it scales as $\langle
T_{\nu}\rangle_{\text{active}}\sim\nu^{1/12}$ for a tension-dominated active
membrane, indicating that the effect of $\nu$ on MFPT is much stronger than
the rigidity $\kappa$.

On the other hand, active noise significantly reduces the effect of $\kappa$
(or $\nu$) on MFPT compared with the thermal noise. For rigidity-dominated
membrane, MFPT with active noise scales as $\langle
T_{\kappa}\rangle_{\text{active}}\sim\kappa^{1/56}$ whereas with thermal noise
it scales as $\langle T_{\kappa}\rangle_{\text{passive}}\sim\kappa^{5/24}$.
Similarly, for tension-dominated membrane, MFPT with active noise scales as
$\langle T_{\nu}\rangle_{\text{active}}\sim\nu^{1/12}$ whereas with thermal
noise it scales as $\langle T_{\nu}\rangle_{\text{passive}}\sim\nu^{1/2}$.

##  VII Discussion and Conclusions

We study the MFPT for one-dimensional membranes under stochastic resetting
with active and passive noises. Starting with the coupled equations for
membrane heights and active noises, we write a Fokker-Planck equation for the
joint distribution and then solve it using the method of characteristics. The
explicit solution of the Fokker-Planck equation for joint distribution
describes how a single height distribution depends on the single point height-
height correlation ($\langle h^{2}\rangle$).

Our study shows that the height-height correlation with active noise grows
much faster than that with thermal noise for both the tension-dominated and
tension-less membranes (Table 1). Across the membrane properties, the tension-
less membrane shows a larger exponent of time than the tension-dominated
membranes shown in Table 1.

We derive a general relation between MFPT and a propagator with resetting
(given in Eq. (19)). Using the relation and the obtained propagators for
heights, we next analytically obtain the MFPT under stochastic resetting with
active (or thermal) noise.

Finally, we demonstrate how $\langle T\rangle$ scales with resetting rate $r$
and target height $h_{0}$ and how it differs between active and passive
systems. Our study reveals that, starting with a very small resetting rate,
$\langle T\rangle$ decreases with increasing $r$, whereas starting with a very
high resetting rate, MFPT decreases with decreasing $r$. Since $\langle
T\rangle$ decreases when $r$ increases to intermediate ranges from both the
extrema of $r$, this indicates that there is an optimal resetting rate at
which MFPT is minimized (shown in Figs. (4) and (3)). This is explained in the
following way: In the absence of resetting, the fluctuating membrane may move
so far away from the target that the first-passage time may be infinitely
long. With a smaller resetting rate, the interface is reset to its initial
height, which may reduce the possibility of moving the membrane far away from
the target, as resetting brings it back to its initial level. With very high
$r$, on the other hand, the membrane is reset so frequently that it moves too
little to reach the target, causing the MFPT to grow. This indicates that at
the intermediate resetting rate (the optimal resetting rate), MFPT becomes
minimal.

With active dynamics, the optimal resetting rate occurs at a smaller resetting
rate than that of passive systems shown in Figs. (4) and (3). In terms of
scaling of the target height $h_{0}$, a tension-dominated passive membrane has
the largest value of the exponent of $h_{0}$ ($\sim h^{2}_{0}$), whereas a
tension-less active membrane has the lowest value of exponent as $h_{0}^{1/7}$
(see Table-1). Our work reveals that with active noise, the scaling exponents
of $\langle T\rangle$ on $\nu$ or $\kappa$ are smaller by an order of
magnitude compared with the passive system (Table-1).

This work may also serve as a benchmark for future studies that may have real-
life applications, such as the fluctuating membrane for a finite system, the
nonzero resetting time, resetting with a finite velocity, etc. Within this
framework, the MFPT of a fluctuating membrane in higher dimensions can be
easily generalized.

## Acknowledgment

TS would like to thank Shamik Gupta for several valuable discussions. The
author thanks Mustansir Barma, Carles Blanch-Mercader, and Pierre Sens for
useful discussions. TS also acknowledges the support provided by a grant from
ITMO Cancer, PSCI.

## Appendix

## Appendix-A: Derivation of the propagator

### Fokker-Planck equation

We consider $\mathbf{p}^{T}=\\{p_{i}\\}$ and $\mathbf{q}^{T}=\\{q_{i}\\}$,
which are the Fourier variables corresponding to $\tilde{h}=\\{h_{i}\\}$ and
$\tilde{v}=\\{v_{i}\\}$, respectively. Let us write the Fourier transform in
the following way

$\widehat{\mathbf{W}}(\mathbf{p}^{T},\mathbf{q}^{T},t|\psi^{0})=\int
d\tilde{h}\int
d\tilde{v}\,\,e^{-i(\mathbf{p}^{T}\tilde{h}+\mathbf{q}^{T}\tilde{v})}\mathbf{W}(\tilde{h},\tilde{v},t|\psi^{0})$
(50)

where $(\mathbf{p}^{T})_{1\times L}$, $(\mathbf{q}^{T})_{1\times L}$, and
$(h)_{L\times 1}$ and $(v)_{L\times 1}$ matrices. Taking into account the
initial conditions, we obtain
$\widehat{\mathbf{W}}_{0}(\mathbf{p^{0}}^{T},\mathbf{q^{0}}^{T},0|\psi^{0})=\int
d\tilde{h}\int
d\tilde{v}\,\,e^{-i(\mathbf{p^{0}}^{T}\tilde{h}+\mathbf{q^{0}}^{T}\tilde{v})}\,\,\delta(\tilde{h}-\tilde{h}^{0})\,\delta(\tilde{v}-\tilde{v}^{0})=e^{-i\,\mathbf{p^{0}}^{T}\tilde{h}^{0}-i\,\mathbf{q^{0}}^{T}\tilde{v}^{0}}.$
We now obtain a Fokker-Planck equation in Fourier space as

$\displaystyle\frac{\partial\mathbf{\widehat{W}}}{\partial t}$
$\displaystyle=-\sum_{i,j}\Lambda_{ij}\,p_{i}\,\,\frac{\partial\mathbf{\widehat{W}}}{\partial
p_{j}}+a\sum_{ij}p_{i}\frac{\partial\mathbf{\widehat{W}}}{\partial q_{i}}$
$\displaystyle-\sum_{ij}\Lambda_{ij}^{\prime}q_{i}\,\frac{\partial\mathbf{\widehat{W}}}{\partial
q_{i}}-\sum_{ij}D_{ij}p_{i}p_{j}\mathbf{\widehat{W}}-\sum_{ij}D^{a}_{ij}q_{i}q_{j}\mathbf{\widehat{W}}.$

###  VII.1 Method of characteristics

We employ the method of characteristics and get

$\frac{d\mathbf{\widehat{W}}}{dt}=\frac{\partial\mathbf{\widehat{W}}}{\partial
t}+\sum_{i}\left(\frac{\partial\mathbf{\widehat{W}}}{\partial
p_{i}}\frac{dp_{i}}{dt}+\frac{\partial\mathbf{\widehat{W}}}{\partial
q_{i}}\frac{dq_{i}}{dt}\right).$ (51)

Along the characteristic line, we have

$\displaystyle\frac{dp_{i}}{dt}=\sum_{j}\Lambda_{ij}p_{j},$ (52)
$\displaystyle\frac{dq_{i}}{dt}=\sum_{j}\Lambda^{\prime}_{ij}q_{j}-a\sum_{ij}p_{i}.$
(53)

Solving the above equations, we obtain

$\mathbf{p}(t)=e^{t\Lambda}\,\mathbf{p^{0}}$ (54)

and

$\mathbf{q}(t)=e^{t\Lambda^{\prime}}\,\mathbf{q^{0}}+a\left(\frac{e^{\Lambda
t}-e^{\Lambda^{\prime}t}}{\Lambda^{\prime}-\Lambda}\right)\mathbf{p}^{0}.$
(55)

In the above equations, $\mathbf{p}(t)$ and $\mathbf{q}(t)$ evolve along the
characteristic line. The total time derivative of FPE is obtained as

$\displaystyle\frac{d\mathbf{\widehat{W}}}{dt}=-\frac{1}{2}\sum_{ij}2D_{ij}\,\,p_{i}\,p_{j}\mathbf{\widehat{W}}-\frac{1}{2}\sum_{ij}2D^{a}_{ij}\,q_{i}\,q_{j}\mathbf{\widehat{W}}$
(56)
$\displaystyle\frac{d(\log{\mathbf{\widehat{W}}})}{dt}=-\frac{1}{2}\left(\mathbf{p}^{T}2D\,\mathbf{p}+\mathbf{q}^{T}2D_{a}\,\mathbf{p}\right).$
(57)

Let us calculate the argument in the above equation as
$\mathbf{p}^{T}2D\mathbf{p}+\mathbf{q}^{T}2D_{a}\mathbf{p}$ in terms of
$\mathbf{p}^{0}$ and $\mathbf{q}^{0}$. Using $\mathbf{q}$ from Eq.(55), we
obtain

$\displaystyle\mathbf{q}^{T}D_{a}\mathbf{q}$
$\displaystyle=(\mathbf{p^{0}})^{T}Q_{1}\mathbf{p}^{0}+(\mathbf{q^{0}})^{T}Q_{2}\mathbf{q}^{0}+2(\mathbf{p^{0}})^{T}Q_{3}\mathbf{q}^{0}$

where

$(Q_{1})_{L\times L}=a^{2}\left[2D_{a}\,\left(\frac{e^{\Lambda
t}-e^{\Lambda^{\prime}t}}{\Lambda^{\prime}-\Lambda}\right)^{2}\right]_{L\times
L},$ (59)

$(Q_{2})_{L\times L}=(2D_{a}\,e^{2\Lambda^{\prime}t})_{L\times L},$ (60)

and

$(Q_{3})_{L\times L}=a\left[2D_{a}\left(\frac{e^{\Lambda
t}-e^{\Lambda^{\prime}t}}{\Lambda^{\prime}-\Lambda}\right)e^{\Lambda^{\prime}t}\right]_{L\times
L}.$ (61)

Similarly Eq. (54) leads to

$\mathbf{p}^{T}2D\mathbf{p}=\mathbf{p^{0}}^{T}(e^{\Lambda t}2De^{\Lambda
t})\,\,\mathbf{p^{0}}$ (62)

Adding Eqs. (LABEL:Cha_q_Matrix) and (62), we obtain

$\displaystyle\mathbf{p}^{T}2D\mathbf{p}+\mathbf{q}^{T}2D_{a}\mathbf{p}=(\mathbf{p^{0}})^{T}$
$\displaystyle(Q_{1}+Q^{\prime}_{1})\,\mathbf{p}^{0}+(\mathbf{q^{0}})^{T}Q_{2}\mathbf{q}^{0}$
(63) $\displaystyle+2(\mathbf{p^{0}})^{T}Q_{3}\,\mathbf{q}^{0}.$

We now substitute Eq. (63) in Eq. (57), and obtain

$\displaystyle\log\left(\frac{\mathbf{\widehat{W}}}{\mathbf{\widehat{W}_{0}}}\right)$
$\displaystyle=-\frac{1}{2}\left[(\mathbf{p^{0}})^{T}R_{1}\,\mathbf{p}^{0}+(\mathbf{q^{0}})^{T}R_{2}\mathbf{q}^{0}+2(\mathbf{p^{0}})^{T}R_{3}\,\mathbf{q}^{0}\right]$

where $\mathbf{\widehat{W}_{0}}$ is the Fourier transform of the initial
height configuration and velocity profiles. In Eq.(LABEL:Sol_Charac1), we have

$\displaystyle
R_{1}=\int^{t}_{0}dt^{\prime}\,(Q_{1}(t^{\prime})+Q^{\prime}_{1}(t^{\prime})),$
(65) $\displaystyle R_{2}=\int^{t}_{0}dt^{\prime}Q_{2}(t^{\prime}),$ (66)
$\displaystyle R_{3}=\int^{t}_{0}dt^{\prime}Q_{3}(t^{\prime}).$ (67)

Using $\mathbf{\widehat{W}_{0}}$ (as shown above) and Eq.(LABEL:Sol_Charac1),
we obtain

$\displaystyle\widehat{\mathbf{W}}$
$\displaystyle=\exp{(-\frac{1}{2}[\mathbf{p^{0}}^{T}R_{1}\mathbf{p}^{0}+\mathbf{q^{0}}^{T}R_{2}\mathbf{q}^{0}+2\mathbf{p^{0}}^{T}R_{3}\mathbf{q}^{0}])}$
$\displaystyle\times\exp{(-i(\mathbf{p^{0}}^{T}\tilde{h}^{0}+\mathbf{q^{0}}^{T}\tilde{v}^{0})))}.$

Below, we express $\mathbf{p^{0}}^{T}$, $\mathbf{q^{0}}^{T}$,
$\mathbf{p^{0}}$, and $\mathbf{q^{0}}$ in terms of $\mathbf{p}^{T}$ and
$\mathbf{q}^{T}$. Therefore, we write the arguments in the above equations as
$\mathbf{p^{0}}^{T}R_{1}\mathbf{p^{0}}=\mathbf{p}^{T}(e^{-\Lambda
t}R_{1}e^{-\Lambda t})\,\mathbf{p}$, and

$\displaystyle\mathbf{q^{0}}^{T}R_{2}\mathbf{q^{0}}$
$\displaystyle=\mathbf{q}^{T}(e^{-2\Lambda^{\prime}t}R_{2})\mathbf{q}+\mathbf{p}^{T}a^{2}e^{-2(\Lambda^{\prime}+\Lambda)t}R_{2}f^{2}\mathbf{p}$
(69)
$\displaystyle-2\mathbf{p}^{T}\left(ae^{-2\Lambda^{\prime}t}R_{2}fe^{-\Lambda
t}\right)\mathbf{q}.$

The other term can be obtained as

$\displaystyle 2\mathbf{p^{0}}^{T}R_{3}\mathbf{q^{0}}$
$\displaystyle=2\mathbf{p}^{T}(e^{-(\Lambda+\Lambda^{\prime})t}R_{3})\mathbf{q}$
$\displaystyle+2\mathbf{p}^{T}$
$\displaystyle(-ae^{-(2\Lambda+\Lambda^{\prime})t}R_{3}f)\mathbf{p}.$ (70)

Combining the above terms, we get

$\displaystyle\exp{\left[-\frac{1}{2}\left(\mathbf{p^{0}}^{T}R_{1}\mathbf{p^{0}}+\mathbf{q^{0}}^{T}R_{2}\mathbf{q^{0}}+2\mathbf{p^{0}}^{T}R_{3}\mathbf{q^{0}}\right)\right]}$
$\displaystyle=\exp{\left[-\frac{1}{2}(\mathbf{p}^{T}\,\mathbf{M_{1}}\mathbf{p}+\mathbf{q}^{T}\mathbf{M_{2}}\mathbf{q}+2\,\mathbf{p}^{T}\mathbf{M_{3}}\mathbf{q})\right]}.$
(71)

The contribution to $\mathbf{p}^{T}\mathbf{p}$ also comes from
$\mathbf{q^{0}}^{T}\mathbf{q^{0}}$ and $\mathbf{p^{0}}^{T}\mathbf{p^{0}}.$
Therefore, we have three terms in $\mathbf{M}_{1}.$

$\displaystyle\mathbf{M}_{1}$ $\displaystyle=(e^{-\Lambda t}R_{1}e^{-\Lambda
t}+a^{2}\,e^{-2(\Lambda+\Lambda^{\prime})t}\,f^{2}(t)R_{2}$ (72)
$\displaystyle-2\,a\,e^{-(2\Lambda+\Lambda^{\prime})t}f(t)\,R_{3})_{L\times
L}.$

Let us now evaluate the integrations for $R_{1}$, $R_{2}$ and $R_{3}$ which
are given in Eq. (67). Thus, we now have

$\displaystyle R_{1}=(2D+\frac{2a^{2}D_{a}}{(\Lambda^{\prime}-\Lambda)^{2}}$
$\displaystyle)(\frac{e^{2\Lambda
t}-1}{2\Lambda})+\frac{2a^{2}D_{a}}{(\Lambda^{\prime}-\Lambda)^{2}}(\frac{e^{2\Lambda^{\prime}t}-1}{2\Lambda^{\prime}})$
(73)
$\displaystyle-4\frac{a^{2}D_{a}}{(\Lambda^{\prime}-\Lambda)^{2}}\frac{(e^{(\Lambda+\Lambda^{\prime})t}-1)}{\Lambda^{\prime}+\Lambda},$
$\displaystyle R_{2}$
$\displaystyle=2D_{a}\,\left(\frac{e^{2\Lambda^{\prime}t}-1}{2\Lambda^{\prime}}\right)$
(74)

and

$\displaystyle R_{3}$
$\displaystyle=a\frac{2D_{a}}{\Lambda^{\prime}-\Lambda}\left(\frac{e^{(\Lambda+\Lambda^{\prime})t}-1}{(\Lambda+\Lambda^{\prime})}-\frac{e^{2\Lambda^{\prime}t}-1}{2\Lambda^{\prime}}\right)$
(75)

Substituting the expressions for $R_{1}$, $R_{2}$ and $R_{3}$ in Eq. (72), we
obtain

$\displaystyle\mathbf{M}_{1}\,\delta_{ij}=2D_{\text{tot}}\left(\frac{1-e^{-2\Lambda
t}}{2\Lambda}\right)+$
$\displaystyle\frac{a^{2}2D_{a}}{(\Lambda^{\prime}-\Lambda)^{2}}\\{\frac{2\,(e^{-(\Lambda+\Lambda^{\prime})t}-1)}{(\Lambda^{\prime}+\Lambda)}\,$
(76)
$\displaystyle+\frac{\left(1-e^{-2\Lambda^{\prime}t}\right)}{2\Lambda^{\prime}}\\}$

where $D_{\text{tot}}=D+\frac{a^{2}D_{a}}{(\Lambda^{\prime}-\Lambda)^{2}}$.
Similarly, we obtain

$\mathbf{M}_{2}\,\delta_{ij}=2D_{a}\left(\frac{1-e^{-2\Lambda^{\prime}t}}{2\Lambda^{\prime}}\right)$
(77)

and

$\mathbf{M}_{3}\,\delta_{ij}=a\frac{2D_{a}}{(\Lambda^{\prime}-\Lambda)}\left(\frac{1-e^{-(\Lambda+\Lambda^{\prime})t}}{\Lambda+\Lambda^{\prime}}-\frac{1-e^{-2\Lambda^{\prime}t}}{2\Lambda^{\prime}}\right).$
(78)

###  VII.2 Propagator

Let us write $\widehat{\mathbf{W}}[\mathbf{p},\mathbf{q},t|\psi^{0}]$ in terms
of $\mathbf{M}_{1}$, $\mathbf{M}_{2}$ and $\mathbf{M}_{3}$ as

$\displaystyle\mathbf{\widehat{W}}[\mathbf{p},\mathbf{q},t|\psi_{0}]=\exp{[-i\,(\mathbf{p}^{T}e^{-\Lambda
t}\tilde{h}^{0}+\mathbf{q}^{T}e^{-\Lambda^{\prime}t}\tilde{v}^{0})]}$
$\displaystyle\exp{(-i\,(-a\,\mathbf{p}^{T}f(t)\,e^{-(\Lambda^{\prime}+\Lambda)t}\tilde{v}^{0})})$
(79)
$\displaystyle\times\exp{(-\frac{1}{2}(\mathbf{p}^{T}\,\mathbf{M_{1}}\mathbf{p}+\mathbf{q}^{T}\mathbf{M_{2}}\mathbf{q}+2\,\mathbf{p}^{T}\mathbf{M_{3}}\mathbf{q}))}$

We now consider the inverse Fourier transform and integrate over $p$ and $q$.
Thus, we obtain

$\displaystyle\mathbf{W}[\tilde{h},\tilde{v},t|\psi_{0}]=\int\frac{d\mathbf{p}}{(2\pi)^{L}}\int\frac{d\mathbf{q}}{(2\pi)^{L}}\,\,e^{i(\mathbf{p}^{T}\tilde{h}+\mathbf{q}^{T}\tilde{v})}\,\,\mathbf{\widehat{W}}[\mathbf{p},\mathbf{q},t|\psi^{0}]$
$\displaystyle=\frac{1/(2\pi)^{L}}{\sqrt{\det{(\mathbf{M}_{1}\mathbf{M}_{2})}}}\exp{\left[-\frac{1}{2}\left(\Delta\tilde{h}\,\mathbf{M}^{-1}_{1}\Delta\tilde{h}\right)-\frac{1}{2}\left(\Delta\tilde{v}^{T}\mathbf{M}^{-1}_{2}\Delta\tilde{v}\right)-\frac{1}{2}\Delta\tilde{h}^{T}\left(\frac{\mathbf{M}_{2}}{\mathbf{M}_{3}^{2}}\right)^{-1}\Delta\tilde{h}+\Delta\tilde{h}^{T}\left(\frac{\mathbf{M}_{2}}{\mathbf{M}_{3}}\right)^{-1}\Delta\tilde{v}\right]}$

where $\Delta\tilde{h}=\tilde{h}-e^{-\Lambda t}\,h^{0}$, and
$\Delta\tilde{v}=\tilde{v}-e^{-\Lambda^{\prime}t}\,v^{0}$. Integrating over
$\tilde{v}$, we obtain the marginal distribution

$\displaystyle\mathbf{W}(\tilde{h},t|h_{0})$ $\displaystyle=\int
d\tilde{v}\int dv_{0}\,\,\mathbf{W}(\tilde{h},\tilde{v},t|\psi_{0})$
$\displaystyle=\frac{\exp{[-\frac{1}{2}(\tilde{h}-e^{-\Lambda
t}h^{0})^{T}\mathbf{M}^{-1}_{1}(\tilde{h}-e^{-\Lambda
t}h^{0})]}}{(2\pi)^{L/2}\sqrt{\det{\mathbf{M}}_{1}}}.$

###  VII.3 Height-height correlation in terms of operator

Let us start with the non-stationary active noise. Setting $t_{0}=0$, we
obtain

$\displaystyle\langle v_{i}(t)v_{j}(t^{\prime})\rangle$
$\displaystyle=e^{-\Lambda^{\prime}(t+t^{\prime})}\int^{t}_{t_{0}}ds\int^{t}_{t_{0}}ds^{\prime}\,e^{\Lambda^{\prime}(s+s^{\prime})}\langle\Gamma_{i}(s)\Gamma_{j}(s^{\prime})\rangle$
$\displaystyle=\delta_{ij}\,D_{a}\,e^{-\Lambda^{\prime}(t+t^{\prime})}\left(\frac{e^{2\Lambda^{\prime}t}-1}{\Lambda^{\prime}}\right)\,\,\,\text{when\,\,}\,t^{\prime}>t.$

Taking into account the other part, i.e., $t>t^{\prime}$, we obtain

$\displaystyle\langle v_{i}(t)v_{j}(t^{\prime})\rangle$
$\displaystyle=\frac{D_{a}}{\Lambda^{\prime}}\left(e^{-\Lambda^{\prime}|t-t^{\prime}|}-e^{-\Lambda^{\prime}(t+t^{\prime})}\right)$
(83)

and

$\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=2D\delta_{ij}\delta(t-t^{\prime}).$
(84)

We first study the height-height correlation when the velocity-velocity
correlation is non-stationary (i.e., $t_{0}=0$). We obtain

$\displaystyle\langle h_{i}(t)h_{j}(t)\rangle=e^{-2\Lambda
t}(\int^{t}_{0}dt^{\prime}e^{\Lambda
t^{\prime}}\int^{t}_{0}dt^{\prime\prime}e^{\Lambda
t^{\prime\prime}}\langle\eta_{i}(t^{\prime})\eta_{j}(t^{\prime\prime})\rangle$
$\displaystyle+a^{2}\,\int^{t}_{0}dt^{\prime}e^{\Lambda
t^{\prime}}\int^{t}_{0}dt^{\prime\prime}e^{\Lambda t^{\prime\prime}}\langle
v_{i}(t^{\prime})v_{j}(t^{\prime\prime})\rangle)$
$\displaystyle=\delta_{ij}\,D_{\text{tot}}\left(\frac{1-e^{-2\Lambda
t}}{\Lambda}\right)+\delta_{ij}\,\frac{2\,a^{2}D_{a}}{(\Lambda^{\prime}-\Lambda)^{2}}\\{\frac{2\,(e^{-(\Lambda+\Lambda^{\prime})t}-1)}{(\Lambda^{\prime}+\Lambda)}\,$
$\displaystyle+\frac{\left(1-e^{-2\Lambda^{\prime}t}\right)}{2\Lambda^{\prime}}\\}$
(85) $\displaystyle=\sum_{ij}(M_{1})_{ij}\,\delta_{ij}.$ (86)

The covariance matrix from the Fokker-Planck equation and the height-height
correlation from the coupled Langevin equations are found to be the same. The
relation in the above equation is obtained with the non-stationary active
noise, which also holds for the stationary-state active noise.

## Appendix-B

###  VII.4 MFPT: Tension-dominated passive membrane

The height-height correlation can be written as

$\langle
h(x,t)^{2}\rangle_{\text{passive}}=\int\frac{dq}{(2\pi)}\int\frac{dq^{\prime}}{(2\pi)}\,\,e^{i(q+q^{\prime})x}\,\langle
h(q,t)h(q^{\prime},t)\rangle.$ (87)

In the above equation, we consider the membrane fluctuations driven by
spatially and temporally uncorrelated thermal noise. The Fourier transform of
the correlation can be expressed as
$\langle\eta(q,t)\eta(q^{\prime},t^{\prime})\rangle=2D(2\pi)\delta(q+q^{\prime})\delta(t-t^{\prime})$.
Using this, we obtain

$\displaystyle\langle
h(x,t)^{2}\rangle_{\text{passive}}=2D\,\int^{\infty}_{-\infty}\frac{dq}{(2\pi)}\,\,\int^{t}_{0}dt^{\prime}\,\,e^{-2\nu
t^{\prime}q^{2}}.$ (88)

We carry out the integration over $q$ and $t^{\prime}$ in the above equation
and obtain

$\displaystyle\langle
h(x,t)^{2}\rangle_{\text{passive}}=D\sqrt{\frac{2t}{\pi\nu}}.$ (89)

Substituting the above equation in Eq. (LABEL:EqPropagator), we obtain

$W_{\text{passive}}(h_{0},t)=\frac{1}{\sqrt{\pi\alpha^{2}_{p}(t)}}\,e^{-\frac{h_{0}^{2}}{\alpha^{2}_{p}(t)}\,}$
(90)

where $\alpha^{2}_{T}(t)=2D\sqrt{\frac{2t}{\pi\nu}}.$ We consider the LT of
the propagator for passive membrane dynamics with thermal noise. The LT of the
above equation can be obtained as

$\widetilde{W}_{\text{passive}}(u_{p},r)=\frac{1}{\sqrt{\pi\alpha^{2}_{p}}}\frac{1}{r^{3/4}}\int^{\infty}_{0}\frac{d\bar{t}}{\bar{t}^{1/4}}\exp{\left(-\bar{t}-\frac{u_{p}^{2}}{\sqrt{\bar{t}}}\right)}$
(91)

where $\bar{t}=rt$, $\alpha_{T}^{2}=2D\sqrt{2/(\pi\nu)}$ and
$u_{p}=r^{1/4}h_{0}/\alpha_{T}$. Therefore, our MFPT can be written as

$\langle
T_{\nu}\rangle_{\text{passive}}=\frac{1}{r}\left(\frac{I_{p}(0)}{I_{p}(u_{p})}-1\right)$
(92)

where

$I_{p}(u_{p})=\int^{\infty}_{0}\frac{d\bar{t}}{\bar{t}^{1/4}}\exp{\left(-\bar{t}-\frac{u_{p}^{2}}{\sqrt{\bar{t}}}\right)}.$
(93)

We carry out the integration for $u_{p}=0$ and obtain
$I_{p}(0)=\Gamma(\frac{3}{4}).$ From the above equation, we obtain

$I_{p}(u_{p})\simeq
I_{p}(0)\left(1-|u_{p}|^{2}\frac{\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})}\right)$
(94)

for $u_{p}\ll 1.$ A simpler form for the MFPT can be expressed as

$\langle
T_{\nu}\rangle_{\text{passive}}\simeq\frac{\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})}\frac{1}{r^{1/2}}\,\,\,\left(\frac{h_{0}}{\alpha_{T}}\right)^{2}\,\,\,\text{for}\,\,\,r\ll\left(\alpha_{T}/h_{0}\right)^{4}.$
(95)

The expression of the MFPT $\langle
T_{\nu}\rangle_{\text{passive}}\rightarrow\infty$ when
$u_{p}\rightarrow\infty$. The MFPT of a passive membrane scales as $\langle
T_{\nu}\rangle_{\text{passive}}\sim\nu^{1/2}.$

###  VII.5 MFPT: Tension-less passive membrane

Similarly, with the thermal noise, we derive the height-height correlation for
the rigidity-dominated membrane as

$\displaystyle\langle h(x,t)^{2}\rangle_{\text{passive}}\,$
$\displaystyle=\frac{D}{3\pi^{3/4}}\,\left[\frac{(2t)^{3}}{\pi\kappa}\right]^{1/4}$
(96)

where we consider $\nu=0.$ Substituting the above equation in equation
(LABEL:EqPropagator), we write the propagator as

$\displaystyle
W_{\text{passive}}(h_{0},t)=\frac{1}{\sqrt{\pi}}\frac{1}{\beta_{T}(t)}\exp{\left(-\frac{h_{0}^{2}}{\beta_{T}(t)^{2}}\right)}$
(97)

where $\beta_{T}(t)=\beta_{T}\,\,t^{3/8}$ with
$\beta_{T}=\sqrt{\frac{2D}{3\pi^{3/4}}}\,\left[\frac{8}{\pi\kappa}\right]^{1/8}.$
The Laplace transform of the above propagator can be written as

$\displaystyle\widetilde{W}_{\text{passive}}(y_{p},r)=\frac{1}{\sqrt{\pi}}\frac{1}{\beta_{T}\,r^{5/8}}J_{p}(y_{p})$
(98)

where $y_{p}=r^{3/8}h_{0}/\beta_{T}$, and

$J_{p}(y_{p})=\int^{\infty}_{0}\frac{d\bar{t}}{\bar{t}^{3/8}}\,e^{-\bar{t}}\,\exp{\left(-\frac{y_{p}^{2}}{\bar{t}^{3/4}}\right)}.$
(99)

Using the definition of the Gamma function, we obtain
$\widetilde{W}_{\text{passive}}(0,r)=\frac{1}{\sqrt{\pi}}\Gamma\left(\frac{5}{8}\right)/(\beta_{T}\,r^{5/8})$
and

$\langle
T_{\kappa}\rangle_{\text{passive}}=\frac{1}{r}\left(\frac{J_{p}(0)}{J_{p}(y_{p})}-1\right).$
(100)

Considering $y_{p}\ll 1$, we obtain $\langle
T_{\kappa}\rangle\simeq\frac{1}{r}\,\,\left[(1-c_{0}|y_{p}|^{5/3}+c_{1}|y_{p}|^{2})^{-1}-1\right]$
where $c_{0}=4.656$, and $c_{1}=6.077.$ For $r\ll(\beta_{T}/h_{0})^{8/3}$ , we
obtain

$\langle
T_{\kappa}\rangle_{\text{passive}}\simeq\frac{c_{0}}{r^{3/8}}\,\,\left(\frac{h_{0}}{\beta_{T}}\right)^{5/3}.$
(101)

The above equation suggests that for the passive system, the MFPT strongly
depends on the rigidity $\kappa$ as $\langle
T_{\kappa}\rangle_{\text{passive}}\sim\kappa^{5/24}.$

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