# Long-term memory induced correction to Arrhenius law

A. Barbier-Chebbah1,2, O. Bénichou2, R. Voituriez2,3, T. Guérin4 1 Decision
and Bayesian Computation, USR 3756 (C3BI/DBC) and Neuroscience Department CNRS
UMR 3751, Institut Pasteur, Université de Paris, CNRS, 75015 Paris, France
2Laboratoire de Physique Théorique de la Matière Condensée, CNRS/UPMC, 4 Place
Jussieu, 75005 Paris, France  3Laboratoire Jean Perrin, CNRS/UPMC, 4 Place
Jussieu, 75005 Paris, France  4Laboratoire Ondes et Matière d’Aquitaine,
CNRS/University of Bordeaux, F-33400 Talence, France

###### Abstract

The Kramers escape problem is a paradigmatic model for the kinetics of rare
events, which are usually characterized by Arrhenius law. So far, analytical
approaches have failed to capture the kinetics of rare events in the important
case of non-Markovian processes with long-term memory, as occurs in the
context of reactions involving proteins, long polymers, or strongly
viscoelastic fluids. Here, based on a minimal model of non-Markovian Gaussian
process with long-term memory, we determine quantitatively the mean FPT to a
rare configuration and provide its asymptotics in the limit of a large energy
barrier $E$. Our analysis unveils a correction to Arrhenius law, induced by
long-term memory, which we determine analytically. This correction, which we
show can be quantitatively significant, takes the form of a second effective
energy barrier $E^{\prime}<E$ and captures the dependence of rare event
kinetics on initial conditions, which is a hallmark of long-term memory.
Altogether, our results quantify the impact of long-term memory on rare event
kinetics, beyond Arrhenius law.

Many physical and chemical processes are controlled by “rare” events,
referring to events that are qualitatively unlikely, but nonetheless important
because their realization has exceptional consequences hanggi1990reaction ;
pollak2005reaction . Such events are ubiquitous in the context of chemical
physics, as exemplified at the molecular scale by the formation or rupture of
bonds hanggi1990reaction  (e.g. in force spectroscopy experiments
bullerjahn2014theory ; bullerjahn2020non ; bullerjahn2016analytical  or
adhesion kinetics jeppesen2001impact ), protein folding ayaz2021non ,
molecular motor dynamics Badoual2002a ; guerin2011 ; Guerin2011c , or more
generally nucleation events. Rare events are also relevant in other contexts,
such as stock market crashes Bouchaud1998  or climate ragone2018computation
or population kamenev2008colored ; dykman2008disease  dynamics. The kinetics
of such events, quantified by the first-passage time (FPT) to a target
configuration, generally follows Arrhenius (also called Kramers, or Eyring-
Kramers) law: the mean waiting time for a rare event is exponentially large
with the energy barrier that has to be crossed to reach the target
configuration hanggi1990reaction . This picture is also valid in non-
equilibrium systems with the definition of a pseudo-potential Freidlin1984 ;
Maier1992 ; bouchet2016generalisation ; delacruz2018minimum . In the weak-
noise limit, the mean FPT is generally obtained by analyzing the dynamics at
the top of the (pseudo-)potential barrier, by expanding around the most
probable path leading to the target configuration. In this limit the waiting
time for a rare event becomes larger than all relaxation times of the
dynamics, and is thus independent of initial conditions.

While the effect of memory on first passage ReviewBray ; metzler2014first ;
lindenberg2019chemical ; Sokolov2003 ; Likthman2006 ; guerin2016mean ;
delorme2015maximum  and rare event kinetics ferrer2021fluid ; ginot2022barrier
; lavacchi2020barrier ; lavacchi2022non ; bullerjahn2020non ;
kappler2018memory ; Caraglio2018 ; carlon2018effect ; medina2018transition ;
goychuk2007anomalous ; Sliusarenko2010 ; arutkin2020extreme ;
levernier2020kinetics ; delorme2017pickands ; goswami2023effects  has been the
object of recent studies, an important open question arises as to whether
Arrhenius law is still valid for stochastic processes (or “reaction
coordinates”) $x(t)$ displaying infinite relaxation times, i.e. with
correlation functions decaying as a power-law rather than exponentially:

$\phi(\tau)\equiv\lim_{t\to\infty}\frac{\langle x(t)x(t+\tau)\rangle}{\langle
x^{2}(t)\rangle}\underset{\tau\to\infty}{\simeq}\frac{A}{\tau^{\alpha}},$ (1)

where $A>0$, $\alpha>0$ and $\langle x(t)\rangle=0$ by convention. Stochastic
processes possessing the property (1) will be called hereafter long-term
memory processes santhanam2008return ; bunde2005long  and arise when their
dynamics results from the evolution of an infinite number of degrees of
freedom. Examples of processes with long-term memory are provided by the
dynamics of polymers Panja2010 , proteins kou2004generalized ; Min2005  or
interfaces ReviewBray , but also earthquakes lennartz2008long  or rainfalls
bunde2013there . It is known that long-term memory induces dispersed kinetics
min2006kramers ; goychuk2009viscoelastic  and correlations between successive
realizations of rare events eichner2007statistics ; santhanam2008return ;
bunde2005long ; its impact on the kinetics of rare events however remains to
be elucidated. In fact, this question was considered in Ref.
goychuk2007anomalous  by means of a generalized Fokker-Planck equation, a
controversial singh2019comment ; bullerjahn2017unified ; bullerjahn2020non
method which leads to the notable prediction that the mean FPT to a rare
configuration is infinite for a class of processes with long-term memory ; in
Ref. Sliusarenko2010 , it was noted that the standard so-called “Wilemski-
Fixman” approximation WILEMSKI1974a  also predicts infinite mean FPTs [when
the exponent $\alpha$ defined in (1) satisfies $\alpha<1$] NoteRef28 .
Nevertheless, these predictions of infinite mean FPTs for processes with long-
term memory seem inconsistent with numerical simulations singh2019comment ;
Sliusarenko2010 ; bullerjahn2017unified  and the mathematical results of Refs.
newell1962zero ; pickands1969upcrossing ; pickands1969asymptotic , which point
to finite mean FPTs. Such contradiction shows that the above mentioned methods
cannot be used to analyse the impact of long-term memory on rare event
kinetics.

Here, on the basis of a simple model of a particle in a potential $V(x)$ at
finite temperature with retarded friction force, we resolve this issue and
quantify the impact of long-term memory on the kinetics of rare events. We
generalize to processes with long-term memory a formalism that was so far
restricted to the analysis of either FPTs in large confining volumes with flat
energy landscapes guerin2016mean , or of rare events without long-term memory
levernier2020kinetics . Our theory predicts finite mean FPTs, and is supported
quantitatively by numerical simulations. In the limit of large energy barriers
– called hereafter rare events limit, we show that Arrhenius law does hold,
with however sub-exponential corrections induced by the long-term memory,
which we determine explicitly. We find that long term memory effectively
induces a second effective energy barrier of size $E^{\prime}=E(1-\alpha)$
(for $\alpha<1$), where $E=V(L)-V(0)$ is the size of the real barrier (see
Fig. 1). We find that the prefactor of this correction, which we explicitly
calculate, is much larger than the prefactor of the leading-order Arrhenius
law, which implies that this correction is significant for a broad range of
energy barriers.

Figure 1: (a) Sketch of the problem. Let $x(t)$ be a random walker in a
potential at temperature $\mathcal{T}$, submitted to a power-law friction
kernel. In this example of long-term memory (meaning that the correlation
function of $x(t)$ decay as a power-law), what is the mean FPT to a target at
$x=L$ that can be reached only by overcoming an energy barrier $E=V(L)-V(0)$ ?
(b) Sketch of the FPT for a single stochastic trajectory of $x(t)$.

Minimal model
We consider a minimal model of non-Markovian process $x(t)$ with long term
memory at temperature $\mathcal{T}$, in a confining potential that is assumed
harmonic, see Fig. 1(a). We assume that $x(t)$ obeys the overdamped
Generalized Langevin Equation (GLE) :

$\int_{0}^{t}dt^{\prime}\ K(t-t^{\prime})\ \dot{x}(t^{\prime})=-k\
x(t)+\xi(t).$ (2)

Here, the 1–dimensional random variable $x(t)$ stands typically for the
position of a particle, $K(t)$ represents the friction kernel, $k$ is the
stiffness of the harmonic potential applied to the particle, and $\xi(t)$ is a
Gaussian thermal force with zero mean whose magnitude is set by the
fluctuation dissipation theorem
$\langle\xi(t)\xi(t^{\prime})\rangle=k_{B}\mathcal{T}K(|t-t^{\prime}|)$. With
these definitions the process $x(t)$ is Gaussian and its stationary
probability density function (pdf) is
$p_{s}(x)=e^{-\frac{kx^{2}}{2k_{B}\mathcal{T}}}/\sqrt{2\pi l^{2}}$, where
$l=\sqrt{k_{B}\mathcal{T}/k}$ is the confinement length. Memory effects are
encoded in the friction kernel $K(t)$, and result typically from complex
interactions of the variable $x(t)$ with other, potentially hidden, degrees of
freedom. The dynamics (2) describes a variety of physical processes: (i) the
motion of a tracer particle in a viscoelastic fluid mason1995optical ;
gisler1998tracer ; mason1997particle , (ii) the motion of a tagged particle
attached to a polymer chain Panja2010 ; Panja2010a ; bullerjahn2011monomer ,
(iii) the dynamics of the distance between two protein residues as
experimentally observed Min2005 . In the following we will mainly focus on
scale invariant friction kernels:

$K(t)=\frac{K_{\alpha}}{\Gamma(1-\alpha)t^{\alpha}},$ (3)

where $0<\alpha<1$, $K_{\alpha}$ is a transport coefficient, and
$\Gamma(\cdot)$ is the gamma function. While the theory presented below could
be applied to other kernels, this choice (3) is relevant to the physical
examples (i),(ii),(iii) above. Furthermore, in absence of target, the
correlation function defined in (1) is
$\phi(t)=E_{\alpha}[-(t/\tau_{d})^{\alpha}]$ goychuk2007anomalous ;
min2006kramers  where $\tau_{d}=(K_{\alpha}/k)^{1/\alpha}$ and
$E_{\alpha}(\cdot)$ is the Mittag-Leffler function. Since
$E_{\alpha}(-u){\sim}1/[\Gamma(1-\alpha)u]$ for large arguments, the choice of
kernel (3) ensures that the process $x(t)$ displays long-term memory as
defined in (1): there is no finite relaxation time in the correlation
function, and $A=K_{\alpha}/[\Gamma(1-\alpha)k]$ (SM, Section A).

If one imposes the initial condition $x(0)=x_{0}$, the average path
$m_{0}(t)\equiv\langle x(t)\rangle_{x(0)=x_{0}}$ and the covariance
$\sigma(t,t^{\prime})\equiv\text{Cov}(x(t),x(t^{\prime}))_{x(0)=x_{0}}$
conditional to $x(0)=x_{0}$ read Eaton1983

$m_{0}(t)=x_{0}\phi(t),\hskip
4.26773pt\sigma(t,t^{\prime})=l^{2}[\phi(|t-t^{\prime}|)-\phi(t)\phi(t^{\prime})].$
(4)

We also define $\psi(t)=\sigma(t,t)$ as the Mean Squared Displacement (MSD) of
$x(t)$. In absence of potential ($k=0$), $x(t)$ is the fractional Brownian
motion of Hurst exponent $H=\alpha/2$ ; for finite $k$ this regime is realized
at short times, when the harmonic force is negligible, as seen from the MSD:

$\psi(t)\underset{t\to 0}{\simeq}\kappa\ t^{2H},\
\kappa=\frac{2k_{B}\mathcal{T}}{K_{\alpha}\Gamma(1+\alpha)},\
H=\frac{\alpha}{2}.$ (5)

Hereafter we study the mean FPT of the process $x(t)$ defined by (2), (3) to a
target threshold $x=L$, with an initial configuration either drawn from the
equilibrium distribution or set by $x(0)=x_{0}$.

Numerical analysis
We have performed numerical simulations of the GLE (2) by using a modified
version of the circulant matrix algorithm davies1987tests  described in Ref.
dietrich_fast_1997 , which is an exact generator of $x(t)$ at sampling times
$t_{n}=n\times dt$ for any value of the time step $dt$. The used values of
$dt$ are indicated in the Supplemental Material (SM, Section D) and are always
smaller than $2\times 10^{-5}\tau_{d}$. We used the two values of $H=\alpha/2$
that are used in classical polymer models : either a semi-flexible chain
($H=3/8$) or a flexible (Rouse) chain without hydrodynamic interactions
($H=1/4$). For each trajectory $\\{x(t_{n})\\}$ we measured the FPT to $L$.
The resulting survival probability $S(t)$ (defined as the probability that the
FPT is larger than $t$) is shown in Fig. 2. Our numerical results are
consistent with the mathematical results of Refs newell1962zero ;
pickands1969upcrossing ; pickands1969asymptotic : an exponential decay of
$S(t)$ in the rare events limit $L\to\infty$, and a stretched exponential
behavior for $L=0$. This numerical analysis thus further supports that the
mean FPT is finite (see Fig. 3).

Figure 2: Survival probabilities $S(t)$ for (a) $H=3/8$ and (b) $H=1/4$, as
measured in numerical simulations. Here $x_{0}$ is drawn from the equilibrium
distribution $p_{s}(x)$. The black line represents $S(t)=e^{-t/\langle
T\rangle}$. Error bars represent 68$\%$ confidence intervals, due to
statistical uncertainties.  Figure 3: Mean FPT when the initial position is
$x_{0}=0$ for (a) $H=3/8$ and (b) $H=1/4$. Symbols: numerical simulations;
dots: numerical integration of Eqs. (6, 8); dashed red line: Arrhenius law at
leading order, Eq. (10); orange full line: refined Arrhenius law (13),
including the corrections due to long-term memory. We have used the values
$\nu_{3/8}=5.26$ and $\nu_{1/4}=5.0$ calculated in Ref. levernier2020kinetics
.

General non-Markovian analysis
We now proceed to the theoretical determination of the mean FPT to $x=L$,
denoted $\langle T\rangle$, with fixed initial condition $x(0)=x_{0}$ [the
case of stationary initial conditions can be obtained by averaging over
$p_{s}(x_{0})$]. Our approach consists in generalizing the tools developed in
Refs. levernier2020kinetics ; guerin2016mean ; levernier2019survival , which,
in the context of rare event kinetics, have been used so far only to analyze
processes with a finite maximal relaxation time levernier2020kinetics . We
describe the main steps of the approach for completeness ; details can be
found in SM (Section B). We start with the following general exact expression
of the mean FPT, derived in Ref. guerin2016mean :

$\langle T\rangle p_{s}(L)=\int_{0}^{\infty}dt\ [p_{\pi}(L,t)-p(L,t)],$ (6)

where we have introduced $p_{\pi}(x,t)$ as the pdf of the process
$x_{\pi}(t)\equiv x(t+T)$, where $T$ is the FPT; $x_{\pi}(t)$ is thus the
process after a first-passage event. To characterize $p_{\pi}(x,t)$, we assume
that the process $x_{\pi}(t)$ is Gaussian (as is $x(t)$), and thus fully
characterized by its first moment $m_{\pi}(t)=\langle x_{\pi}(t)\rangle$, and
covariance $\sigma_{\pi}(t,t^{\prime})\simeq\sigma(t,t^{\prime})$ that is
assumed to be identical to that of the unconditioned process $x(t)$. The
validity of these hypotheses has been checked numerically [Fig. 4(a) and SM,
Section D] and analytically for weakly non-Markovian processes (SM, Sections
E). With these approximations, Eq. (6) becomes

$\langle T\rangle
p_{s}(L)=\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi\psi(t)}}\left[e^{-\frac{[m_{\pi}(t)-L]^{2}}{2\psi(t)}}-e^{-\frac{[x_{0}\phi(t)-L]^{2}}{2\psi(t)}}\right].$
(7)

The so-far unknown quantity $m_{\pi}(t)$ can then be determined self-
consistently by analyzing a generalized version of the renewal equation (see
SM, Section B), leading to

$\displaystyle\int_{0}^{\infty}dt\Bigg{\\{}$
$\displaystyle\frac{e^{-\frac{[m_{\pi}(t)-L]^{2}}{2\psi(t)}}}{\sqrt{2\pi\psi(t)}}\left(m_{\pi}(t+\tau)-\left[m_{\pi}(t)-L\right]\frac{\sigma(t+\tau,t)}{\sigma(t,t)}-L\phi(\tau)\right)$
$\displaystyle-$
$\displaystyle\frac{e^{-\frac{[x_{0}\phi(t)-L]^{2}}{2\psi(t)}}}{\sqrt{2\pi\psi(t)}}\left(x_{0}\phi(t+\tau)-\left[x_{0}\phi(t)-L\right]\frac{\sigma(t+\tau,t)}{\sigma(t,t)}-L\phi(\tau)\right)\Bigg{\\}}=0.$
(8)

This equation generalizes similar equations in Refs. guerin2016mean ;
levernier2020kinetics , which were restricred on the determination of
$p_{\pi}(L,t)$ at short times and thus did not enable the analysis of long-
term memory effects. This integral equation, together with the condition
$m_{\pi}(0)=L$, allows to determine the only unknown $m_{\pi}(t)$ : this
finally gives access to $\langle T\rangle$ thanks to Eq. (7).

General results
This approach first shows unambiguously that the mean FPT is finite. Indeed,
we show in SM (Section B) that the solution to Eq. (8) satisfies at long times

$m_{\pi}(t)\underset{t\to\infty}{\simeq}x_{0}\ \phi(t),$ (9)

which can be checked directly in numerical simulations, see figure 4(b). This
scaling, together with Eq. (6), shows that the mean FPT is finite. This
contradicts the results obtained with the generalized Fokker-Planck equation
goychuk2007anomalous  or with the Wilemski-Fixman approximation WILEMSKI1974a
. The latter amounts to assuming that the process is at all times in an
equilibrium state, and would thus yield $m_{\pi}(t)\simeq L\phi(t)$, leading
to an infinite mean FPT when $\alpha<1$ (as noted earlier in a similar, but
out of equilibrium, situation Sliusarenko2010 ). Beyond this proof of
finiteness, our approach yields a quantitative determination of $\langle
T\rangle$ by solving numerically the integral equation (8) for $m_{\pi}(t)$
and next using Eq. (6) ; this shows quantitative agreement with numerical
simulations in Fig. 3.

Figure 4: (a) Check of the stationary covariance approximation (i.e.
$\sigma_{\pi}(t,t^{\prime})\simeq\sigma(t,t^{\prime})$): comparison between
$\psi_{\pi}(t)=\text{Var}(x_{\pi}(t))$ measured in numerical simulations
(symbols) and $\psi(t)$ (dashed line: $H=3/8$, full line $H=1/4$). (b): Check
of Eq. (48): comparison between the value $m_{\pi}(t)$ in simulations
(symbols) and $x_{0}\phi(t)$ (full line: $x_{0}=l/2$ for $H=3/8$; dashed line
$x_{0}=l$ for $H=1/4$). Note that $m_{\pi}(t)\simeq x_{0}\phi(t)$ is expected
at large times only. (c) Check of the short-time scaling regime for $H=3/8$.
(d) Check of the long-time scaling regime (11) for $H=3/8$. In (a),(c),(d),
the initial position is drawn from an equilibrium distribution, corresponding
to our predictions for $x_{0}=0$. When present, error bars represent $68\%$
confidence intervals.

Rare events limit $L\to\infty$
We now consider the rare event limit to determine explicitly the impact of
long-term memory on rare events kinetics. The mean FPT obtained by the method
of matched asymptotics which we sketch here; calculation details are provided
in SM (Section C). The dynamics involves different time and length scales ;
two can be readily identified: (i) the confinement length $l$ and (ii) the
length $l^{*}=k_{B}\mathcal{T}/F$, where $F=kL$ is the slope of the potential
at $L$. The associated time scales are respectively (i) $\tau_{d}$ and (ii)
the time $t^{*}$ at which the characteristic fluctuations
$\sqrt{\kappa}(t^{*})^{H}$ of the trajectories near the target become
comparable to $l^{*}$, this leads to $t^{*}=(l^{*}/\sqrt{\kappa})^{1/H}$. Note
that in the rare events limit $t^{*}\ll\tau_{d}$.

The leading order term $T_{\text{RE}}$ of $\langle T\rangle$ in the
$L\to\infty$ limit results from the contribution of timescales $t\sim
t^{*}\ll\tau_{d}$ only in (6). Indeed, after a time $t\gg t^{*}$, a particle
initially at $L$ has typically moved away from the target, so that
$p_{\pi}(L,t)$ is exponentially small, whereas $p_{\pi}$ is of order 1 at very
short times $t\sim t^{*}$. In turn, if the starting position is typically not
close from $L$, $p(L,t)$ is exponentially small with $L$ at all times. The
above consideration suggests to look for solutions of the form
$m_{\pi}(t)\simeq L-l^{*}f(t/t^{*})$ ; inserting this ansatz in (8) and taking
the rare event limit leads to an equation for $f$ that depends only on $H$,
justifying our ansatz. The mean FPT at leading order is then obtained as

$\langle T\rangle\underset{L\to\infty}{\sim}\frac{l^{\frac{2}{H}-1}\
\nu_{H}}{L^{\frac{1}{H}-1}\ \kappa^{\frac{1}{2H}}}\times e^{\beta E}\equiv
T_{\text{RE}},$ (10)

where $\nu_{H}=\int_{0}^{\infty}\frac{du}{u^{H}}\ e^{-f^{2}(u)/2u^{2H}}$
depends only on $H$, $E=kL^{2}/2$ is the energy barrier and
$\beta=1/(k_{B}\mathcal{T})$. This leading order result displays the usual
Arrhenius factor $e^{\beta E}$, which is the hallmark of rare event kinetics,
and is compatible with the mathematical results of Pickands
pickands1969upcrossing . Of note, it is controlled only by the short time
behavior of the MSD $\psi(t)$, and is independent of the long time relaxation
of correlations, and thus of long-term memory. It is indeed identical to the
results of Ref. levernier2020kinetics  obtained for non Markovian processes
with the same MSD at short times but finite relaxation time. To prove this
result self-consistently, we need to estimate the contributions to $\langle
T\rangle$ in (6), that are induced by the behaviour of the integrand at time
scales $t\gg t^{*}$. These contributions are expected to be relevant in the
case of long-term memory, due to the slow decay of correlation functions.

Here, the key point is to note that, in addition to the previously identified
timescales $\tau_{d}$ and $t^{*}$, a third relevant timescale for the dynamics
of $x_{\pi}(t)$ is the time $T_{\mathrm{RE}}$ it-self. Indeed, we show in SM
that $m_{\pi}(t,L)$ can be written for $t\gg t^{*}$ :

$m_{\pi}(t)\simeq\begin{cases}L\ \phi_{\pi}(t)&(t^{*}\ll
t=\mathcal{O}(\tau_{d})\ll T_{\text{RE}})\\\
\frac{LA}{T_{\text{RE}}^{\alpha}}\chi\left(\frac{t}{T_{\text{RE}}}\right)&(\tau_{d}\ll
t=\mathcal{O}(T_{\text{RE}}))\end{cases},$ (11)

where $A$ is defined in (1) and $\chi$ and $\phi_{\pi}$ are scaling functions.
The analysis of Eq. (8) at timescales $\tau_{d}$ and $T_{\text{RE}}$,
respectively, enables us to obtain equations for $\phi_{\pi}$ and $\chi$ that
can be solved, leading to

$\phi_{\pi}(t)=\phi(t),\hskip
8.5359pt\chi(y)=\alpha\left(1-\frac{x_{0}}{L}\right)\Gamma(-\alpha,y)e^{y}+\frac{x_{0}}{y^{\alpha}L},$
(12)

where $\Gamma(s,y)=\int_{y}^{\infty}t^{s-1}e^{-t}dt$ is the upper incomplete
gamma function. Finally, inserting the scaling forms for $m_{\pi}(t)$ into Eq.
(6), we obtain

$\langle T\rangle\simeq
T_{\text{RE}}+T_{\text{RE}}^{1-\alpha}\times\frac{A}{l^{2}}L(L-x_{0})\Gamma(1-\alpha).$
(13)

This is the central result of this Letter. It confirms the validity of the
leading order term $T_{\text{RE}}$, which is independent of long-term memory,
and explicitly determines the subleading term, which is induced by long-term
memory, as seen by the factor $A$ that characterizes the long-time decay of
fluctuations. Several remarks are in order: (i) Since $T_{\text{RE}}\propto
e^{\beta E}$, the correction due to long-term memory is of order $e^{\beta
E^{\prime}}$ with an effective energy barrier $E^{\prime}=E(1-\alpha)$. The
smaller the value of $\alpha$ the larger the value of $E^{\prime}$, so that
the convergence to the rare event limit is expected to be slower for small
$\alpha$ (where non-Markovian effects are stronger). (ii) Furthermore, the
pre-exponential factor is clearly much larger for the corrective term than for
the leading order term in the limit $L\to\infty$, so that the corrective term
can be quantitatively important. Indeed, as observed in figure 3, taking into
account this correction is essential to predict the rare event kinetics for
not-too-large values of $L$. (iii) Eq. (13) shows that the subleading
correction depends on the initial position $x_{0}$: because of long-term
memory, initial conditions can thus impact quantitatively rare event kinetics.
(iv) As a further validation of our analysis, the expected scaling behaviors
of $m_{\pi}$ are given in Fig. 4 (c),(d) and hold in the large $L$ limit, with
discrepancies at small times in Fig. 4(c) due to limitations in the choice of
the time step (see SM, Section D for additional parameters).

Conclusion
We have proposed a theoretical analysis of the classical Kramers escape
problem for non Markovian processes with long-term memory. Although our
approach is approximate, it captures the essence of memory effects and allows
for a quantitative determination of the mean FPT to a target, which we
unambiguously show is finite, whereas all existing theoretical approaches so
far incorrectly predicted infinite mean FPTs (for $\alpha<1$). This comes from
the assumption of a system at equilibrium at all times that is implicitly made
in the methods that have been employed so far, namely the Wilemski-Fixman
approximation or the generalized Fokker-Planck equation approach. Such
hypothesis is too strong to take properly into account long-term memory
effects. In our approach, the genuine non-equilibriumness of the system upon a
first passage event manifests itself in the trajectory $m_{\pi}(t)$, whose
behaviour at very long times is affected by long-term memory. In the rare
event limit, we have explicitly determined the correction to Arrhenius laws,
which is due to long-term memory. This takes the form of a second effective
energy barrier of size $E^{\prime}=E(1-\alpha)$, which we show can be
quantitatively significant, and captures the dependence of the kinetics on
initial conditions. It is known that Arrhenius laws can be identified for non-
Gaussian models by considering the linearized dynamics around the target
levernier2020kinetics . Since our study reveals that the effect of long-term
memory on rare event kinetics comes from the slow dynamics at the bottom of
the potential only, we may expect that our main result (13) could be
generalized to non-Gaussian models. Moreover, although we have focused here on
a simple model of a particle with viscoelastic friction at equilibrium at
constant temperature, it is clear that our arguments to identify the mean FPT
could be adapted to active models where the fluctuation-dissipation theorem
does not hold. Indeed, Eq. (7) and (8) would still be valid, and would involve
similarly the properties of the process in absence of target
($A,\phi,p_{s},\kappa,...$), which are in principle still accessible from the
definition of the process in Eq. (2), even if the fluctuation-dissipation
relation does not hold because of active effects Sliusarenko2010  . Last,
because our approach puts forward deviations from Arrhenius law due to long-
term memory, we also anticipate deviations from exponential laws for the
distribution of FPTs, that could be studied by generalizing our approach to
higher moments of the FPT, possibly giving access to the analytical study of
extreme events clustering and dispersed kinetics. Altogether, our results shed
light on the effect of long-term memory on rare event kinetics, beyond
Arrhenius laws.

###### Acknowledgements.

T. G. acknowledges the support of the grant ComplexEncounters,
ANR-21-CE30-0020. Computer time for this study was provided by the computing
facilities MCIA (Mesocentre de Calcul Intensif Aquitain) of the Université de
Bordeaux and of the Université de Pau et des Pays de l’Adour.

Supplemental Material

In this Supplemental Material, we provide

  1. 1.

calculation details to obtain the solution of the GLE equation (without
target) [Section .1].

  2. 2.

a detailed derivation of the equations of the non-Markovian theory [Section
.2]

  3. 3.

calculation details for the asymptotic analysis in the rare event limit
$L\to\infty$ [Section .3].

  4. 4.

Details on simulations and additional simulation data to check the Gaussian
behavior of trajectories in the future of first passage events and our scaling
arguments [Section .4].

  5. 5.

A note on the exactness of the approach for weakly non-Markovian processes
[Section .5].

##  .1 Solution of the Generalized Langevin Equation (without absorbing
target)

Here, we consider the dynamics given by the overdamped GLE

$\displaystyle\int_{0}^{t}dt^{\prime}K(|t-t^{\prime}|)\dot{x}(t^{\prime})=-k\
x(t)+\xi(t),$
$\displaystyle\langle\xi(t)\xi(t^{\prime})\rangle=k_{B}\mathcal{T}\
K(|t-t^{\prime}|).$ (14)

In absence of target, the solution of this equation is well known
goychuk2007anomalous , it is reminded here for the sake of completeness. Since
the above equation is linear, the resulting process $x(t)$ is Gaussian and is
fully characterized by its two first moments. Denoting
$\widetilde{f}(s)=\int_{0}^{\infty}f(t)e^{-st}dt$ the Laplace transform of a
function $f$, we obtain

$\widetilde{x}(s)=\frac{\widetilde{\xi}(s)+x(0)\tilde{K}(s)}{s\tilde{K}(s)+k}.$
(15)

We also write

$\displaystyle\langle\widetilde{\xi}(s)\widetilde{\xi}(s^{\prime})\rangle$
$\displaystyle=k_{B}\mathcal{T}\int_{0}^{\infty}dt\int_{0}^{\infty}dt^{\prime}e^{-(st+s^{\prime}t^{\prime})}K(|t-t^{\prime}|),$
(16)
$\displaystyle=k_{B}\mathcal{T}\int_{0}^{\infty}dt^{\prime}\int_{0}^{\infty}d\tau\
e^{-(s+s^{\prime})t^{\prime}-s\tau}K(\tau)+k_{B}\mathcal{T}\int_{0}^{\infty}dt\int_{0}^{\infty}d\tau^{\prime}\
e^{-(s+s^{\prime})t-s^{\prime}\tau^{\prime}}K(\tau^{\prime}),$ (17)
$\displaystyle=k_{B}\mathcal{T}\frac{\tilde{K}(s)+\tilde{K}(s^{\prime})}{s+s^{\prime}},$
(18)

where Eq. (17) has been obtained by setting $t=t^{\prime}+\tau$ for
$t>t^{\prime}$ and $t^{\prime}=t+\tau^{\prime}$ for $t^{\prime}>t$. Using this
result and (15) yields, for an initially equilibrated initial position
$\langle x(0)^{2}\rangle=k_{B}\mathcal{T}/k$:

$\displaystyle\langle\widetilde{x}(s)\widetilde{x}(s^{\prime})\rangle$
$\displaystyle=\frac{k_{B}\mathcal{T}}{[s\tilde{K}(s)+k][s^{\prime}\tilde{K}(s^{\prime})+k]}\left\\{\frac{\tilde{K}(s)+\tilde{K}(s^{\prime})}{s+s^{\prime}}+\frac{\tilde{K}(s)\tilde{K}(s^{\prime})}{k}\right\\},$
$\displaystyle=\frac{k_{B}\mathcal{T}}{k(s+s^{\prime})}\left\\{\frac{\tilde{K}(s)}{s\tilde{K}(s)+k}+\frac{\tilde{K}(s^{\prime})}{s^{\prime}\tilde{K}(s^{\prime})+k}\right\\}.$
(19)

We may recognize that if one sets

$\displaystyle\langle x(t)x(t^{\prime})\rangle=l^{2}\phi(|t-t^{\prime}|),$
(20)

then, using the same procedure as in Eq. (17),

$\displaystyle\langle\tilde{x}(s)\tilde{x}(s^{\prime})\rangle=\frac{l^{2}}{s+s^{\prime}}[\tilde{\phi}(s)+\tilde{\phi}(s^{\prime})].$
(21)

Comparing the above equation with (.1) leads to

$\displaystyle\tilde{\phi}(s)=\frac{\tilde{K}(s)}{s\tilde{K}(s)+k}.$ (22)

This formula is valid for arbitrary kernel. For the power-law kernel (3) of
the main text, we obtain $\tilde{K}=K_{\alpha}/s^{1-\alpha}$ and $\phi(t)$ is
a Mittag-Leffler function:

$\displaystyle\tilde{\phi}(s)=\frac{K_{\alpha}s^{\alpha}}{s[s^{\alpha}K_{\alpha}+k]},$
$\displaystyle\phi(t)=E_{\alpha}\left(-\left[\frac{t}{\tau_{d}}\right]^{\alpha}\right).$
(23)

The mean and covariance of the process when $x(0)=x_{0}$ is fixed can be
obtained by using general formulas on conditional means and covariances for
Gaussian processes, see e.g. chapter 3 in Ref. Eaton1983 :

$\displaystyle\mathbb{E}(A|Y=y)=\mathbb{E}(A)-\frac{\text{Cov}(A,Y)}{\text{Var}(A)}(\mathbb{E}(Y)-y),$
(24)
$\displaystyle\text{Cov}(A,B|Y=y)=\text{Cov}(A,B)-\frac{\text{Cov}(A,Y)\text{Cov}(B,Y)}{\text{Var}(A)}.$
(25)

These formulas relate conditional averages and covariances to non-conditional
ones, here $\mathbb{E}(A|Y=y)$ is the average of the variable $A$ given that
the variable $Y$ takes the value $y$, and $\text{Cov}(A,B|Y=y)$ is the
covariance of $A,B$ given that $Y=y$. Using these formulas, the average and
the covariance of the process $x(t)$ conditional to $x(0)=x_{0}$ read

$\displaystyle m_{0}(t)\equiv\mathbb{E}(x(t)|x(0)=x_{0})=x_{0}\phi(t),$ (26)
$\displaystyle\sigma(t,t^{\prime})\equiv\text{Cov}(x(t),x(t^{\prime})|x(0)=0)=l^{2}[\phi(|t-t^{\prime}|)-\phi(t)\phi(t^{\prime})].$
(27)

We can also check these expressions by using directly (15).

##  .2 Derivation of the equations of the non-Markovian theory [Eqs.
(7,8,9,10,11)]

Here we derive the equations that will give access to the mean first passage
time (mean FPT) to $x=L$, when the stochastic process starts at $x_{0}$ at
$t=0$. Let us start with a two-point generalized version of the renewal
equation:

$\displaystyle
p(L,t;x_{1},t+t_{1})=\int_{0}^{t}dt^{\prime}F(t^{\prime})p(L,t;x_{1},t+t_{1}|\text{FPT}=t^{\prime}).$
(28)

This exact equation comes from the fact that, if $x$ is observed at position
$L$ at $t$, since the process is non-smooth, it means that $L$ was reached for
the first time at some time $t^{\prime}$, and the above equation is obtained
by partitioning the event of observing $(L,x_{1})$ at times $t,t+t_{1}$ over
the value of the FPT. Here, $p(L,t;x_{1},t+t_{1})$ is the joint probability
density function (pdf) of observing $x=L$ at time $t$ and the position
$x=x_{1}$ at a later time $t+t_{1}$. The fact that the initial position is
fixed is implicitly understood in this notation. Next,
$p(L,t;x_{1},t+t_{1}|\text{FPT}=t^{\prime})$ represents the probability
density of observing $x=L$ at time $t$ and $x=x_{1}$ at a later time $t+t_{1}$
given that the FPT is $t^{\prime}$. Note that, as originally noted in Ref.
Likthman2006 , for non-Markovian processes, it is necessary to keep the
information that the target was reached at $t^{\prime}$ for the first time in
the propagators, this condition is different from the condition that
$x(t^{\prime})=L$ which would hold for Markovian processes.

Now, we introduce the process in the future of the FPT, $x_{\pi}(t)\equiv
x(t+\text{FPT})$ and we denote as $p_{\pi}(y,t)$ its pdf at time $t$ (after
the FPT). By definition,

$\displaystyle p_{\pi}(L,t;x_{1},t+t_{1})=\int_{0}^{\infty}d\tau
F(\tau)p(L,t+\tau;x_{1},t+t_{1}+\tau|\text{FPT}=\tau).$ (29)

We also define the stationary probability density of observing $x=L$ at some
time and $x_{1}$ after a time $t_{1}$ has elapsed:

$\displaystyle
p_{s}(L;x_{1},t_{1})\equiv\lim_{t\to\infty}p(L,t;x_{1},t+t_{1}).$ (30)

We now consider Eq. (28), where we substract $p_{s}(L;x_{1},t_{1})$ on both
sides, leading to

$\displaystyle p(L,t;x_{1},t+t_{1})-p_{s}(L;x_{1},t_{1})=$
$\displaystyle\int_{0}^{t}dt^{\prime}F(t^{\prime})[p(L,t;x_{1},t+t_{1}\text{FPT}=t^{\prime})-p_{s}(L;x_{1},t_{1})]$
$\displaystyle-\int_{t}^{\infty}d\tau F(t^{\prime})p_{s}(L;x_{1},t_{1}),$ (31)

where we have used the fact that $\int_{0}^{\infty}dtF(t)=1$. To proceed
further, we remark that

$\displaystyle\int_{0}^{\infty}dt\int_{t}^{\infty}dt^{\prime}F(t^{\prime})=\int_{0}^{\infty}dt^{\prime}\int_{0}^{t^{\prime}}dtF(t^{\prime})=\int_{0}^{\infty}dt^{\prime}t^{\prime}F(t^{\prime})=\langle
T\rangle.$ (32)

We also note the following equalities:

$\displaystyle\int_{0}^{\infty}dt\int_{0}^{t}$ $\displaystyle
dt^{\prime}F(t^{\prime})[p(L,t;x_{1},t+t_{1}|\text{FPT}=t^{\prime})-p_{s}(L;x_{1},t_{1})],$
$\displaystyle=\int_{0}^{\infty}dt^{\prime}\int_{t^{\prime}}^{\infty}dt\
F(t^{\prime})\
[p(L,t;x_{1},t+t_{1}|\text{FPT}=t^{\prime})-p_{s}(L;x_{1},t_{1})],$ (33)
$\displaystyle=\int_{0}^{\infty}dt^{\prime}\int_{0}^{\infty}du\ F(t^{\prime})\
[p(L,t^{\prime}+u;x_{1},t^{\prime}+t_{1}+u|\text{FPT}=t^{\prime})-p_{s}(L;x_{1},t_{1})],$
(34) $\displaystyle=\int_{0}^{\infty}du\int_{0}^{\infty}dt^{\prime}\
F(t^{\prime})\
[p(L,t^{\prime}+u;x_{1},t^{\prime}+t_{1}+u|\text{FPT}=t^{\prime})-p_{s}(L;x_{1},t_{1})],$
(35) $\displaystyle=\int_{0}^{\infty}du\
[p_{\pi}(L,u;x_{1},u+t_{1})-p_{s}(L;x_{1},t_{1})],$ (36)

where the successive calculation steps are: (i) the inversion of the order of
integration for the variables $(t,t^{\prime})$ in Eq. (33), (ii) the change of
variable $t=u+t^{\prime}$ in Eq. (34), (iii) again a change in the order of
integration between the variables $u,t^{\prime}$ in (35), and (iv) finally the
use of the definition (29) to simplify the integral. Next, using Eqs. (32) and
(36), we see that that integrating Eq. (31) over $t$ leads to

$\displaystyle\int_{0}^{\infty}dt\
[p_{\pi}(L,t;x_{1},t+t_{1})-p(L,t;x_{1},t+t_{1})]=\langle T\rangle
p_{s}(L;x_{1},t+t_{1}).$ (37)

This equation is general and exact, as soon as $p_{s}$ exists, for any
continuous non-smooth stochastic process (even non-Gaussian). Integrating over
$x_{1}$ leads to a general expression for the mean FPT:

$\displaystyle\langle T\rangle p_{s}(L)=\int_{0}^{\infty}dt\
[p_{\pi}(L,t)-p(L,t)].$ (38)

Next, we write
$p_{\pi}(L,t;x_{1},t+t_{1})=p_{\pi}(L,t)p_{\pi}(x_{1},t+t_{1}|L,t)$ (this is
Bayes’ formula). Using this, multiplying Eq. (37) by $x_{1}$ and integrating
over $x_{1}$ yields

$\displaystyle\int_{0}^{\infty}dt[p_{\pi}(L,t)m_{\pi}^{*}(t+t_{1}|L,t)-p(L,t)m_{0}^{*}(t+t_{1}|L,t)]=\langle
T\rangle p_{s}(L)m_{s}^{*}(t_{1}|L,0),$ (39)

where $m_{\pi}^{*}(t+t_{1}|L,t)$ is the conditional average of
$x_{\pi}(t+t_{1})$ given that $x_{\pi}(t)=L$, and (similarly)
$m_{0}^{*}(t+t_{1}|L,t)$ is the conditional average of $x(t+t_{1})$ given that
$x(t)=L$. Finally, $m_{s}^{*}(t_{1})$ is the average of $x(t_{1})$ given that
the system is equilibrated at $t=0$, with the condition $x(0)=L$. Combining
Eqs. (38) and (39), we obtain

$\displaystyle\int_{0}^{\infty}dt\\{p_{\pi}(L,t)[m_{\pi}^{*}(t+t_{1}|L,t)-m_{s}^{*}(t_{1})]-p(L,t)[m_{0}(t+t_{1}|L,t)-m_{s}^{*}(t_{1})]\\}=0.$
(40)

To proceed further, we assume that, in the future of the FPT, the process
$x_{\pi}(t)$ is Gaussian, with a mean $m_{\pi}(t)$ and a covariance
$\sigma_{\pi}(t,t^{\prime})\simeq\sigma(t,t^{\prime})$ that is approximated by
the stationary covariance conditioned to $x=0$ at $t=0$. The next step
consists in using the above equations as closure relations to determine the
mean FPT.

We now write explicit expressions for $m_{\pi}^{*},m_{0}^{*},m_{s}^{*}$. Using
the general formula (24) for conditional averages, where we use
$A=x_{\pi}(t)$, $Y=x_{\pi}(t+t_{1})$ and $y=L$, we obtain

$\displaystyle
m_{\pi}^{*}(t+t_{1}|L,t)=m_{\pi}(t+t_{1})-\frac{\sigma(t+t_{1},t)}{\psi(t)}[m_{\pi}(t)-L],$
(41)

where $\psi(t)=\sigma(t,t)=l^{2}[1-\phi(t)^{2}]$ is the mean square
displacement of the process $x(t)$ conditioned to $x(0)=0$. Similarly,
applying again Eq. (24) for $A=x(t)$, $Y=x(t+t_{1})$ and $y=L$, we obtain

$\displaystyle
m_{0}^{*}(t+t_{1}|L,t)=m_{0}(t+t_{1})-\frac{\sigma(t+t_{1},t)}{\psi(t)}[m_{0}(t)-L].$
(42)

Taking the limit $t\to\infty$ in the above formula enables us to identify
$m_{s}^{*}$:

$\displaystyle m_{s}^{*}(t_{1}|L,0)=L\phi(t_{1}).$ (43)

We also note that, for Gaussian propagators,

$\displaystyle
p_{\pi}(L,t)=\frac{e^{-[L-m_{\pi}(t)]^{2}/2\psi(t)}}{\sqrt{2\pi\psi(t)}},$
$\displaystyle
p(L,t)=\frac{e^{-[L-m_{0}(t)]^{2}/2\psi(t)}}{\sqrt{2\pi\psi(t)}}.$ (44)

Collecting these results, the closure equation (40) for $m_{\pi}(t)$ becomes

$\displaystyle\mathcal{H}(\tau)\equiv$
$\displaystyle\int_{0}^{\infty}dt\Bigg{\\{}\frac{e^{-[L-m_{\pi}(t)]^{2}/2\psi(t)}}{[\psi(t)]^{1/2}}\left[m_{\pi}(t+\tau)-[m_{\pi}(t)-L]\frac{\phi(\tau)-\phi(t)\phi(t+\tau)}{1-\phi^{2}(t)}-L\phi(\tau)\right]$
$\displaystyle-\frac{e^{-[L-x_{0}\phi(t)]^{2}/2\psi(t)}}{[\psi(t)]^{1/2}}\left[x_{0}\phi(t+\tau)-[x_{0}\phi(t)-L]\frac{\phi(\tau)-\phi(t)\phi(t+\tau)}{1-\phi^{2}(t)}-L\phi(\tau)\right]\Bigg{\\}}=0,$
(45)

and the expression (38) for the mean FPT becomes

$\displaystyle\langle T\rangle
p_{s}(L)=\int_{0}^{\infty}dt\left\\{\frac{e^{-[L-m_{\pi}(t)]^{2}/2\psi(t)}}{[2\pi\psi(t)]^{1/2}}-\frac{e^{-(L-x_{0}\phi(t))^{2}/2\psi(t)}}{[2\pi\psi(t)]^{1/2}}\right\\}.$
(46)

### Behavior of $m_{\pi}(t)$ at large times and consequence for the mean FPT

We note that, for large times, $\phi(t)$ becomes a small quantity for large
times. Then we see that the second line of the integrande in Eq. (45) behaves
as

$\displaystyle\frac{e^{-(L-x_{0}\phi(t))^{2}/2\psi(t)}}{[2\pi\psi(t)]^{1/2}}\left[x_{0}\phi(t+\tau)-[x_{0}\phi(t)-L]\frac{\phi(\tau)-\phi(t)\phi(t+\tau)}{1-\phi^{2}(t)}-L\phi(\tau)\right]\underset{t\to\infty}{\simeq}x_{0}(1-\phi(\tau))\phi(t).$
(47)

Since $\phi(t)\simeq A/t^{2H}$ and $H<1/2$, we see that these terms have to be
compensated so that the integral (45) exists; this implies that

$\displaystyle m_{\pi}(t)\underset{t\to\infty}{\simeq}x_{0}\ \phi(t),$ (48)

and this equality should hold at all orders of $t^{-a}$ with $a<1$. If the
behavior (48) holds then the mean FPT predicted by Eq. (46) is finite.

##  .3 Asymptotic analysis in the rare event limit, $L\to\infty$

Here, we analyze the structure of the solution $m_{\pi}(t,L)$ in the limit
$L\to\infty$. As mentioned in the main text, a natural length scale for the
dynamics near the top of the potential is $l^{*}=k_{B}\mathcal{T}/F$, where
$F=kL$ is the slope of the potential. Hence $l^{*}=l^{2}/L$. The associated
time scale $t^{*}$ is the time at which $\psi(t^{*})$ is of order $l^{*}$,
this leads to $t^{*}=(l^{*}/\sqrt{\kappa})^{1/H}$. This suggests the ansatz

$\displaystyle m_{\pi}(t,L)\simeq L-\frac{l^{2}}{L}f(t/t^{*}),\hskip
28.45274ptt^{*}=\left(\frac{l^{2}}{L\sqrt{\kappa}}\right)^{1/H}.$ (49)

Note that $t^{*}\to 0$ when $L\to\infty$. Here $f$ is a scaling function that
is determined by requiring that $\mathcal{H}(\tau=t^{*}v)$, where
$\mathcal{H}$ is defined in Eq. (45), vanishes in the limit $L\to\infty$ (at
fixed $v$):

$\displaystyle\mathcal{H}(t^{*}v)\underset{L\to\infty}{\simeq}\frac{l^{2}(t^{*})^{1-H}}{L\sqrt{\kappa}}\int_{0}^{\infty}\frac{du}{u^{H}}e^{-\frac{f^{2}(u)}{2u^{2H}}}\left[-f(u+v)+f(u)\frac{u^{2H}+(u+v)^{2H}-v^{2H}}{2u^{2H}}+\frac{v^{2H}}{2}\right]=0,$
(50)

where we have used $\phi(\tau)\simeq 1-\tau^{2H}\kappa/(2l^{2})$ for small
$\tau$ (so that $\psi(\tau)\simeq\kappa\tau^{2H}$). Solving this equation
yields the scaling function $f$. Next, we investigate the behavior of
$m_{\pi}(t)$ at time scales larger than $t^{*}$. It is natural to assume that
$m_{\pi}(t)$ admits a regime that varies at the same time scale $\tau_{d}$ as
the original dynamics for $x(t)$, which leads us to the ansatz

$\displaystyle
m_{\pi}(t,L)\simeq\begin{cases}L-\frac{l^{2}}{L}f(t/t^{*})&t=\mathcal{O}(t^{*}),\
(t\ll\tau_{d}),\\\ L\ \phi_{\pi}(t)&t=\mathcal{O}(\tau_{d}),\ (t\gg
t^{*}),\end{cases}$ (51)

where $\phi_{\pi}$ is a scaling function that is independent of $L$. The
linear term in $L$ in factor of $\phi_{\pi}$ is justified by the fact that the
matching with the solution at scale $t^{*}$ can be achieved with the
conditions

$\displaystyle\phi_{\pi}(t)\underset{t\to 0}{\simeq}1-c\ \frac{\kappa\
t^{2H}}{l^{2}},$ $\displaystyle f(u\to\infty)\simeq c\ u^{2H},$ (52)

where $c$ is a numerical constant. The equation for $\phi_{\pi}$ is obtained
by looking at the behavior of $\mathcal{H}(\tau)$ for $L\to\infty$ at fixed
$\tau$, the integrals can in fact be evaluated at times $t^{*}$ (all other
terms are exponentially small) so that we obtain

$\displaystyle\mathcal{H}(\tau)\underset{L\to\infty}{\simeq}\frac{(t^{*})^{1-H}}{\sqrt{\kappa}}\int_{0}^{\infty}du\frac{e^{-f^{2}(u)/2u^{2H}}}{u^{H}}\left[L\
\phi_{\pi}(\tau)-L\ \phi(\tau)\right].$ (53)

Since $\mathcal{H}(\tau)$ has to vanish for all $\tau$ we conclude that
$\phi_{\pi}=\phi$: thus at this time scale $\tau_{d}$ the average trajectory
in the future of the FPT is, at leading order, the same as the trajectory
constrained to $x(0)=L$ starting from an equilibrium configuration. However,
it is obvious that this behavior (51) cannot hold at very long times since at
this stage it is not possible to connect the long-time behavior of
$m_{\pi}=L\phi(t)$ to the already identified behavior given by Eq. (48), where
$m_{\pi}(t)\simeq x_{0}\phi(t)$. Hence, we have to postulate the existence of
at least one additional longer time scales. Let us define now $T_{\text{RE}}$
as

$\displaystyle
T_{\text{RE}}=e^{L^{2}/(2l^{2})}\frac{(t^{*})^{1-H}l}{\kappa^{1/2}}\nu_{H},$
$\displaystyle\ \nu_{H}=\int_{0}^{\infty}du\
\frac{e^{-f^{2}(u)/2u^{2H}}}{u^{H}}.$ (54)

It turns out that $T_{\text{RE}}$ will be the value of the mean FPT at leading
order when $L\to\infty$, but since this is not obvious for our long-term
memory process we use the above equation as a definition for $T_{\text{RE}}$.
Note that $L^{2}/(2l^{2})=E/k_{B}\mathcal{T}$ is the value of the energy
barrier to be crossed to reach the target point. Anticipating the final
result, we postulate that $T_{\text{RE}}$ is also a characteristic time scale
for $m_{\pi}$. Considering this third time scale, the behavior of $m_{\pi}$
reads

$\displaystyle
m_{\pi}(t,L)\simeq\begin{cases}L-\frac{l^{2}}{L}f(t/t^{*})&t=\mathcal{O}(t^{*}),\
(t\ll\tau_{d}),\\\ L\ \phi(t)&t=\mathcal{O}(\tau_{d}),\ (t^{*}\ll t\ll
T_{\text{RE}}),\\\
\frac{LA}{T_{\text{RE}}^{2H}}\chi\left(\frac{t}{T_{\text{RE}}}\right)&t=\mathcal{O}(T_{\text{RE}}),\
(\tau_{d}\ll t)\end{cases}$ (55)

where $\chi$ is a scaling function. The term $LA/T_{\text{RE}}^{2H}$ in factor
of $\chi$ is justified by the fact that the solutions at scales $\tau_{d}$ and
$T_{\text{RE}}$ are matched (i.e. predict the same value for $m_{\pi}$) at the
condition

$\displaystyle\chi(u)\underset{u\to 0}{\simeq}1/u^{2H}.$ (56)

We find the equation for $\chi$ by calculating
$\mathcal{H}(\tau=\overline{\tau}T_{\text{RE}})$ when $L\to\infty$ at fixed
$\overline{\tau}$. The key remark is that since $T_{\text{RE}}$ is
exponentially large with $L$, the integral (45) has two contributions: a first
one coming from $\tau$ of oder $t^{*}$ and a second one coming from
$\tau=O(T_{\text{RE}})$. We note that

$\displaystyle\frac{\phi(\tau)-\phi(t+\tau)\phi(t)}{1-\phi^{2}(t)}\underset{t\ll\tau_{d}\ll\tau}{\simeq}\frac{A}{2\tau^{2H}},$
(57)

so that, with $\tau=T_{\text{RE}}\overline{\tau}$ and $t=ut^{*}$, we have

$\displaystyle
m_{\pi}(t+\tau)=m_{\pi}(ut^{*}+\overline{\tau}T_{\text{RE}})\simeq\frac{LA}{T_{\text{RE}}^{2H}}\chi(\overline{\tau}),$
(58)
$\displaystyle(m_{\pi}(t)-L)\frac{\phi(\tau)-\phi(t+\tau)\phi(t)}{1-\phi^{2}(t)}=-\frac{l^{2}\
f(u)}{L\ T_{\text{RE}}^{2H}}\frac{A}{2\overline{\tau}^{2H}}\ll
m_{\pi}(t+\tau).$ (59)

Following these considerations, we evaluate

$\displaystyle\mathcal{H}(\tau=T_{\text{RE}}\overline{\tau})\underset{L\to\infty}{\simeq}\frac{(t^{*})^{1-H}}{\sqrt{\kappa}}\frac{LA}{T_{\text{RE}}^{2H}}\int_{0}^{\infty}du\frac{e^{-f^{2}(u)/2u^{2H}}}{u^{H}}\left[\chi(\overline{\tau})-\frac{1}{\overline{\tau}^{2H}}\right]$
$\displaystyle+\frac{LA\ T_{\text{RE}}}{l\
T_{\text{RE}}^{2H}}\int_{0}^{\infty}d\overline{t}\
e^{-\frac{L^{2}}{2l^{2}}}\left[\chi(\overline{t}+\overline{\tau})-\frac{\tilde{x}_{0}}{(\overline{t}+\overline{\tau})^{2H}}\right],$
(60)

where $\tilde{x}_{0}=x_{0}/L$ and one keeps $\tilde{x}_{0}$ constant when
taking the limit $L\to\infty$. Equating this expression to zero and using the
definition of $T_{\text{RE}}$ in Eq. (54) we thus obtain

$\displaystyle\chi(\overline{\tau})-\frac{1}{\overline{\tau}^{2H}}+\int_{0}^{\infty}d\overline{t}\
\left[\chi(\overline{t}+\overline{\tau})-\frac{\tilde{x}_{0}}{(\overline{t}+\overline{\tau})^{2H}}\right]=0.$
(61)

This equation can be solved by setting
$G(\overline{\tau})=\chi(\overline{\tau})-\tilde{x}_{0}/\overline{\tau}^{2H}$,
and differentiating with respect to $\overline{\tau}$:

$\displaystyle
G^{\prime}(\overline{\tau})+(1-\tilde{x}_{0})\frac{2H}{\overline{\tau}^{2H+1}}-G(\overline{\tau})=0,$
(62)

where one has assumed that $G(\infty)=0$. The only solution that does not
diverge exponentially for large arguments is

$\displaystyle G(\overline{\tau})=(1-\tilde{x}_{0})2H\
\Gamma(-2H,\overline{\tau})e^{\overline{\tau}},$ $\displaystyle\
\chi(\overline{\tau})=(1-\tilde{x}_{0})2H\
\Gamma(-2H,\overline{\tau})e^{\overline{\tau}}+\frac{\tilde{x}_{0}}{\overline{\tau}^{2H}}.$
(63)

where $\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}dt$ is the upper incomplete
gamma function. We note that, the above expression satisfies the matching
condition Eq. (56), suggesting that our analysis is consistent. We also note
that, when $t\to\infty$, the predictions of Eqs. (55) and (63) coincide with
the behavior (48). This means that the complete structure of $m_{\pi}(t)$ has
been determined, at all time scales.

To evaluate the mean FPT, we introduce two intermediate time scales
$\varepsilon$, $\lambda$ that satisfy

$\displaystyle t^{*}\ll\varepsilon\ll\tau_{d}\ll\lambda\ll T_{\text{RE}}.$
(64)

The mean FPT is evaluated by splitting the integral (46) over the three
intervals $]0,\varepsilon[$, $]\varepsilon,\lambda[$ and $]\lambda,\infty[$,
and by using the appropriate form of $m_{\pi}$ in Eq. (55) for each interval.
This leads to

$\displaystyle\langle T\rangle
p_{s}(L)=\frac{(t^{*})^{1-H}}{\sqrt{\kappa}}\int_{0}^{\varepsilon/t^{*}}du\frac{e^{-f^{2}(u)/2u^{2H}}}{[2\pi
u^{2H}]^{1/2}}+\int_{\varepsilon}^{\lambda}dt\left\\{\frac{e^{-[L(1-\phi)]^{2}/2\psi(t)}}{[2\pi\psi(t)]^{1/2}}-\frac{e^{-(L-x_{0}\phi(t))^{2}/2\psi(t)}}{[2\pi\psi(t)]^{1/2}}\right\\}$
$\displaystyle+\frac{ALT_{\text{RE}}^{1-2H}}{l^{3}\sqrt{2\pi}}e^{-L^{2}/2l^{2}}\int_{\lambda/T_{\text{RE}}}^{\infty}du\
L\left[\chi(u)-\frac{\tilde{x}_{0}}{u^{2H}}\right],$ (65)

where for $t>\lambda$ we have used the fact that $m_{\pi}(t)\ll L$ and
$x_{0}\phi\ll L$, and we have set $t=uT_{\text{RE}}$. Replacing $\chi$ by its
value, and taking the limit $\lambda/T_{\text{RE}}\to 0$ and
$\varepsilon/t^{*}\to\infty$, we finally obtain

$\displaystyle\langle T\rangle\simeq
T_{RE}+T_{RE}^{1-2H}\times\frac{AL(L-x_{0})\Gamma(1-2H)}{l^{2}},$ (66)

which is Eq. (15) in the main text.

##  .4 Details on simulations and additional numerical controls

Here, we present additional numerical results supporting our findings. In Fig.
5 we present additional tests of the validity of the Gaussian approximation
and of the stationary covariance approximation. In Fig. 6 we present a test of
the scaling behavior of $m_{\pi}$ for large $L$. Last, we report the used
values of the time step $dt$ for all simulations of this work in table 1.

Figure 5:  (a) Additional check of the stationary covariance hypothesis. Here, $\psi_{\pi}(t)=\text{var}(x_{\pi}(t))$ and one represents $1-\psi_{\pi}(t)/l^{2}$ to determine whether the stationary covariance approximation is valid at long times (where $\psi_{\pi}(t)\to l^{2}$). Symbols are simulation results (parameter values are indicated in legend) and are compared to $\phi^{2}(t)$ obtained in the stationary covariance approximation (dashed and full lines). (b) Check of the Gaussian approximation. Here, one represents the cumulative distribution function (CDF) of the rescaled variable $\tilde{x}_{\pi}(t)=[x_{\pi}(t)-\langle x_{\pi}(t)\rangle]/\psi_{\pi}^{1/2}(t)$. The red line is the CDF of a normalized Gaussian. Other dashed lines represent simulation results, with parameters indicated in the legend. The collapse of the curves suggests that the stochastic process $x_{\pi}(t)$ is well approximated by a Gaussian process.  Figure 6: Additional checks for scaling behavior of $m_{\pi}(t)$ for $H=1/4$. (a) Check of the short time scaling (49) $m_{\pi}(t)=L-l^{*}f(t/t^{*})$ in the limit $L\to\infty$. Here $f$ is calculated by numerically solving (50). Note that the larger discrepancy between $f$ and the data at short times comes from the finiteness of the time step $\Delta t$ compared to $t^{*}$ (since $t^{*}\propto 1/L^{4}$ here). The fact that one needs to generate trajectories that are longer than $\langle T\rangle\propto e^{L^{2}/2l^{2}}$ prevents us from using smaller time steps for large $L$. (b) Check of the long time scaling regime given by Eq. (13) in the main text. Here the initial position is drawn from an equilibrium distribution, corresponding to our predictions for $x_{0}=0$.  $H$ | $L/l$ | $dt/\tau_{d}$ | Figures
---|---|---|---
3/8 | 0 | $5.96\times 10^{-6}$ | Fig 2(a)
3/8 | 1 | $5.96\times 10^{-6}$ | Figs. 2(a), 3(a), 4(b), 4(d), S1(b)
3/8 | 2 | $5.96\times 10^{-6}$ | Figs. 2(a), 3(a), 4(a), 4(c), 4(d), S1(a)
3/8 | 3 | $7.45\times 10^{-6}$ | Figs. 2(a), 3(a), 4(c), 4(d), S1(b)
3/8 | 3 | $1.49\times 10^{-5}$ | Figs. 4(a), S1(a)
3/8 | 4 | $1.86\times 10^{-5}$ | Figs. 3(a), 4(d)
3/8 | 4 | $7.45\times 10^{-6}$ | 4(c)
1/4 | 0 | $1.86\times 10^{-7}$ | Figs. 2(b)
1/4 | 1 | $1.86\times 10^{-7}$ | Figs. 2(b), 3(b), S1
1/4 | 2 | $7.45\times 10^{-7}$ | Figs. 2(b), 3(b), 4(a), 4(b), S1, S2
1/4 | 3 | $3.72\times 10^{-6}$ | Figs. 2(b), 3(b), 4(a), S1, S2
1/4 | 4 | $1.30\times 10^{-5}$ | Figs. 3(b), S2
Table 1: Value of the time steps used in the simulations.

##  .5 Exactness of the theory at first order for weakly non-Markovian
processes

Let us consider the case of weakly non-Markovian processes, for which the
covariance and mean of the process $x(t)$ are given by Eqs. (26) and (27),
with

$\displaystyle\phi(t)=e^{-\lambda t}+\varepsilon\phi_{1}(t),$ (67)

with $\lambda>0$, $\varepsilon$ is a small parameter, and $\phi_{1}(t)$ is an
arbitrary function. For simplicity, and without loss of generality, we set
$\lambda=1$ and $l=1$. We start with the generalization of Eq. (29) for an
arbitrary number of positions and times $x_{i},t_{i}$:

$\displaystyle p_{\pi}(L,t;$ $\displaystyle
x_{1},t+t_{1};x_{2},t+t_{2};...;x_{N},t+t_{N})=$
$\displaystyle\int_{0}^{\infty}d\tau
F(\tau)p(L,t+\tau;x_{1},t+t_{1}+\tau;x_{2},t+t_{2}+\tau;...;x_{N},t+t_{N}+\tau|\text{FPT}=\tau).$
(68)

Following the approach of Section .2, this equation leads to

$\displaystyle\int_{0}^{\infty}dt[p_{\pi}(0,t;x_{1},t+t_{1};x_{2},t+t_{2};...)-p(0,t;x_{1},t+t_{1};x_{2},t+t_{2};...)]=\langle
T\rangle p_{s}(0;x_{1},t_{1};x_{2},t_{2};...).$ (69)

We may write formally a continuous version of this equation, for all paths
$[y(\tau)]$ with $y(0)=L$:

$\langle T\rangle
P_{\text{s}}([y(\tau)])-\int_{0}^{\infty}dt\left\\{\Pi([y(\tau)],t)-P([y(\tau)],t)\right\\}=0,$
(70)

where $P_{\text{s}}([y(\tau)])$ is the stationary probability to follow the
path $[y(\tau)]$, $\Pi([y(\tau)],t)$ is the probability to follow the path
$[y]$ in the future $t$ of the FPT (ie, the probability density that
$x(\text{FPT}+\tau+t)=y(\tau)$ for all $\tau>0$), and $P([y(\tau)],t)$ is the
probability density that $x(t+\tau)=y(\tau)$ for all $\tau>0$. Using Bayes’
formula, we can write (70) as

$\int_{0}^{\infty}dt\left\\{\Pi([y(\tau)],t|y(0)=L)p_{\pi}(L,t)-P([y(\tau)],t|y(0)=L)p(L,t)\right\\}-\langle
T\rangle p_{s}(L)P_{\text{s}}([y(\tau)]|y(0)=L)=0,$ (71)

which is valid for all paths $[y(\tau)]$ (if $y(0)\neq L$ the above equation
is simply $0=0$). Let us define a functional $\mathcal{F}([k])$ as the value
of the above expression when multiplied by $e^{\int_{0}^{\infty}d\tau
k(\tau)y(\tau)}$ and integrated over all paths $y$. In principle
$\mathcal{F}([k])$ should vanish for all functions $k(\tau)$. Let us evaluate
$\mathcal{F}$ for a distribution of paths $\Pi$ that satisfies our hypotheses,
i.e. by assuming that the process in the future of the first passage time is
Gaussian with mean $m_{\pi}(t)$ and with the stationary covariance
approximation. Using formulas for the moment generating function of Gaussian
processes, we find

$\displaystyle\mathcal{F}([k(\tau)])=\langle T\rangle
p_{s}(L)e^{\int_{0}^{\infty}d\tau
k(\tau)m_{s}^{*}(\tau)}e^{\frac{1}{2}\int_{0}^{\infty}d\tau\int_{0}^{\infty}d\tau^{\prime}k(\tau)k(\tau^{\prime})\sigma(\tau,\tau^{\prime})}$
$\displaystyle-\int_{0}^{\infty}dt\left[p_{\pi}(0,t)e^{\int_{0}^{\infty}d\tau
k(\tau)m_{\pi}^{*}(t+\tau|L,t)}-p(0,t)e^{\int_{0}^{\infty}d\tau
k(\tau)m_{0}^{*}(t+\tau|L,t)}\right]e^{\int_{0}^{\infty}d\tau\int_{0}^{\infty}d\tau^{\prime}\frac{k(\tau)k(\tau^{\prime})}{2}\sigma(t+\tau,t+\tau^{\prime}|t)},$
(72)

where we remind that $\langle T\rangle$ is evaluated with Eq. (46), and

$\displaystyle\sigma(t+\tau,t+\tau^{\prime}|t)=\sigma(t+\tau,t+\tau^{\prime})-\frac{\sigma(t+\tau,t)\sigma(t+\tau^{\prime},t)}{\sigma(t,t)}.$
(73)

If one could find a function $m_{\pi}(t)$ so that $\mathcal{F}([k(\tau)])$
vanishes for all $k(\tau)$, it would mean that the theory is exact. It does
not seem to be the case in general. However, when $\varepsilon\to 0$, assuming
that

$\displaystyle m_{\pi}(t)=Le^{-\lambda t}+\varepsilon\mu_{1}(t),$ (74)

we can evaluate (72) as

$\displaystyle\mathcal{F}([k(\tau)])=-\varepsilon\int_{0}^{\infty}d\tau
k(\tau)Q_{1}(\tau)\times
e^{\int_{0}^{\infty}du\int_{0}^{\infty}du^{\prime}k(u)k(u^{\prime})\frac{1}{2}\sigma(u,u^{\prime})}+\mathcal{O}(\varepsilon^{2}),$
(75)

where

$\displaystyle
Q_{1}(\tau)=\int_{0}^{\infty}\frac{dt}{\sqrt{2\pi(1-e^{-2t})}}\Bigg{\\{}$
$\displaystyle
e^{-\frac{[L(1-e^{-t})]^{2}}{2(1-e^{-2t})}}\left[\mu_{1}(t+\tau)-\mu_{1}(t)e^{-\tau}-Le^{-t}S_{1}(t,\tau)-L\phi_{1}(\tau)\right]$
$\displaystyle-e^{-\frac{(L-x_{0}e^{-t})^{2}}{2(1-e^{-2t})}}\left[m_{1}^{*}(t,\tau)-L\phi_{1}(\tau)\right]\Bigg{\\}},$
(76)

where we have defined $S_{1}$ and $m_{1}^{*}$ such that

$\displaystyle\frac{\sigma(t+\tau,t)}{\sigma(t,t)}=e^{-\tau}+\varepsilon
S_{1}(t,\tau)+\mathcal{O}(\varepsilon^{2}),$ $\displaystyle
m_{0}^{*}(t+\tau|L,t)=Le^{-\tau}+\varepsilon
m_{1}^{*}(t,\tau)+\mathcal{O}(\varepsilon^{2}).$ (77)

Note that, to obtain (75), it is important to remark that

$\displaystyle\sigma(t+\tau,t+\tau^{\prime}|t)=\sigma(\tau,\tau^{\prime})+\mathcal{O}(\varepsilon^{2}).$
(78)

We observe that the equality $Q_{1}(\tau)=0$ for all $\tau$ can be realized by
a proper choice of $\mu_{1}$, so that $\mathcal{F}([k(\tau)])$ vanishes at
order $\varepsilon$ for all functions $k(\tau)$. This suggests that our theory
is exact at order $\varepsilon$.

## References

  * (1) Hänggi, P., Talkner, P. & Borkovec, M.  Reaction-rate theory: fifty years after kramers.  _Rev. Mod. Phys._ 62, 251 (1990).
  * (2) Pollak, E. & Talkner, P.  Reaction rate theory: What it was, where is it today, and where is it going?  _Chaos: An Interdisciplinary Journal of Nonlinear Science_ 15, 026116 (2005).
  * (3) Bullerjahn, J. T., Sturm, S. & Kroy, K.  Theory of rapid force spectroscopy.  _Nat. Comm._ 5 (2014).
  * (4) Bullerjahn, J. T., Sturm, S. & Kroy, K.  Non-markov bond model for dynamic force spectroscopy.  _J. Chem. Phys._ 152, 064104 (2020).
  * (5) Bullerjahn, J. & Kroy, K.  Analytical catch-slip bond model for arbitrary forces and loading rates.  _Phys. Rev. E_ 93, 012404 (2016).
  * (6) Jeppesen, C. _et al._ Impact of polymer tether length on multiple ligand-receptor bond formation.  _Science_ 293, 465–468 (2001).
  * (7) Ayaz, C. _et al._ Non-markovian modeling of protein folding.  _Proc. Natl. Acad. Sc. USA_ 118, e2023856118 (2021).
  * (8) Badoual, M., Jülicher, F. & Prost, J.  Bidirectional cooperative motion of molecular motors.  _Proc Natl Acad Sci USA_ 99, 6696–6701 (2002).
  * (9) Guérin, T., Prost, J. & Joanny, J.-F.  Motion reversal of molecular motor assemblies due to weak noise.  _Phys Rev Lett_ 106, 068101 (2011).
  * (10) Guerin, T., Prost, J. & Joanny, J. F.  Bidirectional motion of motor assemblies and the weak-noise escape problem.  _Phys Rev E_ 84, 041901 (2011).
  * (11) Bouchaud, J. P. & Cont, R.  A Langevin approach to stock market fluctuations and crashes.  _Eur Phys J B_ 6, 543–550 (1998).
  * (12) Ragone, F., Wouters, J. & Bouchet, F.  Computation of extreme heat waves in climate models using a large deviation algorithm.  _Proc. Natl. Acad. Sci. USA_ 115, 24–29 (2018).
  * (13) Kamenev, A., Meerson, B. & Shklovskii, B.  How colored environmental noise affects population extinction.  _Phys. Rev. Lett._ 101, 268103 (2008).
  * (14) Dykman, M. I., Schwartz, I. B. & Landsman, A. S.  Disease extinction in the presence of random vaccination.  _Phys. Rev. Lett._ 101, 078101 (2008).
  * (15) Freidlin, M. I. & Wentzell, A. D.  _Random Perturbations of Dynamical Systems_ (Springer-Verlage, New-York, Berlin, 1984).
  * (16) Maier, R. & Stein, D.  Transition-rate theory for nongradient drift fields.  _Phys Rev Lett_ 69, 3691–3695 (1992).
  * (17) Bouchet, F. & Reygner, J.  Generalisation of the Eyring–Kramers transition rate formula to irreversible diffusion processes.  In _Annales Henri Poincaré_ , vol. 17, 3499–3532 (Springer, 2016).
  * (18) de la Cruz, R., Perez-Carrasco, R., Guerrero, P., Alarcon, T. & Page, K. M.  Minimum action path theory reveals the details of stochastic transitions out of oscillatory states.  _Phys. Rev. Lett._ 120, 128102 (2018).
  * (19) Bray, A. J., Majumdar, S. N. & Schehr, G.  Persistence and first-passage properties in nonequilibrium systems.  _Adv. Phys._ 62, 225–361 (2013).
  * (20) Metzler, R., Redner, S. & Oshanin, G.  _First-passage phenomena and their applications_ (World Scientific, 2014).
  * (21) Lindenberg, K., Metzler, R. & Oshanin, G.  _Chemical Kinetics: beyond the textbook_ (World Scientific, 2019).
  * (22) Sokolov, I. M.  Cyclization of a polymer: first-passage problem for a non-markovian process.  _Phys Rev Lett_ 90, 080601 (2003).
  * (23) Likthman, A. E. & Marques, C. M.  First-passage problem for the rouse polymer chain: An exact solution.  _Europhys. Lett._ 75, 971–977 (2006).
  * (24) Guérin, T., Levernier, N., Bénichou, O. & Voituriez, R.  Mean first-passage times of non-markovian random walkers in confinement.  _Nature_ 534, 356–359 (2016).
  * (25) Delorme, M. & Wiese, K. J.  Maximum of a fractional brownian motion: analytic results from perturbation theory.  _Phys. Rev. Lett._ 115, 210601 (2015).
  * (26) Ferrer, B. R., Gomez-Solano, J. R. & Arzola, A. V.  Fluid viscoelasticity triggers fast transitions of a brownian particle in a double well optical potential.  _Phys. Rev. Lett._ 126, 108001 (2021).
  * (27) Ginot, F., Caspers, J., Krüger, M. & Bechinger, C.  Barrier crossing in a viscoelastic bath.  _Phys. Rev. Lett._ 128, 028001 (2022).
  * (28) Lavacchi, L., Kappler, J. & Netz, R. R.  Barrier crossing in the presence of multi-exponential memory functions with unequal friction amplitudes and memory times.  _Europhys. Lett._ 131, 40004 (2020).
  * (29) Lavacchi, L., Daldrop, J. O. & Netz, R. R.  Non-Arrhenius barrier crossing dynamics of non-equilibrium non-Markovian systems.  _Europhys. Lett._ 139, 51001 (2022).
  * (30) Kappler, J., Daldrop, J. O., Brünig, F. N., Boehle, M. D. & Netz, R. R.  Memory-induced acceleration and slowdown of barrier crossing.  _J. Chem. Phys._ 148, 014903 (2018).
  * (31) Caraglio, M., Put, S., Carlon, E. & Vanderzande, C.  The influence of absorbing boundary conditions on the transition path time statistics.  _Phys. Chem. Chem. Phys._ 20, 25676–25682 (2018).
  * (32) Carlon, E., Orland, H., Sakaue, T. & Vanderzande, C.  Effect of memory and active forces on transition path time distributions.  _J. Phys. Chem. B_ (2018).
  * (33) Medina, E., Satija, R. & Makarov, D. E.  Transition path times in non-Markovian activated rate processes.  _J. Phys. Chem. B_ (2018).
  * (34) Goychuk, I. & Hänggi, P.  Anomalous escape governed by thermal 1/f noise.  _Phys. Rev. Lett._ 99, 200601 (2007).
  * (35) Sliusarenko, O. Y., Gonchar, V. Y., Chechkin, A. V., Sokolov, I. M. & Metzler, R.  Kramers-like escape driven by fractional gaussian noise.  _Phys. Rev. E_ 81, 041119 (2010).
  * (36) Arutkin, M., Walter, B. & Wiese, K. J.  Extreme events for fractional brownian motion with drift: Theory and numerical validation.  _Phys. Rev. E_ 102, 022102 (2020).
  * (37) Levernier, N., Bénichou, O., Voituriez, R. & Guérin, T.  Kinetics of rare events for non-markovian stationary processes and application to polymer dynamics.  _Phys. Rev. Res._ 2, 012057 (2020).
  * (38) Delorme, M., Rosso, A. & Wiese, K. J.  Pickands’ constant at first order in an expansion around brownian motion.  _J. Phys. A: Math. Theor._ 50, 16LT04 (2017).
  * (39) Goswami, K. &Metzler, R.,  Effects of active noise on transition-path dynamics.  _Journal of Physics: Complexity_ 4, 025005 (2023).
  * (40) Panja, D.  Anomalous polymer dynamics is non-Markovian: memory effects and the generalized Langevin equation formulation.  _J. Stat. Mech.: Theory Exp._ 2010, P06011 (2010).
  * (41) Kou, S. & Xie, X. S.  Generalized Langevin equation with fractional gaussian noise: subdiffusion within a single protein molecule.  _Phys. Rev. Lett._ 93, 180603 (2004).
  * (42) Min, W., Luo, G., Cherayil, B. J., Kou, S. & Xie, X. S.  Observation of a power-law memory kernel for fluctuations within a single protein molecule.  _Phys. Rev. Lett._ 94, 198302 (2005).
  * (43) Lennartz, S., Livina, V., Bunde, A. & Havlin, S.  Long-term memory in earthquakes and the distribution of interoccurrence times.  _Europhys. Lett._ 81, 69001 (2008).
  * (44) Bunde, A., Büntgen, U., Ludescher, J., Luterbacher, J. & Von Storch, H.  Is there memory in precipitation?  _Nature Climate Change_ 3, 174–175 (2013).
  * (45) Min, W. & Xie, X. S.  Kramers model with a power-law friction kernel: Dispersed kinetics and dynamic disorder of biochemical reactions.  _Phys. Rev. E_ 73, 010902 (2006).
  * (46) Goychuk, I.  Viscoelastic subdiffusion: From anomalous to normal.  _Phys. Rev. E_ 80, 046125 (2009).
  * (47) Eichner, J. F., Kantelhardt, J. W., Bunde, A. & Havlin, S.  Statistics of return intervals in long-term correlated records.  _Phys. Rev. E_ 75, 011128 (2007).
  * (48) Santhanam, M. & Kantz, H.  Return interval distribution of extreme events and long-term memory.  _Phys. Rev. E_ 78, 051113 (2008).
  * (49) Bunde, A., Eichner, J. F., Kantelhardt, J. W. & Havlin, S.  Long-term memory: A natural mechanism for the clustering of extreme events and anomalous residual times in climate records.  _Phys. Rev. Lett._ 94, 048701 (2005).
  * (50) Singh, R.  Comment on “anomalous escape governed by thermal 1/f noise”.  _Phys. Rev. Lett._ 123, 238901 (2019).
  * (51) Bullerjahn, J. T.  _A Unified Theory for Single-molecule Force Spectroscopy Experiments and Simulations_.  Ph.D. thesis, Universität Leipzig (2017).
  * (52) Wilemski, G. & Fixman, M.  Diffusion-controlled intrachain reactions of polymers. 1. theory.  _J. Chem. Phys._ 60, 866–877 (1974).
  * (53) Note that in the non-equilibrium model of Ref. Sliusarenko2010 , the correlation function decays as $\langle x(t)x(0)\rangle\sim 1/t^{\alpha}$ with $\alpha=2-2H$, where $H$ is defined so that $\langle x^{2}\rangle\propto t^{2H}$ in absence of potential. As a consequence, the long-term memory property $\alpha<1$ corresponds to $H>1/2$, with an infinite mean FPT predicted by the Wilemski-Fixman approximation.
  * (54) Newell, G. F. & Rosenblatt, M.  Zero crossing probabilities for Gaussian stationary processes.  _The Annals of Mathematical Statistics_ 33, 1306–1313 (1962).
  * (55) Pickands, J.  Upcrossing probabilities for stationary gaussian processes.  _Transactions of the American Mathematical Society_ 145, 51–73 (1969).
  * (56) Pickands, J.  Asymptotic properties of the maximum in a stationary gaussian process.  _Transactions of the American Mathematical Society_ 145, 75–86 (1969).
  * (57) Mason, T. G. & Weitz, D.  Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids.  _Phys. Rev. Lett._ 74, 1250 (1995).
  * (58) Gisler, T. & Weitz, D. A.  Tracer microrheology in complex fluids.  _Current opinion in colloid & interface science_ 3, 586–592 (1998).
  * (59) Mason, T., Ganesan, K., Van Zanten, J., Wirtz, D. & Kuo, S.  Particle tracking microrheology of complex fluids.  _Phys. Rev. Lett._ 79, 3282 (1997).
  * (60) Panja, D.  Generalized Langevin equation formulation for anomalous polymer dynamics.  _J. Stat. Mech. - Theor. Exp._ (2010).
  * (61) Bullerjahn, J. T., Sturm, S., Wolff, L. & Kroy, K.  Monomer dynamics of a wormlike chain.  _Europhys. Lett._ 96, 48005 (2011).
  * (62) Davies, R. B. & Harte, D.  Tests for hurst effect.  _Biometrika_ 74, 95–101 (1987).
  * (63) Dietrich, C. R. & Newsam, G. N.  Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix.  _SIAM J. Sci. Comp._ 18, 1088–1107 (1997).
  * (64) Levernier, N., Dolgushev, M., Bénichou, O., Voituriez, R. & Guérin, T.  Survival probability of stochastic processes beyond persistence exponents.  _Nat. Comm._ 10, 1–7 (2019).
  * (65) Krug, J. _et al._ Persistence exponents for fluctuating interfaces.  _Phys. Rev. E_ 56, 2702–2712 (1997).
  * (66) Molchan, G.  Maximum of a fractional Brownian motion: Probabilities of small values.  _Commun. Math. Phys._ 205, 97–111 (1999).
  * (67) Eaton, M. L.  _Multivariate Statistics, A Vector Space Approach_ , vol. 53 (Institute of Mathematical Statistics Beachwood, Ohio, USA, 1983).