# Survival probability of stochastic processes beyond persistence exponents

N. Levernier  NCCR Chemical Biology, Departments of Biochemistry and
Theoretical Physics, University of Geneva, Geneva, Switzerland  M. Dolgushev
Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne
Université, 4 Place Jussieu, 75005 Paris, France  O. Bénichou  Laboratoire de
Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place
Jussieu, 75005 Paris, France  R. Voituriez  Laboratoire de Physique Théorique
de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005
Paris, France  Laboratoire Jean Perrin, CNRS/Sorbonne Université, 4 Place
Jussieu, 75005 Paris, France  T. Guérin  Laboratoire Ondes et Matière
d’Aquitaine, University of Bordeaux, Unité Mixte de Recherche 5798, CNRS,
F-33400 Talence, France

###### Abstract

For many stochastic processes, the probability $S(t)$ of not-having reached a
target in unbounded space up to time $t$ follows a slow algebraic decay at
long times, $S(t)\sim S_{0}/t^{\theta}$. This is typically the case of
symmetric compact (i.e. recurrent) random walks. While the persistence
exponent $\theta$ has been studied at length, the prefactor $S_{0}$, which is
quantitatively essential, remains poorly characterized, especially for non-
Markovian processes. Here we derive explicit expressions for $S_{0}$ for a
compact random walk in unbounded space by establishing an analytic relation
with the mean first-passage time of the same random walk in a large confining
volume. Our analytical results for $S_{0}$ are in good agreement with
numerical simulations, even for strongly correlated processes such as
Fractional Brownian Motion, and thus provide a refined understanding of the
statistics of longest first-passage events in unbounded space.

Introduction

In order to determine the time it takes for a random walker to find a target,
or the probability that a stochastic signal has not reached a threshold up to
time $t$, it is required to analyse the First-Passage Time (FPT) statistics.
This has attracted considerable attention from physicists and mathematicians
in the last decades Redner:2001a ; Condamin2007 ; pal2017first ;
grebenkov2016universal ; benichou2010optimal ; vaccario2015first  notably
because of the relevance of FPT related quantities in contexts as varied as
diffusion controlled reactions, finance, search processes, or biophysics
metzler2014first ; Berg1985 .

A single-target first-passage problem is entirely characterized by the so-
called “survival probability” $S(t)$ (the probability that the target has not
been reached up to time $t$), or equivalently by the FPT distribution
$F(t)=-\partial_{t}S(t)$. For a symmetric random walk in a confined domain,
the mean FPT is in general finite and has been studied at length. This led
recently to explicit results for broad classes of stochastic processes
Condamin2005 ; Condamin2007 ; Benichou2008 ; Schuss2007 ; Guerin2012a . The
opposite case of unconfined random walks is drastically different. In this
case, either the walker has a finite probability of never finding the target
(non-compact random walks), or it reaches it with probability one (compact
random walk) and the survival probability decays algebraically with time,
$S(t)\sim S_{0}/t^{\theta}$, with $\theta$ the persistence exponent that does
not depend on the initial distance to the target. In this case the mean FPT is
often infinite so that the relevant observable to quantify FPT statistics is
the long time algebraic decay of the probability $S(t)$ that the target has
not been reached up to $t$. This, additional to the fact that $\theta$ can be
non-trivial for non-Markovian random walks, has triggered a considerable
amount of work to characterize the persistence exponent $\theta$ in a wide
number of models of non-equilibrium statistical mechanics. Indeed, $S(t)$ is
an essential observable to quantify the kinetics of transport controlled
reactions and the dynamics of coarsening in phase transitions in general
Majumdar1999 ; ReviewBray .

However, if one aims to evaluate the time $t$ to wait for observing a first-
passage event with a given likelihood, or to determine the dependence of the
survival probability on the initial distance to the target, one needs to know
the prefactor $S_{0}$, which turns out to be much less characterized than the
persistence exponent $\theta$. Even for Markovian random walks this problem is
not trivial Aurzada2015 , as exemplified by recent studies for one dimensional
Levy flights Majumdar2017 , while only scaling relations for $S_{0}$ (with the
initial distance to the target) are known meroz2011distribution  in fractal
domains. However, if the dynamics of the random walker results from
interactions with other degrees of freedom, the process becomes non-Markovian
and the determination of $S_{0}$ becomes much more involved VanKampen1992 . In
this case, the only explicit results are derived from perturbation expansion
around Markovian processes delorme2015maximum ; delorme2016perturbative , or
have been obtained for particular processes such as “run and tumble” motion
(driven by telegraphic noise Masoliver1986 ) or the random acceleration
process Burkhardt1993 . For long range correlated processes, such as
fractional Brownian Motion, the existence of $S_{0}$ is not even established
rigourously Aurzada2011 ; Aurzada2015 , and it has been found that
straightforward adaptation of Markovian methods can lead to order-of-magnitude
overestimations of $S_{0}$ and even to erroneous scalings Sanders2012 .

In this article, we rely on a non-perturbative strategy to determine $S_{0}$,
which is of crucial interest to quantify the statistics of long FPT events.
Our main result is a relation between the prefactor $S_{0}$ in the long time
survival probability in free space and the mean FPT for the same process in a
large confining volume. Our formula thus shows how to make use of the wealth
of explicit results obtained recently on first-passage properties in
confinement Condamin2005 ; Condamin2007 ; Benichou2008 ; Schuss2007 ;
Guerin2012a ; guerin2016mean  to determine the decay of the free-space
survival probability. This formula is shown to be robust and holds for
Markovian or non-Markovian processes with stationary increments, that are
scale invariant at long times with diverging moments of the position, in one
or higher spatial dimensions, and also for processes displaying transient
aging (i. e. processes with finite memory time, whose initial state is not
stationary, see below). This theory is confirmed by simulations for a variety
of stochastic processes, including highly correlated ones such as Fractional
Brownian Motion.

Results

Markovian case. We consider a symmetric random walker of position
$\mathbf{r}(t)$ moving on an infinite discrete lattice (potentially fractal)
of dimension $d_{f}$ [see Fig. 1(a) for the continuous space counterpart] in
continuous time $t$, in absence of external field. The initial position is
$\mathbf{r}_{0}$. We assume that the increments are stationary (no aging),
which means in particular that
$\sigma(t,\tau)\equiv\langle|\mathbf{r}(t+\tau)-\mathbf{r}(t)|^{2}\rangle$ is
independent of the elapsed time $t$. Note that in the case of fractal spaces,
we use the standard “chemical” distance defined as the minimal number of steps
to link two points on the lattice. We define the walk-dimension $d_{w}$ such
that $\sigma(t,\tau)\propto\tau^{2/d_{w}}$ for $\tau\to\infty$. Note that (i)
this scale invariance is assumed only at long times, and that (ii) it implies
that all even moments of the position diverge with time. We assume
$d_{w}>d_{f}$ so that the process is compact DEGENNES1982 ; benAvraham2000
(and eventually reaches any point with probability one). We also introduce the
Hurst exponent $H=1/d_{w}$.

We first consider the case of Markovian (memoryless) random walks. One can
then define a propagator $p(\mathbf{r},t|\mathbf{r}_{0})$, which represents
the probability to observe the walker at site $\mathbf{r}$ at time $t$ given
that it started at $\mathbf{r}_{0}$ at initial time. Note that $p$ is defined
in absence of target. We now add an absorbing target at site $\mathbf{r}=0$
(different from $\mathbf{r}_{0}$). We start our analysis with the standard
renewal equation VanKampen1992 ; Redner:2001a ; hughes1995random :

$p(\mathbf{0},t|\mathbf{r}_{0})=\int_{0}^{t}d\tau
F(\tau;\mathbf{r}_{0})p(\mathbf{0},t-\tau|\mathbf{0}),$ (1)

which relates the propagator $p$ to the FPT distribution $F$ that depends on
$\mathbf{r}_{0}$. This equation is obtained by partitioning over the FPT to
the target, and can be rewritten in Laplace space as

$\tilde{p}(\mathbf{0},s|\mathbf{r}_{0})=\tilde{F}(s;\mathbf{r}_{0})\tilde{p}(\mathbf{0},s|\mathbf{0}),$
(2)

where $\tilde{F}(s)=\int_{0}^{\infty}dtF(t)e^{-st}$ stands for the Laplace
transform of $F(t)$. Here, we only focus on the long-time behavior of $F(t)$,
that can be obtained by expanding Eq. (2) for small $s$. Scale invariance at
long times implies benAvraham2000  that for any site $\mathbf{r}$

$\displaystyle
p(\mathbf{0},t|\mathbf{r})\underset{t\to\infty}{\sim}K/t^{d_{f}/d_{w}},$ (3)

where the notation “$\sim$” represents asymptotic equivalence, and $K$ is a
positive coefficient. Note that $K$ is known to be position independent and is
well characterized (at least numerically) for a large class of stochastic
processes, including diffusion in a wide class of fractals
grabner1997functional ; kron2004asymptotics ; weber2010random ;
meroz2011distribution . We find that the small-$s$ behavior of the propagator
is

$\displaystyle\frac{K\,\Gamma(1-\frac{d_{f}}{d_{w}})}{s^{1-d_{f}/d_{w}}}-\tilde{p}(\mathbf{0},s|\mathbf{r})\underset{s\to
0}{\sim}\int_{0}^{\infty}dt\left[\frac{K}{t^{d_{f}/d_{w}}}-p(\mathbf{0},t|\mathbf{r})\right],$
(4)

where $\Gamma(\cdot)$ is the Gamma function. Eqs. (2) and (4) (written for
$\mathbf{r}=\mathbf{0}$ and $\mathbf{r}=\mathbf{r}_{0}$) lead to

$1-\tilde{F}(s;\mathbf{r}_{0})\underset{s\to
0}{\sim}\int_{0}^{\infty}dt\left[p(\mathbf{0},t|\mathbf{0})-p(\mathbf{0},t|\mathbf{r}_{0})\right]\frac{s^{1-d_{f}/d_{w}}}{K\,\Gamma\left(1-\frac{d_{f}}{d_{w}}\right)}.$
(5)

Taking the inverse Laplace transform (and using $F(t)=-\dot{S}$) leads to
$S(t)\sim S_{0}/t^{\theta}$ with $\theta=1-d_{f}/d_{w}$ (as found in
meroz2011distribution ), and to

$\displaystyle S_{0}=\frac{\sin(\pi
d_{f}/d_{w})}{K\pi}\int_{0}^{\infty}dt\left[p(\mathbf{0},t|\mathbf{0})-p(\mathbf{0},t|\mathbf{r}_{0})\right].$
(6)

This expression is exact and characterizes the decay of the survival
probability of unconfined scale invariant Markovian random walks.

We now consider the target search problem for the same random walk, with the
only difference that it takes place in a confining volume $V$ (that is equal
to the number of sites $N$ in our discrete formulation) [see Fig. 1(b)]. For
this problem, the mean FPT $\langle\mathbf{T}\rangle$ is in general finite and
it is known that it scales linearly with the volume and reads Condamin2005 ;
Condamin2007

$\frac{\langle\mathbf{T}\rangle}{V}\underset{V\to\infty}{\sim}\int_{0}^{\infty}dt\left[p(\mathbf{0},t|\mathbf{0})-p(\mathbf{0},t|\mathbf{r}_{0})\right].$
(7)

We recognize in the above expression the time integral of propagators
appearing in Eq. (6), leading to

$S_{0}=\frac{\sin(\pi d_{f}/d_{w})}{\pi\ K}\overline{T},\hskip
14.22636pt\text{with}\hskip
14.22636pt\overline{T}=\underset{V\to\infty}{\lim}\langle\mathbf{T}\rangle/V.$
(8)

Hence, for compact Markovian random walks, we have identified a
proportionality relation between the prefactor $S_{0}$ that characterizes the
long time survival probability in free space and the rescaled mean FPT to the
target in unconfined space. The proportionality coefficient involves the walk-
dimension $d_{w}$ and the coefficient $K$ which characterizes the long time
decay of the propagator [see Eq. (3)]. Formula (8) is the key result of this
paper. As we proceed to show, it is very robust and is not limited to
Markovian walks.

As a first application, consider the case of scale invariant Markovian random
walks (such as diffusion on fractals), for which it was shown benichou2008zero
that $\overline{T}\simeq r_{0}^{d_{w}-d_{f}}$, where the mean waiting time on
a given site is taken as unity, and $r_{0}$ is the initial source-target
(chemical) distance. Inserting this formula into Eq. (8) thus leads to

$\displaystyle S_{0}\simeq\frac{\sin(\pi d_{f}/d_{w})r_{0}^{d_{w}-d_{f}}}{\pi\
K}.$ (9)

In this case, we thus recover the scaling result of Ref. meroz2011distribution
but in addition obtain the value of the prefactor. We have checked this
relation for the Sierpinski gasket: simulation results are shown on Fig. 2(a).
The long time persistence is perfectly described by our formula without any
fitting parameter for different source-target distances, confirming the
validity of our approach (see SI for other examples).

As a second application, we can consider the 1-dimensional Lévy stable process
of index $\alpha$, which is defined as the only Markovian process whose jump
distribution is given by $p(\Delta
x,t)=1/(2\pi)\int_{-\infty}^{\infty}e^{i\Delta x.k-t|k|^{\alpha}}dk$. This
process, defined in continuous space, is the continuous time limit of the Lévy
Flight with same index $\alpha$. Its walk dimension is $d_{w}=\alpha$ and it
is compact for $\alpha>1$, so that the first passage to a point target is well
defined (note that we consider here the first arrival at the target, and not
the first crossing event al:2003 ; Tejedor2011 ). For such a process, the
prefactor $S_{0}$ for an unconfined random walk starting at a distance $r_{0}$
from the target has been shown to be
$S_{0}=\alpha\sin(\pi\alpha/2)\sin(\pi/\alpha)\Gamma(2-\alpha)r_{0}^{\alpha-1}/(\pi\Gamma(1/\alpha)(\alpha-1))$
Blumenthal1961 . By computing the rescaled MFPT in confinement with Eq. (7),
one can check that the relation (8), which can be readily generalised to
continuous space, is still verified for this process.

Extension to non-Markovian processes. We now relax the Markov property and
generalize our theory to the case of non-Markovian processes, i. e. displaying
memory. In the following, we argue that the relation (8) yields much more
accurate results for $S_{0}$ than Markov approximations; it is exact for
processes with finite memory time, and is very precise (even though not exact)
for strongly correlated processes such as the Fractional Brownian Motion. As
the mean FPT in confinement has recently been characterized for non-Markovian
Gaussian processes guerin2016mean , this equation (8) provides a means to
estimate $S(t)$ at long times, beyond persistence exponents, for a wide class
of random walks with memory.

For simplicity, we consider 1-dimensional processes and we switch to
continuous space description. The stochastic trajectories $x(t)$ are assumed
to be continuous but non-smooth (the method in fact also applies to compact
and not continuous processes, such as $1d$ Levy stable processes of index
$\alpha>1$ as discussed above), mathematically meaning that
$\langle\dot{x}^{2}\rangle=\infty$ and physically corresponding to very rough
trajectories, similar to those of Brownian motion. We assume that the
increments of the walk are stationary (meaning that there is no aging, even
transient 111In particular the case of continuous time random walks (CTRWs) is
not directly covered by our analysis; persistence exponents and prefactors for
CTRWs can be obtained from the subordination principle). This hypothesis is
known to have two consequences: (i) the persistence exponent for the
unconfined problem is exactly given by $\theta=1-1/d_{w}$
levernier2018universal ; Molchan1999 ; ReviewBray ; Krug1997 ; Aurzada2011 ;
Aurzada2015 ; (ii) the mean FPT for the confined problem varies linearly with
the confinement volume $V$, so that $\overline{T}$ is finite and has been
identified as guerin2016mean :

$\overline{T}=\int_{0}^{\infty}dt\,[p_{\pi}(0,t)-p(0,t)].$ (10)

Here, $p(0,t)$ is the probability density of $x=0$ at a time $t$ after the
initial state (where $x(0)=r_{0}$), and $p_{\pi}(0,t)$ denotes the probability
density of finding the walker on the target at a time $t$ after the first-
passage:

$\displaystyle
p_{\pi}(0,t)=\int_{0}^{\infty}d\tau\,p(0,t+\tau|\mathrm{FPT}=\tau)F(\tau;r_{0}),$
(11)

where $p(0,t+\tau|\mathrm{FPT}=\tau)$ is the probability density of $x=0$ at
time $t+\tau$, given that the FPT is $\tau$.

The starting point to relate $\overline{T}$ to $S_{0}$ consists in writing the
generalization of Eq. (1) to non-smooth non-Markovian processes:

$\displaystyle p(0,t)=\int_{0}^{t}d\tau
F(\tau;r_{0})p(0,t|\mathrm{FPT}=\tau).$ (12)

To proceed further, we insert into Eq. (10) the expressions (11) and (12) of
$p_{\pi}(0,t)$ and $p(0,t)$:

$\displaystyle\overline{T}=\int_{0}^{\infty}dt\,\Big{[}$
$\displaystyle\,\int_{0}^{\infty}d\tau\,p(0,t+\tau\,|\,\text{FPT}=\tau)\,F(\tau;r_{0})$
$\displaystyle-\int_{0}^{t}d\tau\,p(0,t\,|\,\text{FPT}=\tau)\,F(\tau;r_{0})\Big{]}.$
(13)

To avoid diverging integrals in the change of variables $t=u+\tau$, we replace
$\int_{0}^{\infty}dt(...)$ by
$\underset{A\to\infty}{\lim}\;\;\int_{0}^{A}dt(...)$, so that

$\overline{T}=\underset{A\to\infty}{\lim}\int_{0}^{A}dt\,\int_{A-t}^{\infty}d\tau
F(\tau;r_{0})p(0,t+\tau|\text{ FPT}=\tau).$ (14)

Setting $t=uA$ and $\tau=vA$, we note that when $A\to\infty$, only the large
time behavior are involved in these integrals, where one can use the
asymptotics $F(Av;r_{0})\sim S_{0}\theta/(Av)^{1+\theta}$ and

$\displaystyle
p(0,(u+v)A|\text{FPT}=vA)\underset{A\to\infty}{\sim}\frac{KG(u/v)}{(Au)^{1/d_{w}}},$
(15)

which is a form imposed by dimensional analysis. As previously, $K$ is the
constant which characterizes the long time behavior of the one point
probability distribution function [i.e.
$p(x,t)\underset{t\to\infty}{\sim}K/t^{1/d_{w}}$], $G$ is a scaling function,
with $G(\infty)=1$, that does not depend on the geometrical parameters of the
problem. Inserting these asymptotic behaviors into Eq. (14), we get:

$\overline{T}=\underset{A\to\infty}{\lim}A^{1-\theta-1/d_{w}}\,K\,\int_{0}^{1}du\,\int_{1-u}^{\infty}dv\,\frac{\theta\;S_{0}}{v^{\theta+1}}\frac{G(u/v)}{u^{1/d_{w}}}.$
(16)

The fact that the above integral exists and is finite leads to the (known)
relation $\theta=1-1/d_{w}$. This finally leads to the exact relation:

$\displaystyle
S_{0}=\frac{\overline{T}}{K(1-1/d_{w})}\left(\,\int_{0}^{1}du\,\int_{1-u}^{\infty}dv\,\frac{G(u/v)}{u^{1/d_{w}}v^{2-1/d_{w}}}\right)^{-1}.$
(17)

We stress that the dependency of $S_{0}$ on the source-to-target distance,
even when not-trivial, is entirely contained in the term $\overline{T}$.
Indeed, the scaling function $G$ depends only on the large scale properties of
the random walk and not on the geometrical parameters.

While the exact determination of $G$ is a challenging task, the following
decoupling approximation turns out to be very accurate. In this approximation,
the return probability to the target at a time $t$ after the first passage
time is independent of the actual value of the FPT, which leads to
$p(0,t+\tau|\text{FPT}=\tau)\simeq p_{\pi}(t)$ for self-consistence reason.
Within this decoupling approximation, $G\simeq 1$ and we obtain

$S_{0}\simeq\frac{\sin(\pi/d_{w})}{K\pi}\overline{T},$ (18)

which generalizes Eq. (8) to non-Markovian processes. We now comment on the
validity of this key relation.

First, we stress that Eq. (18) is exact for processes with finite memory time
(i.e. for which the correlation function of increments decays exponentially at
long times). This comes from the very definition of the function $G$, which
involves only large time scales in Eq. (15), over which this finite memory
time becomes irrelevant. This case is illustrated here by considering a
Gaussian process whose Mean Square Displacement function
$\psi(t)=\langle[x(t+\tau)-x(\tau)]^{2}\rangle$ is given by
$\psi(t)=Dt+B(1-e^{-\lambda t})$. This “bidiffusive” process involves two
diffusive behaviors at long and short time scales, and presents only one
relaxation time $\lambda^{-1}$. This is typically relevant to tracers moving
in viscoelastic Maxwell fluids grimm2011brownian , nematics turiv2013  or
solutions of non-adsorbing polymers ochab2011scale . We also consider the
effect of multiple relaxation times with the case that $x(t)$ is the position
of the first monomer of a flexible polymer chain with $N$ monomers, in the
most simple (Rouse, bead-spring) polymer model. We use recently obtained
estimates of $\overline{T}$ in Ref. guerin2016mean  to obtain estimates of
$S_{0}$ through Eq. (18) and compare with numerical simulations on Fig.
2(b),(c). We also compare with a pseudo-Markovian approximation (using Eq. (6)
with effective “propagators” PM ). Our prediction for $S_{0}$ is in good
agreement with numerical simulations, and shows that even if the memory time
is finite, memory effects are strong.

Second, it is showed in SI that Eq. (18) is also exact at first order in
$\varepsilon=H-1/2$ for the fractional Brownian Motion (FBM), which is an
emblematic example of processes with infinite memory time. The FBM is used in
fields as varied as hydrology mandelbrot1968noah , finance cutland1995stock
and biophysics ernst2012fractional ; burnecki2012universal . This Gaussian
process is characterized by
$\left\langle[x(t+\tau)-x(t)]^{2}\right\rangle=\kappa\tau^{2H}$, with $0<H<1$.

Third, in the strongly non-Markovian regime, where $\varepsilon$ cannot be
considered as small, it turns out that Eq. (18) provides a very accurate
approximation [Fig. 2(e)] of $S_{0}$, which takes the explicit form

$\displaystyle S_{0}=\beta_{H}\sin(\pi
H)\sqrt{\frac{2}{\pi}}\left(\frac{r_{0}}{\kappa^{1/2}}\right)^{\frac{1}{H}-1}$
(19)

where $\beta_{H}$ is a function of $H$ analyzed in Ref. guerin2016mean . It
can indeed be seen on Fig. 2(e) that Eq. (18) correctly predicts the long time
behavior of $S(t)$ when $H=0.34$. For this value non-Markovian effects are
strong, as can be seen by comparing with the prediction of the pseudo-
Markovian approximation, which is wrong by more than one order of magnitude
[Fig. 2(e), dashed line]. The value of $S_{0}$ is slightly underestimated in
the decoupling approximation, but can be made more precise by evaluating the
scaling function $G$ (see SI).

Furthermore, our approach also holds in dimension higher than one, even for
strongly correlated non-Markovian processes. Indeed, the $d-$dimensional
version of Eq. (18) [i.e. Eq. (8)] correctly predicts (but slightly
underestimates) the value of $S_{0}$ for an example of two dimensional FBM
[Fig. 2(f)]. In this example, the target radius $a$ is not zero even if the
$a\to 0$ limit is well defined for compact processes; the dependence of
$S_{0}$ on the target radius is predicted to be the same as that of
$\overline{T}$, which is available in the non-Markovian theory of Ref.
guerin2016mean . Finally, in the case of processes with finite memory, we find
that Eq. (18) also holds for non-stationary initial conditions. This is
illustrated by considering the case of a flexible phantom polymer for which
all monomers are placed initially at $r_{0}$ (instead of having the shape of a
random equilibrium coil for stationary initial conditions). This non-
stationary initial condition induces transiently aging dynamics, and $S_{0}$
is changed with respect to the case of stationary initial conditions, but is
still predicted correctly by Eq. (18) [see Fig. 2(d)].

Finally, let us mention the case of the 1-dimensional run and tumble process,
where a particle switches between phases of constant velocities $\pm v$ with
rate $\alpha$. This process is smooth and is a priori not covered by our
analysis. However, our relation (18) between $S_{0}$ and $T/V$ is still exact,
as is made clear by comparing the results for the mean FPT in confinement
Masoliver1986  and in semi-infinite space Malakar2018 . This agreement holds
even for non-stationary initial conditions, where the probability $p$ that the
initial velocity is positive differs from $1/2$: in this case, one can obtain
$S_{0}=(r_{0}+pv/\alpha)\sqrt{2\alpha/(\pi v^{2})}=\overline{T}/(K\pi)$, with
$K=\sqrt{\alpha/(2\pi v^{2})}$, and we can check that our relation still holds
Masoliver1986 ; Malakar2018 . Furthermore, it also holds in the case of
partially reflecting targets, as can be deduced from the results of
Angelani2015 . This suggests that our analysis can be extended to smooth non-
Markovian processes with partial absorption as well.

Discussion

The determination of the survival probability $S(t)$, and in particular its
dependence on the initial distance to the target, requires the knowledge of
its prefactor $S_{0}$, which has remained an elusive quantity up to now. In
this article, we have bridged this gap by identifying a general relation
between the long time persistence and the mean FPT in confinement. The latter
can be calculated with various recently introduced methods, for a large class
of Markovian Condamin2007 ; Benichou2008 ; Schuss2007  and non-Markovian
random walks guerin2016mean . Our theory holds for compact, unbiased walks
with stationary increments that are scale invariant at long times (without
confinement), with moments of the position that diverge with time. Our main
result is Eq. (8), which is exact for both Markovian processes (such as
diffusion in fractals) and for non Markovian processes with finite memory time
(for which memory effects are nevertheless quantitatively non-negligible). For
long-ranged correlated processes such as FBM our formula provides a good
approximation of $S_{0}$ in one or higher dimensions, and is found to be exact
at first order in a perturbation expansion around Brownian motion. Together,
our results thus improve our understanding of the impact of memory on the
statistics of long first passage events.

Acknowledgments. This work was supported by ERC grant FPTOpt-277998. 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.

Data and code availability. The numerical data presented in Figure 2, as well
as the code that generated these data, are available from the corresponding
author on reasonable request.

Competing interests statement. The Authors declare no competing interests.

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Figure 1: First-passage problem with or without confinement. Two first passage
problems in which a random walker starting from a given site (green square)
reaches a target (red disk) at the end of a stochastic trajectory: (a) in free
space, (b) in a confined reflecting domain. Sample trajectories for fractional
Brownian motion ($H=0.45$) are shown.

Figure 2: Survival probability $S(t)$ for various stochastic processes. In all
graphs, symbols are the results of stochastic simulations (detailed in SI),
continuous lines give the theoretical predictions [Eq. (18)], and dashed line
represent the predictions of the pseudo-Markovian approximation PM . (a)
$S(t)$ for a random walk on the Sierpinski gasket for two values of the
initial (chemical) source-target distance. Here, $d_{f}=\ln 3/\ln 2$,
$d_{w}=\ln 5/\ln 2$, and $K\simeq 0.30$ weber2010random ;
meroz2011distribution . Simulations are shown for a fractal of generation
$11$. Continuous lines are the predictions of Eq. (9). (b) $S(t)$ for a
1-dimensional “bidiffusive” Gaussian process of MSD $\psi(t)=t+30(1-e^{-t})$.
(c) $S(t)$ for a one-dimensional Rouse chain with $N=20$ monomers, for various
source-to-target distance $r_{0}$ (indicated in the legend in units of the
monomer length). (d) $S(t)$ for the same system with $N=15$ and $r_{0}=3$,
comparing stationary initial conditions (the other monomers being initially at
equilibrium) or non-stationary ones (for which all monomers start at the same
position $r_{0}$). (e) $S(t)$ for a one dimensional FBM of MSD
$\psi(t)=t^{2H}$ with Hurst exponent $H=0.34$. (f) Two-dimensional FBM of MSD
$\psi(t)=t^{2H}$ in each spatial direction with $H=0.35$. The target is a disk
of radius $a=1$ and $r_{0}$ is the distance to the target center. For
(b),(c),(d),(e),(f) the continuous lines represent our predictions [Eq. (18)],
in which $\overline{T}$ is calculated by using the theories of Refs.
Guerin2012a ; guerin2016mean ; Guerin2013 ; in (b) and (c) the only hypothesis
to predict $\overline{T}$ is that the distribution of supplementary degrees of
freedoms at the FPT is Gaussian, in (e) and (f) we use the additional
“stationary covariance” approximation. In (d), for non-stationary initial
conditions, $\overline{T}$ is measured in simulations in confined space. A
table that compares the values of $S_{0}$ in the theory and in the simulations
is given in SI.