# Stiffness of random walks with reflecting boundary conditions

Sascha Kaldasch  Andreas Engel  Universität Oldenburg, Institut für Physik,
26111 Oldenburg, Germany

###### Abstract

We study the distribution of occupation times for a one-dimensional random
walk restricted to a finite interval by reflecting boundary conditions. At
short times the classical bimodal distribution due to Lévy is reproduced with
walkers staying mostly either left or right to the initial point. With
increasing time, however, the boundaries suppress large excursions from the
starting point, and the distribution becomes unimodal converging to a
$\delta$-distribution in the long time limit. An approximate spectral analysis
of the underlying Fokker-Planck equation yields results in excellent agreement
with numerical simulations.

###### pacs:

02.50.Ey, 05.40.-a, 05.40.Fb

##  I Introduction

Random walks are central to the theory of stochastic processes, both because
they can be analyzed analytically in great detail and because they are
ubiquitous in nature. Despite their simplicity they have some intriguing
properties and exhibit several counterintuitive features. A prominent example
is the surprising “stiffness” of an unbiased one-dimensional random walk: in a
fixed time interval the walker will most of the time stay either left or right
to its starting point. Trajectories that stay half of the time on either side
of the initial point have the smallest probability in apparent contrast to the
fact that jumps to the left and to the right are equally likely.

More precisely, the probability density $P_{T}(S)$ for the fraction $S$ of
overall time $T$ the walker spent to the right (or left) of the starting point
is given by

$P_{T}(S)=\frac{1}{\pi\sqrt{S(1-S)}},$ (1)

cf. Fig. 1. This remarkable result was established long ago by Paul Lévy Levy
and gave rise to a plethora of discussions and generalizations in the
mathematics and physics literature, see, e.g., FellerI ; FellerII ; Yor ;
Redner ; Majumdar  and references therein. Examples include the generalization
to cases with deterministic Akahori ; Dassios  and random drift fields MaCo
as well as to anomalous diffusion Barkai , the investigation of different
large deviation properties of $P_{T}(S)$ MaBr ; Barkai2 ; BuTo , and the
possibility of dynamical phase transitions NyTo1 ; NyTo2 .

Figure 1: Distribution of the fraction of time $S$ an unrestricted symmetric
one-dimensional random walker spends to the right of the starting point.

From an intuitive point of view one may suspect that the large values of
$P_{T}(S)$ near $S=0,1$ arise because the walker may travel arbitrarily long
distances away from the starting point. This is corroborated by other well-
known properties of one-dimensional random walks like the fact that although
return to the origin is certain, the mean time for it diverges, by the
distribution of the number of returns to the starting point FellerI , by the
statistics of successive returns McFa ; Redner ; Barkai , as well as by the
asymmetry of the random walk Weiss . The shape of the distribution $P_{T}(S)$
should therefore change qualitatively if reflecting boundary conditions
restrict the random walk to a finite interval.

To test this conjecture we analyze in the present paper the distribution
$P_{T}(S)$ for an unbiased one-dimensional random walk with reflecting
boundary conditions symmetric to the starting point. Although various
properties of random walks on finite intervals have been investigated in the
past FellerIII ; Grebenkov ; BuTo ; Barkai  the details of the shape
transformation in $P_{T}(S)$ have not been elucidated so far. The problem may
be mapped onto a Sturm-Liouville eigenvalue problem that cannot completely be
solved analytically. Nevertheless, we provide a highly accurate approximate
solution that is in perfect agreement with results from numerical simulations.
We find that for small $T$ the distribution is still of the form shown in Fig.
1 but changes to an unimodal distribution with maximum at $S=1/2$ with
increasing duration $T$ of the walk. For $T\to\infty$ it approaches a
$\delta$-distribution around $S=1/2$ in accordance with equilibrium
statistical mechanics Barkai .

The paper is organized as follows. In section II we introduce the basic
notation and establish the central Fokker-Planck equation for the joint
probability distribution of the walker position $x$ and the time fraction $S$.
In section III we show how to map the solution of this equation to a Sturm-
Liouville eigenvalue problem, that we analyze in section IV. Section V
discusses the numerical determination of the eigenvalues. In section VI we
analytically extract the asymptotics of $P_{T}(S)$ for large $T$. In section
VII we present results for $P_{T}(S)$ for various values of $T$ and compare
them with numerical simulations. Finally, section VIII contains some
conclusions.

##  II Basic Equations

We consider the time interval $0\leq t\leq T$ of a one-dimensional random walk
$x(t)$ with reflecting boundary conditions at $x=\pm a$ that started at
$x(0)=0$. The main quantity of interest is the fraction of time

$S_{T}:=\frac{1}{T}\int_{0}^{T}\\!\\!dt\,\theta\big{(}x(t)\big{)}$ (2)

the walker spent at positive values of $x$. Here $\theta$ denotes the
Heaviside step function $\theta(x)=1$ for $x>1$ and $\theta(x)=0$ otherwise.

With $x$ also $S_{T}$ is a random quantity. It is characterized by a
probability density function $P_{T}(S)$ such that $P_{T}(S)dS$ gives the
probability for $S\leq S_{T}\leq S+dS$.

The stochastic dynamics of the system are described by two coupled Langevin
equations,

$\displaystyle\dot{x}$ $\displaystyle=\xi(t)$ (3) $\displaystyle\dot{S}_{T}$
$\displaystyle=\frac{1}{T}\theta\big{(}x(t)\big{)},$ (4)

where $\xi(t)$ denotes Gaussian white noise with expectation values

$\langle\xi(t)\rangle\equiv
0\qquad\mathrm{and}\qquad\langle\xi(t)\xi(t^{\prime})\rangle=2D\delta(t-t^{\prime}).$
(5)

The diffusion constant $D$ characterizes the noise strength.

We measure $x$ in units of $a$ and rescale time such that $D=1/2$. The only
remaining parameter in the problem is then the (dimensionless) time $T$. For
$T\ll 1$ the walker has hardly a chance to feel the boundaries and $P_{T}(S)$
should be similar to the form shown in Fig. 1. With $T$ increasing towards
values of order one the boundary conditions become more and more relevant and
the shape of $P_{T}(S)$ has to change accordingly. Finally, for large $T$ the
walker has explored the whole interval $-1\leq x\leq 1$ evenly and we expect

$P_{T}(S)\to\delta\left(S-\frac{1}{2}\right).$ (6)

The set of Langevin equations (3) and (4) is equivalent to the following
Fokker-Planck equation for the joint probability density function $P(x,S,t)$
vanKampen

$\partial_{t}P(x,S,t)=-\frac{1}{T}\theta(x)\partial_{S}P(x,S,t)+\frac{1}{2}\partial_{x}^{2}P(x,S,t).$
(7)

This equation is complemented by zero-flux boundary conditions at $x=\pm 1$,

$\partial_{x}P(x,S,t)\Big{|}_{x=\pm 1}=0\qquad\forall t>0,\forall S\in[0,1],$
(8)

and the initial condition

$P(x,S,0)=\delta(x)\delta(S).$ (9)

From the symmetry of the problem it is clear that

$P(-x,1-S,t)=P(x,S,t).$ (10)

Our central quantity of interest $P_{T}(S)$ is obtained from the solution
$P(x,S,t)$ of the Fokker-Planck equation (7) by marginalization in $x$:

$P_{T}(S)=\int_{-1}^{1}\\!\\!dx\,P(x,S,T).$ (11)

##  III Solution of the Fokker-Planck equation

Since $P(x,S,t)$ is defined on a finite interval of $S$ values it may be
written as a Fourier series of the form

$P(x,S,t)=\sum_{p}\psi_{p}(x,t)\,e^{2\pi ipS}.$ (12)

The sum runs over all integer values of $p$ and the function $\psi_{p}(x,t)$
is given by

$\psi_{p}(x,t)=\int_{0}^{1}\\!\\!dS\,P(x,S,t)\,e^{-2\pi ipS}.$ (13)

Since $P(x,S,t)$ is real we have

$\psi_{-p}(x,t)=\psi_{p}^{*}(x,t)$ (14)

which together with (10) results in

$\psi_{p}(x,t)=\psi_{p}^{*}(-x,t).$ (15)

Multiplying (7) by $e^{-2\pi ipS}$ and integrating over $S$ we get

$\partial_{t}\psi_{p}(x,t)=-\frac{2\pi
ip}{T}\theta(x)\psi_{p}(x,t)+\frac{1}{2}\partial_{x}^{2}\psi_{p}(x,t).$ (16)

The boundary conditions translate to

$\partial_{x}\psi_{p}(x,t)\Big{|}_{x=\pm 1}=0\qquad\forall t>0,\forall
p\in\mathbb{Z}$ (17)

and the initial condition requires

$\psi_{p}(x,0)=\delta(x)\qquad\forall p\in\mathbb{Z}.$ (18)

Using the abbreviation

$\bar{p}:=\frac{2\pi p}{T}$ (19)

we solve (16) with the help of the separation ansatz

$\psi_{p}(x,t)=e^{-\frac{1}{2}(E+i\bar{p})t}\,\varphi(x).$ (20)

Plugging (20) into (16) yields

$\varphi^{\prime\prime}(x)+\big{(}E-i\bar{p}\,\text{sign}(x)\big{)}\varphi(x)=0$
(21)

subject to the boundary conditions

$\varphi^{\prime}(x=\pm 1)=0.$ (22)

Here $\text{sign}(x)$ is the sign function

$\text{sign}(x)=\begin{cases}1&x>0\\\ -1&x\leq 0\end{cases},$

and the prime denotes differentiation with respect to $x$. Alternatively, one
may derive (21) from the path measure $P[x(\cdot)]$ using the Feynman-Kac
formula, see Example 2 in Kac49 . From (15) we have

$\varphi(x)=\varphi^{*}(-x).$ (23)

We expect for each $p$ a denumerable set of discrete eigenvalues $E^{(p)}_{n}$
and corresponding eigenfunctions $\varphi^{(p)}_{n}(x)$ solving (21). Standard
arguments show that the spectrum is non-degenerate and that eigenfunctions
corresponding to different eigenvalues are orthogonal:

$\int_{-1}^{1}\\!\\!dx\,\varphi^{(p)}_{n}(x)\,\varphi^{(p)}_{m}(x)=:a^{(p)}_{n}\,\delta_{nm}.$
(24)

The general solution to eq. (16) is hence of the form

$\psi_{p}(x,t)=\sum_{n=1}^{\infty}c^{(p)}_{n}\,\varphi^{(p)}_{n}(x)\,e^{-\frac{1}{2}(E^{(p)}_{n}+i\bar{p})t}.$
(25)

It has to be kept in mind that due to the imaginary term
$i\bar{p}\,\text{sign}(x)$ in eq. (21) $E^{(p)}_{n}$ and
$\varphi^{(p)}_{n}(x)$ and consequently also the normalization and expansion
coefficients $a^{(p)}_{n}$ and $c^{(p)}_{n}$, respectively, will in general be
complex.

The $c^{(p)}_{n}$ may be determined from the initial condition (18):

$\delta(x)=\psi_{p}(x,0)=\sum_{n=1}^{\infty}c^{(p)}_{n}\,\varphi^{(p)}_{n}(x).$
(26)

Multiplying with $\varphi^{(p)}_{m}(x)$, integrating over $x$ and using the
orthogonality (24) yields

$d_{m}^{(p)}:=\varphi^{(p)}_{m}(0)=c_{m}^{(p)}\,a_{m}^{(p)}$ (27)

Moreover, in view of (11) we do not need the complete function $\psi_{p}(x,t)$
to finally determine $P_{T}(S)$. It is sufficient to know

$\chi_{p}(T):=\int_{-1}^{1}\\!\\!dx\,\psi_{p}(x,T)$ (28)

from which we get using (12)

$P_{T}(S)=\sum_{p}\chi_{p}(T)\,e^{2\pi ipS}.$ (29)

Defining

$b_{m}^{(p)}:=\int_{-1}^{1}\\!\\!dx\,\varphi^{(p)}_{m}(x),$ (30)

combining (25) with (27) and (30), and observing (19) we end up with

$\chi_{p}(T)=(-1)^{p}\,\sum_{n=1}^{\infty}\frac{b^{(p)}_{n}d^{(p)}_{n}}{a^{(p)}_{n}}\,e^{-\frac{1}{2}E^{(p)}_{n}T}.$
(31)

Whenever no confusion may arise we will suppress the superscript $p$ at
$E,\varphi(x),a,b$ and $d$ in the following to lighten the notation.

##  IV The eigenvalue problem

To complete the determination of $P_{T}(S)$ via (31) and (29) we need to solve
the eigenvalue problem (21)

$\varphi_{n}^{\prime\prime}(x)+\big{(}E_{n}-i\bar{p}\,\text{sign}(x)\big{)}\varphi_{n}(x)=0$
(32)

for all integer values of $p$. The real part of $E_{n}$ must always be
nonnegative. To see this we multiply (32) with $\varphi_{n}^{*}(x)$, integrate
over $x$ and take the real part to find

$\Re(E_{n})\int_{-1}^{1}\\!\\!dx\,|\varphi_{n}(x)|^{2}=\int_{-1}^{1}\\!\\!dx\,|\varphi^{\prime}_{n}(x)|^{2}.$
(33)

The integral on the l.h.s. of this equation must be positive for
$\varphi_{n}(x)$ to be an eigenfunction, the one on the r.h.s. is nonnegative.
This proves the assertion. Similar arguments show that $E=0$ is possible only
for $\bar{p}=0$.

Figure 2: Blue lines show the probability density function $p(x,t)$ for the
position $x$ of the walker for $t=0.05,0.1,0.25,0.5$ and $t=1$ (middle top to
bottom). The red line is the stationary distribution $p_{\mathrm{st}}(x)\equiv
1/2$ reached for $t\to\infty$.

In fact, for $p=0$ the whole eigenvalue problem is equivalent to a standard
exercise in quantum mechanics LLIIIpar22  and may be solved analytically. The
result reads

$\psi_{0}(x,t)=\frac{1}{2}+\sum_{n=1}^{\infty}\cos(n\pi
x)\,e^{-\frac{1}{2}n^{2}\pi^{2}t}.$ (34)

Note that from (13) it follows that

$\psi_{0}(x,t)=\int_{0}^{1}\\!\\!dS\,P(x,S,t)=:p(x,t).$ (35)

Hence, (34) describes the time-evolution of the probability density function
$p(x,t)$ for the position of the walker. Fig.2 shows a few snapshots.
Moreover, Eq. (28) implies

$\chi_{0}(t)\equiv 1$ (36)

which via (29) ensures the normalization of $P_{T}(S)$ for all $T$.

Figure 3: Left: Left hand side of equation (39) for real $E$ and $\bar{p}=1$.
Middle: Blowup for small values of $E$ and $\bar{p}=1.2,1.3855,1.6$ (left top
to bottom). Right: Blowup around the second pair of roots for
$\bar{p}=8.2,8.3862,8.6$ (left top to bottom).

For $p\neq 0$ (32) is a linear ordinary differential equation with piecewise
constant coefficients. It is solved for $x<0$ and $x>0$ separately and the
solutions are then matched at $x=0$ such that $\varphi_{n}$ and
$\frac{d}{dx}\ln\varphi_{n}$ are continuous there. With the abbreviations

$k_{n}:=\sqrt{E_{n}+i\bar{p}}\qquad\mathrm{and}\qquad
q_{n}:=\sqrt{E_{n}-i\bar{p}}$ (37)

the result is

$\varphi_{n}(x)=\begin{cases}\displaystyle\sqrt{\frac{\cos q_{n}}{\cos
k_{n}}}\,\cos k_{n}(x+1)&-1\leq x\leq 0\\\ \displaystyle\sqrt{\frac{\cos
k_{n}}{\cos q_{n}}}\,\cos q_{n}(x-1)&0\leq x\leq 1\\\ \end{cases}$ (38)

where $E_{n}$ has to fulfill

$\sqrt{E_{n}+i\bar{p}}\,\tan\sqrt{E_{n}+i\bar{p}}+\sqrt{E_{n}-i\bar{p}}\,\tan\sqrt{E_{n}-i\bar{p}}=0.$
(39)

The prefactors of the cosine functions in (38) have been chosen such that (23)
holds. From (38) the coefficients $a_{n},b_{n}$ and $d_{n}$ can be determined
from their respective definitions (24), (30), (27). We find

$\displaystyle b_{n}\,d_{n}$ $\displaystyle=\frac{\cos q_{n}\sin
k_{n}}{k_{n}}+\frac{\cos k_{n}\sin q_{n}}{q_{n}}$ (40) $\displaystyle a_{n}$
$\displaystyle=\frac{1}{2}\left(\frac{\cos q_{n}}{\cos k_{n}}+\frac{\cos
k_{n}}{\cos q_{n}}+b_{n}\,d_{n}\right).$ (41)

##  V Numerical determination of the eigenvalues

Eq. (39) can only be solved numerically. Nevertheless a few prior
consideration are in order. Taking the complex conjugate of (32) and using the
fact that $\text{sign}(x)$ is an odd function of $x$ we see that with
$E_{n},\varphi_{n}(x)$ also $E^{*}_{n},\varphi_{n}^{*}(-x)$ is an admissible
solution. Complex eigenvalues hence come in pairs of conjugates entailing the
same for the associated constants $a_{n},b_{n}$ and $c_{n}$.

Moreover, for real $E_{n}$ the imaginary part of eq. (39) is identically zero
rem1 . Fig. 3 shows plots of the l. h. s. of (39) for real $E$. From the left
plot we infer that the gap between successive real roots $E_{n}$ is much
larger than 1. Keeping in mind that we are interested in values of $T$ of
order 1 only the contributions from the first two roots will play a noticeable
role in the superposition (31). This is corroborated by comparison with our
numerical simulations, see section VII below. The higher eigenvalues are only
important for the very short time dynamics in which $P_{T}(S)$ has to
transform from the initial condition $\delta(S)$ to the shape shown in Fig. 1.

Figure 4: The two solutions of the eigenvalue equation (39) with the smallest
real part as function of $\bar{p}$. The real part is shown in red, the
imaginary one in blue. At $\bar{p}=\bar{p}_{\mathrm{c1}}\simeq 1.38$ the two
real solutions merge and give birth to a pair of complex conjugated solutions.

The other two plots of Fig. 3 show that with increasing $\bar{p}$ the real
roots disappear successively to give way to pairs of complex conjugate
solutions for $E_{n}$. At the critical values $\bar{p}_{\mathrm{ci}}$ of
$\bar{p}$ where these transitions occur the l. h. s. of (39) and its
derivative with respect to $E_{n}$ both vanish. For the first two bifurcation
values we find $\bar{p}_{\mathrm{c1}}=1.385577425$ and
$\bar{p}_{\mathrm{c2}}=8.386237648$, respectively. The first bifurcation of
this type is shown in detail in Fig. 4.

Figure 5: Blue lines are plots of $P_{T}(S)$ obtained as described in the main
text. Symbols are results of numerical simulations with time step $\Delta
t=5\cdot 10^{-4}$ averaged over $2.5\cdot 10^{6}$ realizations. Statistical
errors are smaller than the symbol size. Left: $T=1,2,2.5$ (middle bottom to
top). Middle: $T=3,3.5,4$ (middle bottom to top); note the reduced scale for
$P_{T}$. Right: $T=5,10,50$ (middle bottom to top).

.

These considerations make clear how to get rather accurate approximate results
for $P_{T}(S)$. First, depending on the desired resolution for $S$ the maximal
number $p_{\mathrm{max}}$ of Fourier modes is chosen. Then for given value of
$T$ one checks for each $p$ whether $\bar{p}$ is smaller or larger than
$\bar{p}_{\mathrm{c1}}$. For $\bar{p}<\bar{p}_{\mathrm{c1}}$ one determines
the two lowest real solutions $E_{1}$ and $E_{2}$ numerically from (39) and
calculates the corresponding values of $a,b$, and $d$. If
$\bar{p}>\bar{p}_{\mathrm{c1}}$ it is sufficient to determine one eigenvalue
$E_{1}$ with its coefficients $a,b$, and $d$; $E_{2}$ and its corresponding
coefficients then follow by complex conjugation. Because of (14) it is
sufficient to consider positive values of $p$ only. Plots obtained in this way
are shown in section VII below together with results from numerical
simulations.

##  VI Asymptotics for large $T$

The asymptotics of $\chi_{p}(T)$ as given by (31) for large $T$ is not
completely obvious since $T$ enters the eigenvalue equation (39) via
$\bar{p}$, cf. Eq. (19), and therefore the $E_{n}$ depend on $T$. For
$T\to\infty$ we have $\bar{p}\to 0$ giving rise to $E_{1}\to 0$. An expansion
of (39) in $\bar{p}$ and $E_{1}$ yields the asymptotic behaviour

$E_{1}\sim\frac{\bar{p}^{2}}{3}.$ (42)

For large $T$ we therefore have $E_{1}=O(1/T^{2})$ and consequently
$E_{1}\,T\to 0$. At the same time we find from (40) and (41) for $\bar{q}\to
0$ by using (37) and (42)

$\frac{b_{1}\,d_{1}}{a_{1}}\to 1.$ (43)

Hence (31) results in $\zeta_{p}(T)\to(-1)^{p}$ for $T\to\infty$ which implies

$P_{T}(S)\to\delta\left(S-\frac{1}{2}\right)\qquad\mathrm{for}\qquad
T\to\infty$ (44)

as expected.

##  VII Results

Results for $P_{T}(S)$ obtained along the lines of sections III-V for
different values of $T$ are shown in Fig. 5. We have chosen
$p_{\mathrm{max}}=128$ as number of Fourier modes which is sufficient to
resolve the important details of $P_{T}(S)$. To suppress spurious oscillations
in particular in the almost constant parts of $P_{T}(S)$ for small $T$ we have
additionally smoothed the results with a Gaussian filter of width
$\sigma=0.01$.

The left plot of Fig. 5 shows that for $T\lesssim 2$ there are only small
modifications in $P_{T}(S)$ as compared to the case without boundary
conditions shown in Fig. 1. Only few realizations are able to reach the
boundaries at $x=\pm 1$ and to get back and cross the starting point to
contribute to the slight increase of $P_{T}(S)$ near $S=1/2$. Note that, in
marked contrast, for these values of $T$ the distribution of the walker itself
is already near to the stationary state, cf. Fig. 2. The equilibration of
$p(x,t)$ therefore occurs mostly separately in the regions of positive and
negative $x$ without many crossings of the starting point.

For $2.5\leq T\leq 4$ the main reshaping of $P_{T}(S)$ from a bimodal
distribution with maxima at $S=0,1$ to a unimodal one with maximum at $S=1/2$
takes place. This is demonstrated by the middle part of Fig. 5. For the values
of $T$ shown the walker typically not only reaches one of the boundaries but
also has enough time to get back to the starting point. As a result more and
more of the realizations cross the starting point and contribute to the
growing central maximum of $P_{T}(S)$. At the same time it becomes
increasingly unlikely for the walker to stay in only the left or the right
half of the allowed interval resulting in a steady decrease of $P_{T}(S)$ near
the boundary values $S=0$ and $S=1$.

Finally, for $T\geq 5$ most realizations have visited both boundaries and
crossed the origin several times. Accordingly, $P_{T}(S)$ approaches a
Gaussian centered at $S=1/2$ that becomes sharper with increasing $T$ to
eventually approach the limit given by (44). The right part of Fig. 5 shows
some intermediate stages.

All results obtained are in excellent agreement with numerical simulations
shown by the symbols in Fig. 5. This a posteriori validates the restriction of
the expansion (31) to the two leading eigenvalues.

##  VIII Conclusion

In conclusion we have shown that the somewhat surprising shape of the
distribution $P_{T}(S)$ for the fraction $S$ of total time $T$ an unrestricted
random walker spends to the right of the starting point as shown in Fig. 1 is
indeed due to the possibility of very large excursions away from this point.
When restricting the walker to a finite interval by reflecting boundary
conditions these excursions are precluded and $P_{T}(S)$ assumes a unimodal
form with maximum at $S=1/2$ for sufficiently large $T$. Unlike the case
without boundaries no complete analytical solution seems possible. However,
rather accurate approximate results may be obtained on the basis of just the
two leading eigenvalues of the corresponding Fokker-Planck operator. The
results are in excellent agreement with numerical simulations, and the
emerging picture is consistent with physical intuition.

###### Acknowledgements.

We are grateful to Eli Barkai and Hugo Touchette for pointing out pertinent
references.

## References

  * (1) P. Lévy, Compositio Mathematica 7, 283 (1939)
  * (2) W. Feller, An introduction to probability theory and its applications, volume I, 3rd edition (Wiley, New York, 1968), ch. III
  * (3) W. Feller, An introduction to probability theory and its applications, volume II, 2nd edition (Wiley, New York, 1971), ch. XII
  * (4) R. Mansuy and M. Yor, Aspects of Brownian Motion (Springer, Berlin, 2008), ch. 8.
  * (5) S. Redner, A guide to first-passage processes (Cambridge University Press, Cambridge, 2001)
  * (6) S. N. Majumdar, Current Science, 89, 2076 (2005)
  * (7) J. Akahori, Ann. Appl. Prob. 2, 383 (1995)
  * (8) A. Dassios, Ann. Appl. Prob. 5, 389 (1995)
  * (9) S. N. Majumdar and A. Comtet, Phys. Rev. Lett. 89, 060601 (2002)
  * (10) E. Barkai, J. Stat. Phys. 123, 883 (2006)
  * (11) W. Wang, J. H. P. Schulz, W. Deng, and E. Barkai, Phys. Rev. E 98, 042139 (2018)
  * (12) S. N. Majumdar and A. J. Bray, Phys. Rev. E 65, 051112 (2002)
  * (13) J. du Buisson and H. Touchette, Phys. Rev. E 102, 012148 (2020)
  * (14) P. T. Nyawo and H. Touchette, EPL 116, 50009 (2016)
  * (15) P. T. Nyawo and H. Touchette, Phys. Rev. E 98, 052103 (2018)
  * (16) J. A. McFadden, IRE Trans. Inf. Theor. 2, 146 (1956)
  * (17) G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amsterdam, 1994), section 5.6d
  * (18) W. Feller, An introduction to probability theory and its applications, volume II, 2nd edition (Wiley, New York, 1971) ch. XVIII
  * (19) D. S. Grebenkov, Phys. Rev. E 76, 041139 (2007)
  * (20) N. G. van Kampen, Stochastic processes in physics and chemistry (Elsevier, Amsterdam, 2006)
  * (21) M. Kac, Trans. Amer. Math. Soc. 65, 1 (1949)
  * (22) L. D. Landau and E. M. Lifshitz, Course of theoretical physics, III: Quantum mechanics : non-relativistic theory (Butterworth-Heinemann, Oxford, 2005), §22
  * (23) This is one of the virtues of separating $i\bar{p}$ from $E$ in the ansatz (20).