# A First Look at First-Passage Processes

S. Redner<EMAIL_ADDRESS>Santa Fe Institute, 1399 Hyde Park Road, Santa
Fe, New Mexico 87501, USA

###### Abstract

These notes are based on the lectures that I gave (virtually) at the Bruneck
Summer School in 2021 on first-passage processes and some applications of the
basic theory. I begin by defining what is a first-passage process and
presenting the connection between the first-passage probability and the
familiar occupation probability. Some basic features of first passage on the
semi-infinite line and a finite interval are then discussed, such as splitting
probabilities and first-passage times. I also treat the fundamental connection
between first passage and electrostatics. A number of applications of first-
passage processes are then presented, including the hitting probability for a
sphere in greater than two dimensions, reaction rate theory and its extension
to receptors on a cell surface, first-passage inside an infinite absorbing
wedge in two dimensions, stochastic hunting processes in one dimension, the
survival of a diffusing particle in an expanding interval, and finally the
dynamics of the classic birth-death process.

††journal: Physica A

##  1 What is a First-Passage Process?

The _first-passage probability_ is defined as the probability that a diffusing
particle or a random walk _first_ reaches a given site (or set of sites) at a
specified time. Typical examples of first-passage processes include:
fluorescence quenching, in which light emission by a fluorescent molecule
stops when it reacts with a quencher; integrate-and-fire neurons, in which a
neuron fires when a fluctuating voltage level first reaches a specified level;
and the execution of buy/sell orders when a stock price first reaches a
threshold. To appreciate why first-passage phenomena might be relevant
practically, consider the following example. You are an investor who buys
stock in a company at a price of $100. Suppose that this price fluctuates
daily by $\pm\$1$. You will sell if the stock price reaches $200 and if the
stock price reaches $0, the company has gone bankrupt and you’ve lost all your
investment. What it the probability of doubling your investment or losing your
entire investment? How long will it take before one of these two events
occurs? These are the types of questions that are the purview of first-passage
phenomena. Much of the material covered here is discussed more detail in this
monograph [1], and in other general reviews and texts on probability theory
and stochastic processes [2, 3, 4, 5].

##  2 First-Passage and Occupation Probabilities

Let’s start by deriving the formal relation between the first-passage
probability and the familiar occupation probability. For concreteness,
consider a random walk in discrete space and discrete time. We define
$P(\mathbf{r},t)$ as the occupation probability; this is the probability that
a random walk is at site $\mathbf{r}$ at time $t$ when it starts at the
origin. Similarly, let $F(\mathbf{r},t)$ be the first-passage probability,
namely, the probability that the random walk _first_ visits $\mathbf{r}$ at
time $t$ with the same initial condition. Clearly $F(\mathbf{r},t)$ decays
more rapidly in time than $P(\mathbf{r},t)$ because once a random walk reaches
$\mathbf{r}$, there can be no further contribution to $F(\mathbf{r},t)$,
although the same walk may still contribute to $P(\mathbf{r},t)$.

Figure 1: Schematic diagrammatic relation between the occupation probability
of a random walk (whose propagation is represented by a wavy line) and the
first-passage probability (straight line).

It is convenient to write $P(\mathbf{r},t)$ in terms of $F(\mathbf{r},t)$ and
then invert this relation to find $F(\mathbf{r},t)$. For a random walk to be
located at $\mathbf{r}$ at time $t$, the walk must first reach $\mathbf{r}$ at
some earlier time step $t^{\prime}$ and then return to $\mathbf{r}$ after
$t-t^{\prime}$ additional steps (Fig. 1). This connection between
$F(\mathbf{r},t)$ and $P(\mathbf{r},t)$ may be expressed as the convolution

$P(\mathbf{r},t)=\delta_{\mathbf{r},0}\delta_{t,0}+\sum_{t^{\prime}\leq
t}F(\mathbf{r},t^{\prime})\,P(0,t-t^{\prime})\,,$ (2.1)

where $\delta_{t,0}$ is the Kronecker delta function. This equation expresses
the fact that if a random walk is at $\mathbf{r}$ at time $t$, it must have
first reached $\mathbf{r}$ at some earlier time $t^{\prime}$ (which could even
be $t$). If the walk reached $\mathbf{r}$ at a time earlier than $t$, then it
must return to $\mathbf{r}$ (and any number of such returns could occur) in
the remaining time $t-t^{\prime}$. The probability for this set of events is
expressed by $P(0,t-t^{\prime})$. The delta function term accounts for the
initial condition that the walk starts at $\mathbf{r}=0$.

The above convolution is most conveniently solved by introducing the
generating functions,

$\displaystyle
P(\mathbf{r},z)=\sum_{t=0}^{\infty}P(\mathbf{r},t)z^{t}\,,\qquad
F(\mathbf{r},z)=\sum_{t=0}^{\infty}F(\mathbf{r},t)z^{t}\,.$

For a random walk in continuous time, we would merely replace the sum over
discrete time in Eq. (2.1) by an integral and then use the Laplace transform.
However, the asymptotic results would be identical. To solve for the first-
passage probability, we multiply Eq. (2.1) by $z^{t}$ and sum over all $t$. We
thereby find that the generating functions for $P$ and $F$ are related by

$P(\mathbf{r},z)=\delta_{\mathbf{r},0}+F(\mathbf{r},z)P(0,z)\,.$ (2.2)

Thus we obtain the fundamental connection between the generating functions

$F(\mathbf{r},z)=\begin{cases}{\displaystyle\frac{P(\mathbf{r},z)}{P(0,z)}}&\mathbf{r}\neq
0\,,\\\\[11.38109pt] {\displaystyle
1-\frac{1}{P(0,z)}},&\mathbf{r}=0\,.\end{cases}$ (2.3)

The important point is that the first-passage probability can be determined
solely from the occupation probability. Many profound results about random
walks in infinite space can be obtained from the fundamental relation (2.3)
(see, e.g., [6, 7, 8, 9]). Our focus here will be on random walks or diffusion
in confined geometries that reflect important physical constraints.

##  3 The Infinite Half Line

Suppose that a diffusing particle starts at $x_{0}>0$ on the infinite half
line $[0,\infty]$ and is absorbed when it reaches the origin. Does the
particle ever reach the origin? If so, when does this particle _first_ reach
the origin? To answer these questions, we have to solve the diffusion equation
for the concentration $c(x,t)$, subject to the initial condition
$c(x,t\\!=\\!0)=\delta(x-x_{0})$, and the boundary condition
$c(x\\!=\\!0,t\\!>\\!0)=0$; the latter enforces the absorption of the particle
when it reaches the origin. Finally we have to compute the diffusive flux to
the origin. A standard approach to solve this problem is to first take the
Laplace transform of the diffusion equation and then solve for the Green’s
function of this transformed equation. Then one inverts the Laplace transform
of the Green’s function to finally obtain the diffusive flux to the origin,
which coincides with the first-passage probability to the origin.

Figure 2: Concentration profile of a diffusing particle on the absorbing
infinite half line $(0,\infty)$ at $Dt=10$, with $x_{0}=2$ (solid curve). Also
shown are the component Gaussian (blue dashed) and image anti-Gaussian (red
dot-dash) and their superposition in the physical region $x>0$.

A more fun way to solve this problem is by invoking the familiar image method
from electrostatics. In this method, a diffusing particle that starts at
$x_{0}>0$ and is subject to an absorbing boundary condition at $x=0$ is
equivalent to removing the boundary altogether and introducing an image
“antiparticle” that is initially at $x=-x_{0}$. Because the concentration is
clearly equal to zero at the origin by symmetry, this image antiparticle
effectively imposes the absorbing boundary condition $c(x\\!=\\!0,t)=0$. The
initial particle and the image antiparticle both diffuse freely on
$[-\infty,\infty]$ and their superposition gives a resultant concentration
$c(x,t)$ for $x>0$ that solves the original problem.

Hence the concentration for a diffusing particle on the positive half-line is
the sum of a Gaussian centered at $x_{0}$ and an anti-Gaussian centered at
$-x_{0}$:

$c(x,t)=\frac{1}{\sqrt{4\pi
Dt}}\left[e^{-(x-x_{0})^{2}/4Dt}-e^{-(x+x_{0})^{2}/4Dt}\right].$ (3.1)

This concentration profile has a linear dependence on $x$ near the origin and
a Gaussian tail for $x/\sqrt{Dt}\gg 1$, as illustrated in Fig. 2. Because the
initial condition is normalized, the first-passage probability to the origin
at time $t$ is just the diffusive flux to this point. From the above
expression for $c(x,t)$, we find

$\displaystyle F(0,t)=+D\frac{\partial c(x,t)}{\partial
x}\bigg{|}_{x=0}=\frac{x_{0}}{\sqrt{4\pi Dt^{3}}}\,\,e^{-x_{0}^{2}/4Dt}\qquad
t\to\infty\,.$ (3.2)

This fundamental and simple formula has a number of striking implications:

  1. 1.

The particle is sure to reach to the origin because
$\int_{0}^{\infty}F(0,t)\,dt=1$. That is, _eventual absorption is certain_.

  2. 2.

The _average_ time for the particle to reach the origin is infinite! This fact
arises because the first-passage probability has the long-time algebraic tail
$F(0,t)\to x_{0}/\sqrt{4\pi Dt^{3}}$ for $t\gg x_{0}^{2}/D$. This dichotomy
between hitting the origin with certainty but taking an infinite average time
to do so underlies many of the intriguing features of one-dimensional
diffusion.

  3. 3.

The _typical_ time to reach the origin is finite. We can define the term
typical time in a precise way as follows: As a preliminary, define the typical
position of the particle, $x_{T}$, as

$\displaystyle\int_{x_{T}}^{\infty}c(x,t)\,dx=\tfrac{1}{2}\,.$ (3.3)

That is, one half of the total probability lies in the range beyond $x_{T}$
and one half lies in the range $[0,x_{T}]$. Substituting the concentration
profile (3.1) into the above integral and performing the integral leads to the
transcendental equation

$\displaystyle\text{erfc}\left(\frac{x_{T}-x_{0}}{\sqrt{4Dt}}\right)-\text{erfc}\left(\frac{x_{T}+x_{0}}{\sqrt{4Dt}}\right)=1\,.$
(3.4)

This equation can only be solved numerically and the salient feature is that
$x_{T}$ monotonically decreases with time and reaches zero at a time that is
roughly $1.1\,x_{0}^{2}/D$; this defines the typical hitting time.

  4. 4.

Even though the average time to reach the origin is infinite, the number of
times the origin is reached in a time $t$ is proportional to $\sqrt{t}$. This
result follows directly from the probability distribution of a freely
diffusing particle. At large times, the bulk of Gaussian distribution for a
particle that starts at $x_{0}$ will spread past the origin. Each site that is
within the Gaussian envelope will have been visited of the order of $\sqrt{t}$
times. This last fact will be relevant for our discussion of the reaction rate
of the sphere in Sec. 6.

##  4 The Finite Interval

We now turn to the first-passage properties in a finite interval. The reason
for focusing on the finite interval is that the basic first-passage questions
in this geometry have many profound implications. Moreover, the interval
geometry is sufficiently simple that many results can be readily derived. Let
us begin by outlining the basic questions that we will address. We consider a
diffusing particle that starts at some point $x_{0}$ within the interval
$[0,L]$, with absorbing boundary conditions at both ends of the interval.
Eventually the particle is absorbed, and our goal is to characterize the time
dependence of this absorption. Basic first-passage questions include:

  1. 1.

What is the time dependence of the survival probability $S(t)$? This is the
probability that a diffusing particle does not touch either absorbing boundary
before time $t$.

  2. 2.

What is the time dependence of the first-passage, or exit probabilities, to
either 0 or to $L$ as a function of $x_{0}$? Integrating these probabilities
over all time gives the eventual hitting, or _splitting probability_ to a
specified boundary. What is the dependence of the splitting probability to 0
or to $L$ as a function of the starting position?

  3. 3.

What is the average exit time, that is, the average time until the particle
hits either of the absorbing boundaries as a function of starting position?
What are the _conditional_ exit times, that is, the average time to hit a
specified boundary (without ever touching the other boundary) as a function of
the starting position?

To answer the first question, we need to solve the diffusion equation in the
interval, subject to the initial condition that a particle starts at $x_{0}$,
and with absorbing boundary conditions at both ends. This is a standard
exercise and the result for the concentration is

$c(x,t)=\sum_{n\geq 1}^{\infty}\frac{2}{L}\sin\frac{n\pi
x_{0}}{L}\,\sin\frac{n\pi x}{L}\;e^{-n^{2}\pi^{2}\,Dt/L^{2}}~{}.$ (4.1)

Since the large-$n$ eigenmodes decay more rapidly in time, the most slowly
decaying eigenmode dominates in the long-time limit. As a result, the survival
probability, which is the spatial integral of the concentration over the
interval, asymptotically decays as

$S(t)\sim e^{-D\pi^{2}t/L^{2}}\,.$ (4.2)

For answering the second question, it is mathematically simpler to work in the
Laplace transform domain. Applying the Laplace transform to the diffusion
equation recasts it as

$sc(x,s)-c(x,t=0)=Dc^{\prime\prime}(x,s)\,,$ (4.3)

where the prime denotes differentiation with respect to $x$. The argument $s$
indicates that $c(x,s)$ is the Laplace transform. Within the standard Green’s
function approach, the homogeneous equation in each subdomain $(0,x_{0})$ and
$(x_{0},L)$ has elementary solutions of the form
$c(x,s)=A\exp(x\sqrt{{s/D}})+B\exp(-x\sqrt{{s/D}})$, with the constants $A$
and $B$ determined by the boundary conditions. Because the absorbing boundary
condition at $x=0$ mandates an antisymmetric combination of exponentials and
because the form of the Green’s function as $x\to L$ must be identical to that
as $x\to 0$, we can be immediately write

$\displaystyle\begin{split}c_{<}(x,s)&=A\sinh\left(\sqrt{\frac{s}{D}}\,x\right)\hskip
36.135ptx<x_{0}\,,\\\\[5.69054pt]
c_{>}(x,s)&=B\sinh\left(\sqrt{\frac{s}{D}}(L-x)\right)\quad
x>x_{0}\,,\end{split}$ (4.4)

for the subdomain Green’s functions $c_{<}$ and $c_{>}$, where $A$ and $B$ are
constants.

We now impose the continuity condition $c_{<}(x_{0},s)=c_{>}(x_{0},s)$ and the
jump condition that is obtained by integrating Eq. (4.3) over an infinitesimal
interval that includes $x_{0}$:

$c^{\prime}(x,s)\big{|}_{x=x_{0}^{+}}-c^{\prime}(x,s)\big{|}_{x=x_{0}^{-}}=-1/D\,,$

to finally obtain

$c(x,s)=\frac{\displaystyle\sinh\left(\sqrt{\frac{s}{D}}\,x_{<}\right)\sinh\left(\sqrt{\frac{s}{D}}(L-x_{>})\right)}{\displaystyle\sqrt{sD}\sinh\left(\sqrt{\frac{s}{D}}\,L\right)}\,.$
(4.5)

From this Green’s function, the Laplace transform of the fluxes at $x=0$ and
at $x=L$ are

$\displaystyle j_{-}(s|x_{0})$ $\displaystyle\equiv+D\frac{\partial
c(x,s)}{\partial
x}\Bigg{|}_{x=0}={\displaystyle\sinh\left({\sqrt{\frac{s}{D}}}(L-x_{0})\right)}\bigg{/}{\displaystyle\sinh\left({\sqrt{\frac{s}{D}}}\,L\right)}\,.$
(4.6a) $\displaystyle j_{+}(s|x_{0})$ $\displaystyle\equiv-D\frac{\partial
c(x,s)}{\partial
x}\Bigg{|}_{x=L}={\displaystyle\sinh\left(\sqrt{\frac{s}{D}}\,x_{0}\right)}\bigg{/}{\displaystyle\sinh\left(\sqrt{\frac{s}{D}}\,L\right)}\,.$
(4.6b)

The subsidiary argument $x_{0}$ in $j$ emphasizes that the flux depends on the
initial particle position. Since the initial condition is normalized, the
magnitude of the flux to each boundary is identical to the respective first-
passage probabilities.

For $s=0$, these Laplace transforms are just the time-integrated first-passage
probabilities to 0 and at $L$. These quantities therefore coincide with the
respective splitting probabilities, ${\mathcal{E}}_{-}(x_{0})$ and
${\mathcal{E}}_{+}(x_{0})$, namely, the probabilities to eventually hit the
left and the right ends of the interval as a function of the initial position
$x_{0}$:

$\displaystyle\begin{split}{\mathcal{E}}_{-}(x_{0})&=j_{-}(s\\!=\\!0|x_{0})=1-\frac{x_{0}}{L}\,,\\\\[5.69054pt]
{\mathcal{E}}_{+}(x_{0})&=j_{+}(s\\!=\\!0|x_{0})=\frac{x_{0}}{L}\,.\end{split}$
(4.7)

Thus the splitting probabilities are given by an amazingly simple formula—the
probability of reaching one endpoint is just the fractional distance to the
other endpoint!

It is instructive to also derive these splitting probabilities by the backward
Kolmogorov approach [1, 2]. The word backward reflects the feature that the
initial condition becomes the dependent variable, rather than the current
position of the particle. As we shall see, this method provides a powerful
tool for determining first-passage properties. Physically, we obtain the
eventual hitting probability ${\mathcal{E}}_{+}(x_{0})$ to the right boundary
by summing the probabilities for all paths that start at $x_{0}$ and reach $L$
without touching $0$. Thus

${\mathcal{E}}_{+}(x_{0})=\sum_{\text{paths}}\Pi_{x_{0}\to L}\,,$ (4.8)

where $\Pi_{x_{0}\to L}$ denotes the probability of a path from $x_{0}$ to $L$
that avoids $0$. As illustrated in Fig. 3, the sum over all such paths can be
decomposed into the outcome after one step and the sum over all path
remainders from the intermediate point $x^{\prime}$ to $L$. This gives

$\displaystyle{\mathcal{E}}(x_{0})=\tfrac{1}{2}\sum_{\text{paths}^{\prime}}\Pi_{x_{0}+\delta
x\to L}+\tfrac{1}{2}\,\sum_{\text{paths}^{\prime\prime}}\Pi_{x_{0}-\delta x\to
L}=\tfrac{1}{2}[{\mathcal{E}}_{+}(x_{0}\\!+\\!\delta
x)+{\mathcal{E}}_{+}(x_{0}\\!-\\!\delta x)]\,.$ (4.9)

Here, paths′ and paths′′ indicate, respectively, all paths that start at
$x_{0}\pm\delta x$ and reach $L$ without touching 0.

Figure 3: Schematic decomposition of a random walk path from $x_{0}$ to $L$
into the outcome after one step and the remainder from $x^{\prime}$ to $L$.
The factors $1/2$ account for the probabilities associated with the first step
of the decomposed paths.

The right-hand side of (4.9) reduces to
$\Delta^{(2)}{\mathcal{E}}_{\pm}(x_{0})=0$, where $\Delta^{(2)}$ is the
discrete second-difference operator, $\Delta^{(2)}f(x)\equiv f(x-\delta
x)-2f(x)+f(x+\delta x)$. This difference equation is subject to the boundary
conditions ${\mathcal{E}}_{+}(0)=0$, ${\mathcal{E}}_{+}(L)=1$. The solution is
simply

$\displaystyle{\mathcal{E}}_{+}(x_{0})=\frac{x}{L}\,,$ (4.10)

and correspondingly ${\mathcal{E}}_{-}(x)=1-\frac{x}{L}$.

It is worth mentioning that there is an even simpler way to determine the exit
probabilities by the martingale method [10]. This solution relies on the fact
that the motion of the random walk in the interval is a “fair game” at any
time. Thus the average position of the walk is time independent. At $t=0$,
this average position is $\langle x\rangle=x_{0}$. At infinite time, the walk
is either at the left end or the right end of the interval, with respective
probabilities ${\mathcal{E}}_{+}(x_{0})$ or ${\mathcal{E}}_{-}(x_{0})$. Thus
the average position at infinite time is $\langle
x\rangle=0\times{\mathcal{E}}_{-}(x_{0})+L\times{\mathcal{E}}_{+}(x_{0})$.
Since the initial average position equals the final average position, we
immediately recover (4.10).

For a biased random walk with a probability $p$ of hopping to the right and
probability $q$ of hopping to the left, the analog of Eq. (4.9) for the
splitting probability is

$\displaystyle{\mathcal{E}}(x_{0})=p{\mathcal{E}}_{+}(x_{0}\\!+\\!\delta
x)+q{\mathcal{E}}_{+}(x_{0}\\!-\\!\delta x)\,,$ (4.11)

with solution

$\displaystyle{\mathcal{E}}_{+}(x_{0})=\frac{1-e^{-vx_{0}/D}}{1-e^{-vL/D}}\equiv\frac{1-e^{-u_{0}P\\!e}}{1-e^{-P\\!e}}\,,$
(4.12)

where $v=(p-q)\delta x$, $D=\delta x^{2}/2$, $u_{0}=x_{0}/L$, and $P\\!e=vL/D$
is the _Péclet_ number, which is a dimensionless measure of the influence of
the bias relative to diffusive fluctuations. When the Péclet number is large,
the exit probability clearly reflects the strong influence of the bias (Fig.
4).

Figure 4: Dependence of the exit probability to $x=L$ on $u_{0}=x_{0}/L$ in
the interval $[0,L]$ for various Péclet numbers $P\\!e$.

We now extend the backward Kolmogorov approach to determine the average exit
time from the finite interval. We distinguish between the unconditional
average exit time, namely, the average time for a particle to reach _either_
end of the interval, and the conditional average exit time, namely, the
average time for a particle to reach, say, the right end of the interval
without ever touching the other end.

In close analogy with (4.8), the unconditional time satisfies

$t(x_{0})=\sum_{\text{paths}}(\Pi t)_{x_{0}\to\pm L}\,.$ (4.13)

That is, to compute the average unconditional exit time, we take the time for
a path to go from $x_{0}$ to $L$ times the probability of this path and sum
over all possible paths. Using the same decomposition that which led to (4.9),
the unconditional exit satisfies

$\displaystyle t(x_{0})$
$\displaystyle=\tfrac{1}{2}\\!\\!\sum_{\text{paths}^{\prime}}\\!\\!\Pi_{x_{0}+\delta
x\to\pm L}(t_{x_{0}+dx\to\pm
L}\\!+\\!dt)+\tfrac{1}{2}\\!\\!\sum_{\text{paths}^{\prime\prime}}\\!\\!\Pi_{x_{0}-\delta
x\to\pm L}(t_{x_{0}-dx\to\pm L}\\!+\\!dt)$
$\displaystyle=\tfrac{1}{2}[t(x_{0}\\!+\\!\delta x)+t(x_{0}\\!-\\!\delta
x)]+dt\,.$ (4.14)

In the continuum limit, (4) simplifies to $D\,t^{\prime\prime}=-1$, subject to
the boundary conditions $t(0)=t(L)=0$. The solution is (see Fig. 5)

$t(x_{0})=\frac{x_{0}(L-x_{0})}{2D}\,.$ (4.15)

Notice that this exit time is of the order of $L$ for a particle that starts a
distance of the order of one from an absorbing boundary and of the order of
$L^{2}$ for a particle that starts near the middle of the interval.

Figure 5: Unconditional and conditional average exit times from the finite
interval $[0,L]$, normalized by $L^{2}/2D$, as a function of the dimensionless
initial position $x_{0}/L$.

Now let’s calculate the _conditional_ exit time to the right boundary when
starting from $x_{0}$, $t_{+}(x_{0})$. By definition this conditional time is

$\displaystyle t_{+}(x_{0})=\frac{\sum_{\text{paths}_{+}}(\Pi t)_{x_{0}\to
L}}{\sum_{\text{paths}_{+}}\Pi_{x_{0}\to L}}=\frac{\sum_{\text{paths}_{+}}(\Pi
t)_{x_{0}\to L}}{{\mathcal{E}}_{+}(x_{0})}\,.$ (4.16)

That is, the conditional exit time is the time for a path to $x_{0}=L$ without
touching 0 times the probability of this path, summed over all such allowable
paths. Here the subscript + on the word paths indicates that only paths that
go from $x_{0}$ to $L$ without touching 0 are included. Since the total
probability of all these restricted paths is less than 1, we need to divide by
this total probability, ${\mathcal{E}}_{+}(x_{0})$, to obtain the properly
normalized conditional exit time. Thus, for the quantity
${\mathcal{E}}_{+}(x_{0})t_{+}(x_{0})$, we have

$\displaystyle{\mathcal{E}}_{+}(x_{0})t_{+}(x_{0})$
$\displaystyle=\sum_{\text{paths}_{+}}(\Pi t)_{x_{0}\to L}$
$\displaystyle=\tfrac{1}{2}\\!\\!\sum_{\text{paths}^{\prime}_{+}}\Pi(t_{x_{0}+dx\to
L}+dt)+\tfrac{1}{2}\\!\\!\sum_{\text{paths}^{\prime\prime}_{+}}\Pi(t_{x_{0}-dx\to
L}+dt)$
$\displaystyle=\tfrac{1}{2}\left[{\mathcal{E}}_{+}(x_{0}\\!+\\!dx)t_{+}(x_{0}\\!+\\!dx)\right]+\tfrac{1}{2}\left[{\mathcal{E}}_{+}(x_{0}\\!-\\!dx)t_{+}(x_{0}\\!-\\!dx)\right]$
$\displaystyle\hskip 158.99377pt+{\mathcal{E}}_{+}(x_{0})\,dt\,.$ (4.17)

In the continuum limit, the above equation reduces to
$D({\mathcal{E}}_{+}t_{+})^{\prime\prime}=-{\mathcal{E}}_{+}$, with solution

$\displaystyle t_{+}(x_{0})=\frac{L^{2}-x_{0}^{2}}{6D}\,.$ (4.18)

From this result, we also immediately find $t_{-}(x_{0})=t_{+}(L-x_{0})$. The
dependences of all the exit times on the starting position are illustrated in
Fig. 5.

##  5 Connection with Electrostatics

One of the alluring features of first-passage processes is its intimate
connection to electrostatics. By this connection, one can recast an
electrostatic problem in a given geometry as a first-passage problem in the
same geometry. With this perspective, it is possible to solve seemingly
difficult first-passage problems in a simple way by this electrostatic
connection.

To illustrate the basic principle, consider the following general problem.
Suppose that a diffusing particle starts at some point $\mathbf{r}_{0}$ inside
an arbitrary bounded domain. At the boundary of this domain, the particle is
absorbed. Eventually, all of the initial probability is absorbed on the
boundary and we ask: what is the exit probability at some arbitrary point
$\mathbf{r}_{B}$ on the domain boundary? Formally, we have to solve the
diffusion equation in this domain, subject to the appropriate initial and
boundary conditions:

$\displaystyle\begin{cases}\displaystyle{\frac{\partial
c(\mathbf{r},t)}{\partial t}}=D\nabla^{2}c(\mathbf{r},t)\\\\[5.69054pt]
c(\mathbf{r},t=0)=\delta(\mathbf{r}-\mathbf{r}_{0})\\\\[2.84526pt]
c(\mathbf{r},t)\big{|}_{\mathbf{r}_{B}}=0\end{cases}$ (5.1)

Then the exit probability at the boundary point $\mathbf{r}_{B}$ is given by

$\displaystyle\mathcal{E}(\mathbf{r}_{B})=-D\int_{0}^{\infty}\frac{\partial
c(\mathbf{r},t)}{\partial\hat{\mathbf{n}}}\bigg{|}_{\mathbf{r}_{B}}\,dt\,,$
(5.2)

where $\hat{\mathbf{n}}$ is the outward normal to the surface of the domain at
$\mathbf{r}_{B}$.

Let’s look critically at this calculation. We are attempting to solve a
partial differential equation in some domain (which may well be difficult),
then take the result of this calculation and integrate over all time. That is,
we really don’t need that exit probability at all times, but merely the time
integral of the exit probability. This observation suggests that it will be
useful to take the original problem (5.1) an integrate it over all time. To
simplify what emerges, we also define the time integrated concentration,
$\mathcal{C}(\mathbf{r})\equiv\int_{0}^{\infty}c(\mathbf{r},t)\,dt$.
Performing this time integration on Eq. (5.1) leads to

$\displaystyle\begin{cases}-\delta(\mathbf{r}-\mathbf{r}_{0})=D\nabla^{2}\mathcal{C}(\mathbf{r})\\\
\mathcal{C}(\mathbf{r})\big{|}_{\mathbf{r}_{B}}=0\end{cases}$

The delta function on the left-hand side is what remains when we integrate the
time derivative in (5.1) over all time. At $t=\infty$ the concentration is
zero, while at $t=0$, we merely have the initial condition. But notice that in
terms of the time-integrated concentration, the exit probability may be
written as

$\displaystyle\mathcal{E}(\mathbf{r}_{B})=-D\frac{\partial\mathcal{C}(\mathbf{r},t)}{\partial\hat{\mathbf{n}}}\bigg{|}_{\mathbf{r}_{B}}\,.$
(5.3)

Thus we arrive at the fundamental result:

$\displaystyle\mathcal{E}(\mathbf{r}_{B})$ $\displaystyle=\text{the\ electric\
field\ at\ }\mathbf{r}_{B}\text{\ on the surface of a grounded conductor,}$
$\displaystyle\hskip 14.45377pt\text{when a point charge of magnitude\
}1/(D\Omega_{d})\text{\ is placed at\ }\mathbf{r}_{0}\,.$

Here $\Omega_{d}$ is the surface area of a $d$-dimensional sphere; this factor
is needed to convert the prefactor $D$ in the diffusion equation to the
correct prefactor in the Laplace equation. Thus a given first-passage problem
can be expressed as an equivalent electrostatic problem in the same geometry.

##  6 Hitting a Sphere and Reaction Rate Theory

What is the probability $H(r)$ that a diffusing particle eventually hits a
sphere of radius $a$, when the particle starts at a distance $r>a$ from the
origin? One can determine this hitting probability in the standard way by
solving the diffusion equation exterior to the sphere, computing the flux to
the sphere, and then integrating over all time. However, it is much simpler to
use the connection between first passage and electrostatics. Indeed, by a
direct extension of Eq. (4.9) to three dimensions, the hitting probability
satisfies (here for a discrete random walk in Cartesian coordinates for
simplicity)

$\displaystyle H(x,y)$
$\displaystyle=\tfrac{1}{6}\big{[}H(x+1,y,z)+H(x-1,y,z)+H(x,y+1,z)$
$\displaystyle\hskip 36.135pt+H(x,y-1,z)+H(x,y,z+1)+H(x,y,z-1)\big{]}\,.$
(6.1)

Let us now take the continuum limit so that we can work in spherical
coordinates. Then the above discrete difference equation becomes
$\nabla^{2}H=0$, subject to the boundary conditions $H(r\\!=\\!a)=1$ and
$H(\infty)=0$. The solution is

$\displaystyle H(r)=\frac{a}{r}\,.$ (6.2)

Amazingly simple!

Now let’s treat a related problem that is fundamental in chemical kinetics.
Suppose that there is an initially uniform concentration of particles exterior
to an absorbing sphere of radius $a$. A fundamental kinetic characteristic of
the absorbing sphere is its _reaction rate_ $k$, namely, the efficiency at
which this sphere captures particles. Formally, $k$ is defined as the number
of particles absorbed per unit time divided by the initial concentration. This
normalization ensures that the reaction rate is a quantity that is intrinsic
to the system. By dimensional analysis, the reaction rate as defined above has
units of (length)${}^{d}/$time. Moreover the reaction rate can only be a
function of intrinsic parameters of the system, namely, the diffusion
coefficient $D$ and the sphere radius $a$. Since the units of $D$ are
length2/time, we infer that $k$ must have units of $D\,a^{d-2}$. Thus the
reaction rate $k=A\,D\,a^{d-2}$, where $A$ is a constant of the order of 1.
This example shows the power of dimensional analysis in obtaining a non-
trivial physical property of multi-particle system.

This simple result has some surprising implications. First, in three
dimensions, the reaction rate is _linear_ in $a$; it is _not_ proportional to
the cross-sectional area of the sphere. Second for $d<2$, the reaction rate
increases when the radius of the absorbing sphere is decreased! This
nonsensical result indicates that there is a basic problem with classic
chemical kinetics for $d<2$. This pathology arises because the concentration
field exterior to the absorber never reaches a steady state for $d<2$.
Instead, the absorption rate of the sphere is time dependent. In contrast, for
$d>2$, a steady state concentration field does arise. Once this steady state
is reached, it is a trivial exercise to compute the steady-state density by
solving the Laplace equation rather than the diffusion equation, and thereby
obtain the steady-state flux to the sphere. From these steps, one finds the
exact reaction rate

$\displaystyle k=4\pi Da\,.$ (6.3)

Armed with the basic result for the reaction rate, we now turn to a much more
profound problem that is fundamental to living systems, namely, how many
receptors should there be on the surface of a cell? One might expect that much
of the cell surface should be covered by receptors so that its detection
efficiency is high. On the other hand, one can imagine that receptors are
complex and evolutionarily expensive machines. Based on cost considerations
only, it would be advantageous for a cell to minimize the number of receptors.
What is the appropriate balance between these two competing attributes? As a
first step to address this question, we want to compute the reaction rate of a
cell that is sensitive to its environment only at the locations of the
receptors. We model the cell as a sphere of radius $a$ in which most of the
surface is reflecting. However, on the sphere surface there are also $N$
circular domains of radius $s$ that are absorbing (Fig. 6). We view these $N$
absorbing circles as the receptors on the cell surface. What is the reaction
rate of this toy model of a cell? If most of the sphere surface is reflecting,
one might anticipate that the reaction efficiency of the cell will be poor.
Surprisingly, the reaction efficiency of the sphere with $N$ absorbing
receptors is almost as good as a perfectly absorbing sphere, even when the
area fraction covered by the receptors is vanishingly small! This realization
is the brilliant insight of the article by Berg and Purcell that was far ahead
of its time [11]. Here I outline their argument.

Figure 6: Cartoon of a cell surface with 4 receptors (gray ovals). A diffusing
particle first hits the cell (green trajectory) and then makes 3 non-
independent hits of the surface (blue), before an independent hit (red). The
particle lands on a receptor after one more non-independent hit.

The first step in their argument relies on the feature that if a diffusing
particle hits the surface of a sphere, it will hit again many times before
diffusing away; this point was discussed above in Sec. 3. There are two types
of subsequent hitting events: (i) the particle rises a distance less than $s$
above the surface before hitting it again, and (ii) the particle rises a
distance greater than $s$ above the surface before hitting it again (Fig. 6).
In the former case, if the particle initially misses a receptor, it will
likely miss upon the second encounter. In the latter case, if the particle
misses a receptor initially, we have no information about whether the second
encounter will hit or miss a receptor. Thus the rise distance $s$ demarcates
the regime of _dependent_ subsequent hits and _independent_ subsequent hits.
It is the latter events that are relevant for estimating the reaction rate. We
thus use the height $s$ as the criterion for determining the number of times
that a particle independently hits the cell surface before diffusing away.

When a particle is a distance $s$ above the cell surface, the probability that
it eventually hits the cell again is, using Eq. (6.2),

$\displaystyle p_{s}=\frac{a}{a+s}\,.$ (6.4a) Thus the probability that the
diffusing particle independently hits the cell $n$ times before diffusing away
is $p_{s}^{n}(1-p_{s})$. Correspondingly, the average number of independent
hits to the surface is $\displaystyle\langle
n\rangle=\sum_{n>0}np_{s}^{n}\;(1-p_{s})=(1-p_{s})^{-1}=\frac{a+s}{s}\sim\frac{a}{s}\,.$
(6.4b)

The probability for a diffusing particle to not land on a receptor in a single
independent hitting event is

$\displaystyle\beta=1-\frac{N\pi s^{2}}{4\pi a^{2}}\,,$ (6.4c)

namely, the area fraction of the surface that is not covered by receptors.
Thus the probability $p_{\rm escape}$ that a diffusing particle that reaches
the cell but never lands on a receptor is the probability that the particle
always misses a receptor in each of its independent hitting attempts. This is

$\displaystyle p_{\rm escape}=\sum_{n>0}(\beta
p_{s})^{n}(1-p_{s})=\frac{1-p_{s}}{1-\beta p_{s}}=\frac{4a}{4a+Ns}\,.$ (6.4d)

The probability that a diffusing particle ultimately hits a receptor thus is

$\displaystyle p_{\rm hit}=1-p_{\rm escape}=\frac{Ns}{4a+Ns}\,.$ (6.4e)

The final result for the reaction rate of a cell of radius $a$ that is covered
by $N$ receptors, each of radius $s$, is

$\displaystyle k=4\pi Da\times\frac{Ns}{4a+Ns}\equiv 4\pi
Da\times\text{efficiency}\,.$ (6.5)

To understand the implication of this result, let’s use some numbers that
typify a cell: $a=1$ micron, $s=10$ nanometers, and $N=10^{4}$. The area
fraction covered by the receptors is roughly $10^{-3}$, but the absorption
efficiency of the cell is roughly 2/3! Evidently, Mother Nature is very smart
to not waste resources on endowing a cell with too many receptors.

##  7 Wedge Domains

We now turn to the first-passage properties of diffusion in a two-dimensional
wedge domain with absorption when the particle hits the wedge boundary. One of
our motivations for studying this system is that first passage in the wedge
can be mapped onto a simple diffusive capture problem in one dimension that
we’ll treat in the next section. We will obtain first-passage properties in
the wedge geometry both by direct solution of the diffusion equation and also,
for two dimensions, in a more aesthetically pleasing fashion by conformal
transformation techniques, in conjunction with the electrostatic formulation.
We also present a heuristic extension of the electrostatic approach that
allows us to infer, with little additional computational effort, time-
dependent first-passage properties in the wedge from corresponding time-
integrated properties.

### Solution to the Diffusion Equation

We first solve the diffusion equation in the wedge to determine the survival
probability of a diffusing particle. While the exact Green’s function for this
system is well known [12, 13], we adopt the strategy of choosing an initial
condition that allows us to eliminate angular variables and deal with an
effective radial problem. This simplification is appropriate if we are
interested only in asymptotic first-passage properties.

The diffusion equation for the two-dimensional wedge geometry in plane polar
co-ordinates is

$\frac{\partial c}{\partial t}=D\left(\frac{\partial^{2}c}{\partial
r^{2}}+\frac{1}{r}\frac{\partial c}{\partial
r}+\frac{1}{r^{2}}\frac{\partial^{2}c}{\partial\theta^{2}}\right),$ (7.1)

where $c=c(r,\theta,t)$ is the particle concentration at $(r,\theta)$ at time
$t$, $D$ is the diffusion coefficient, and the boundary conditions are $c=0$
at $\theta=0,\Theta$, where $\Theta$ is the wedge opening angle. To reduce
this two-dimensional problem to an effective one-dimensional radial problem,
note that the exact Green’s function can be written as an eigenfunction
expansion in which the angular dependence is a sum of sine waves of the form
$\sin(n\pi\theta/\Theta)$, such that an integral number of half-wavelengths
fit within $(0,\Theta)$ to satisfy the absorbing boundary conditions [13]. In
this series, each sine wave is multiplied by a conjugate decaying function of
time, in which the decay rate increases with $n$. In the long time limit, only
the lowest term in this expansion dominates the survival probability.
Consequently, we obtain the long-time behavior by choosing an initial
condition whose angular dependence is a half sine-wave in the wedge. This
ensures that the time-dependent problem will contain only this single term in
the Fourier series.

We therefore define $c(r,\theta,t=0)=\pi\delta(r-r_{0})/(2\Theta
r_{0})\times\sin(\pi\theta/\Theta)$. With this initial distribution function
and after the Laplace transform is applied, the diffusion equation (7.1)
becomes

$\displaystyle sc(r,\theta,s)-\frac{\pi}{2\Theta
r_{0}}\delta(r-r_{0})\sin(\pi\theta/\Theta)=D\left(\frac{\partial
c^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial c}{\partial
r}+\frac{1}{r^{2}}\frac{\partial^{2}c}{\partial\theta^{2}}\right)\,,$

where $c=c(r,\theta,s)$. Substituting in the ansatz
$c(r,\theta,s)=R(r,s)\sin(\pi\theta/\Theta)$, the angular dependence may now
be separated and reduces the system to an effective one-dimensional radial
problem. By introducing the dimensionless co-ordinate $x=r\sqrt{s/D}$, we find
the modified Bessel equation for the remaining radial co-ordinate,

$R^{\prime\prime}(x,s)+\frac{1}{x}R^{\prime}(x,s)-\left(1+\frac{\nu^{2}}{x^{2}}\right)R(x,s)=-\frac{\nu}{2Dx_{0}}\delta(x-x_{0}),$
(7.2)

where $\nu=\pi/\Theta$ and the prime now denotes differentiation with respect
to $x$.

The general solution for $x\neq x_{0}$ is a superposition of modified Bessel
functions of order $\nu$. Since the domain is unbounded, the interior Green’s
function ($x<x_{0}$) involves only $I_{\nu}$, since $K_{\nu}$ diverges as
$x\to 0$, while the exterior Green’s function ($x>x_{0}$) involves only
$K_{\nu}$, since $I_{\nu}$ diverges as $x\to\infty$. By imposing continuity at
$x=x_{0}$, we find that the Green’s function has the symmetric form
$R(x,s)=AI_{\nu}(x_{<})K_{\nu}(x_{>})$, with the constant $A$ determined by
the joining condition that arises from integrating Eq. (7.2) over an
infinitesimal radial range that includes $r_{0}$. This gives

$\displaystyle
R_{>}^{\prime}\mid_{x=x_{0}}-R_{<}^{\prime}\mid_{x=x_{0}}=-\frac{\nu}{2Dx_{0}}\,,$

from which $A=\nu/2D$. Therefore the radial Green’s function in the wedge is

$R(x,s)=\frac{\nu}{2D}\,I_{\nu}(x_{<})\,K_{\nu}(x_{>}),$ (7.3)

and its Laplace inverse has the relatively simple closed form [12, 13]

$R(r,t)=\frac{\nu}{4Dt}\,\,e^{-(r^{2}+r_{0}^{2})/4Dt}\,\,I_{\nu}\left(\frac{rr_{0}}{2Dt}\right).$
(7.4)

With this radial Green’s function, the asymptotic survival probability is

$S(t)\sim\int_{0}^{\Theta}\sin(\nu\theta)\,d\theta\,\int_{0}^{\infty}r\,R(r,t)\,dr.$
(7.5)

We can estimate this integral by noting that the radial distance over which
the concentration is appreciable extends to the order of $\sqrt{Dt}$. This
provides a cutoff $r\approx\sqrt{Dt}$ in the radial integral in Eq. (7.5),
within which the Gaussian factors in $R(r,t)$ can be replaced by one. Using
the small-argument expansion of the Bessel function, we then obtain

$\displaystyle
S(t)=\int_{0}^{\Theta}\\!\\!\\!\sin(\nu\theta)\,d\theta\,\int_{0}^{\infty}\\!\\!\\!r\,R(r,t)\,dr\propto\int_{0}^{\sqrt{Dt}}\\!\frac{1}{Dt}\left(\frac{rr_{0}}{Dt}\right)^{\nu}\,r\,dr\sim\left(\frac{r_{0}}{\sqrt{Dt}}\right)^{\pi/\Theta}.$
(7.6)

The basic result is that the survival probability of a diffusing particle in a
wedge of opening angle $\Theta$ decays with time as

$S(t)\sim t^{-\pi/2\Theta}\equiv t^{-\alpha}.$ (7.7)

The striking feature of this formula is that the exponent $\alpha$ depends on
the wedge opening angle in a non-trivial way, with $\alpha\to\infty$ as
$\Theta\to 0$ and $\alpha\to 1/4$ for $\Theta\to 2\pi$.

### Conformal Transformations and Electrostatic Methods

Let’s now solve the same wedge problem by exploiting conformal
transformations, together with the connection between first passage and
electrostatics. To set the stage for the wedge geometry, consider the first-
passage probability for a diffusing particle in two dimensions to an absorbing
infinite line. This problem may also be solved elegantly by the electrostatic
formulation. In this approach, the time-integrated concentration
$\mathcal{C}(x,y)=\int_{0}^{\infty}c(x,y,t)\,dt$ obeys the Laplace equation

$\displaystyle\nabla^{2}\mathcal{C}(z)=-\frac{1}{2\pi D}\;\delta(z-z_{0})\,,$

where $z=(x,y)$ is the complex co-ordinate and the factor $1/(2\pi D)$ ensures
the correct normalization. Using the image method for two-dimensional
electrostatics, we find that the complex potential is

$\mathcal{C}(z)=\frac{1}{2\pi D}\ln\frac{z-z_{0}}{z-z_{0}^{*}}=\frac{1}{2\pi
D}\ln\frac{x-x_{0}+i(y-y_{0})}{x-x_{0}+i(y+y_{0})},$ (7.8)

where the asterisk denotes complex conjugation. Finally, the time-integrated
flux that is absorbed at $x$ coincides with the electric field at this point.
This is

$\displaystyle{\mathcal{E}}(x|x_{0},y_{0})$
$\displaystyle=-D\,\frac{\partial\mathcal{C}(x,y)}{\partial y}\bigg{|}_{y=0}$
$\displaystyle=\frac{1}{2\pi}\left(\frac{1}{y_{0}+i(x-x_{0})}+\frac{1}{y_{0}-i(x-x_{0})}\right)$
$\displaystyle=\frac{1}{\pi}\frac{y_{0}}{(x-x_{0})^{2}+y_{0}^{2}}\,.$ (7.9)

We now use a conformal transformation to extend the result for the hitting
probability to the infinite line to the hitting probability in the wedge.
Consider the transformation $w=f(z)=z^{\pi/\Theta}$ that maps the interior of
the wedge of opening angle $\Theta$ to the upper half plane. In complex co-
ordinates, the electrostatic potential in the wedge is

$\mathcal{C}(z)=\frac{1}{2\pi}\ln\frac{z^{\pi/\Theta}-z_{0}^{\pi/\Theta}}{z^{\pi/\Theta}-(z_{0}^{*})^{\pi/\Theta}}.$
(7.10)

From this expression and using the analogy between electrostatics and first
passage, we can extract time-integrated first-passage properties in the wedge.
For example, the probability of being absorbed at a distance $x$ from the
wedge apex, when a particle begins at a unit distance from the apex along the
wedge bisector, is just the electric field at this point

$\displaystyle{\mathcal{E}}(x|x_{0},y_{0})=\frac{1}{\Theta}\frac{x^{\pi/\Theta-1}}{1+x^{2\pi/\Theta}}\to
x^{-(1+\pi/\Theta)}\qquad\text{for~{}}x\to\infty\,.$ (7.11)

Although the electrostatic formulation ostensibly gives only time-integrated
first-passage properties, we can adapt it to also give time-dependent
features. This adaptation is based on the following re-interpretation of the
equivalence between electrostatics and diffusion: an electrostatic system with
a point charge and specified boundary conditions is identical to a diffusive
system in the same geometry and boundary conditions, in which a continuous
source of particles is fed in at the location of the charge starting at time
$t=\\-\infty$. Suppose now that the particle source is “turned on” at $t=0$.
Then, in a near zone that extends out to a distance of the order of
$\sqrt{Dt}$ from the source, the concentration has sufficient time to reach
its steady-state value. Within this zone, the diffusive solution converges to
the Laplacian solution. Outside this zone, however, the concentration is close
to zero. This almost-Laplacian solution provides the time integral of the
survival probability up to time $t$. We can then deduce the survival
probability by differentiating the concentration that is due to this finite-
duration source.

Thus suppose that a constant source of diffusing particles at $z_{0}$ inside
the absorbing wedge is turned on at $t=0$. Within the region where the
concentration has had time to reach the steady state, $|z_{0}|<|z|<\sqrt{Dt}$,
the density profile is approximately equal to the Laplacian solution,
$\mathcal{C}(z)\sim|z|^{-\pi/\Theta}$. We can neglect the angular dependence
of $\mathcal{C}(z)$ in this zone, as this dependence is immaterial for the
survival probability. Conversely, for $|z|>\sqrt{Dt}$, the particle
concentration is vanishingly small because a particle is unlikely to diffuse
such a large distance. From the analogy between electrostatics and first
passage, the near-zone density profile is just the same as the time integral
of the diffusive concentration. Thus, by using the equivalence between the
spatial integral of this near-zone concentration in the wedge and the time
integral of the survival probability, we have

$\int_{0}^{t}S(t^{\prime})\,dt^{\prime}\approx\int_{0}^{\sqrt{Dt}}r^{-\pi/\Theta}\,r\,dr\sim
t^{1-\pi/2\Theta}.$ (7.12)

Since the total density injected into the system equals $t$, the survival
probability in the wedge is roughly
$\int_{0}^{t}S(t^{\prime})\,dt^{\prime}/t\sim t^{1-\pi/2\Theta}/t$, which
gives $S(t)\sim t^{-\pi/2\Theta}$.

##  8 Stochastic Hunting in One Dimension

What is the time dependence of the survival probability of a diffusing lamb
that is hunted by $N$ diffusing lions? We define this survival probability as
$S_{N}(t)$. This toy problem is most interesting in a one-dimensional geometry
where all the lions are located to one side of the lamb. It is known [14] that
this survival probability asymptotically decays algebraically with time,

$\displaystyle S_{N}(t)\sim t^{-\beta_{N}}\,,$ (8.1)

and the goal is to compute the decay exponent $\beta_{N}$. As we shall
discuss, the decay exponent is known for $N=1$ and $N=2$ only:
$\beta_{1}=\frac{1}{2}$, and $\beta_{2}=\frac{3}{4}$. For $N\geq 3$,
$\beta_{N}$ grows slowly with $N$, and numerical simulations give
$\beta_{3}\approx 0.91$, $\beta_{4}\approx 1.03$, and $\beta_{10}\approx 1.4$.
The focus of this section is to derive $\beta_{2}=\frac{3}{4}$ by a simple
geometric approach and to develop some analytical understanding of the
dependence of $\beta_{N}$ on $N$ for $N\to\infty$.

Let us begin by treating a lamb that starts at $x_{0}>0$ and a single lion
that starts at $x=0$. For simplicity, the diffusivities of the lamb and the
lion are assumed to both equal $D$. The separation between the lamb and the
lion thus diffuses with diffusion coefficient $2D$. When this separation
reaches zero, the lamb has been eaten. This problem is just the classic first-
passage problem on the positive infinite line, except that the diffusion
coefficient is $2D$. The probability that the lamb survives until time $t$ is
the same as the first-passage time being greater than $t$. This probability
therefore is

$\displaystyle S_{1}(t)$
$\displaystyle=\int_{t}^{\infty}\\!\frac{x_{0}}{\sqrt{8\pi Dt^{\prime
3}}}\;e^{-x_{0}^{2}/8Dt^{\prime}}\,dt^{\prime}=\text{erf}\biggl{(}\frac{x_{0}}{\sqrt{8Dt}}\biggr{)}\sim\frac{x_{0}}{\sqrt{8\pi
Dt}}\,.$ (8.2)

Thus the survival probability of the lamb asymptotically decays as $t^{-1/2}$.
While the lamb is sure to die, its average lifetime is infinite. Thus a single
diffusing lion is not a particularly good hunter and it might starve before
eating the lamb.

What happens when there are $N=2$ lions? We again assume that the lions start
from the origin while the lamb starts at $x_{0}>0$, and that the diffusivities
of all particles are the same. Let us label the positions of the lions as
$x_{1}$ and $x_{2}$, and the position of the lamb as $x_{3}$ The lamb survives
up to time $t$ if the conditions $x_{3}>x_{2}$ and $x_{3}>x_{1}$ always hold.
We can give an insightful geometric interpretation of this problem by viewing
the motion of the three particles on the line as equivalent to the motion of a
single effective particle in three dimensions with coordinates
$(x_{1},x_{2},x_{3})$. The constraints $x_{3}>x_{2}$ and $x_{3}>x_{1}$ mean
that the effective particle in three-space remains to the left of the plane
$x_{2}=x_{3}$ and behind the plane $x_{1}=x_{3}$ (Fig. 7(a)). This allowed
region is a wedge of opening angle $\Theta$ that is defined by the
intersection of these two planes. If the particle hits one of the planes, then
one of the lions has eaten the lamb.

Figure 7: (a) The allowed region for the effective particle that corresponds
to the motion of the lamb and two lions. (b) A view of the constraint planes
perpendicular to the (1,1,1) axis. The constraint plane $x_{1}=x_{2}$ is also
shown

This mapping therefore provides the lamb survival probability, since the
survival probability of a diffusing particle within this absorbing wedge
asymptotically decays as $S_{\rm wedge}(t)\sim t^{-\pi/2\Theta}$. What is the
opening angle of this wedge? We can determine this angle in a simple way by
also including the plane $x_{1}=x_{2}$ and then viewing the system along the
$(1,1,1)$ axis (Fig. 7(b)). It is then clear that the wedge angle is $2\pi/3$.
Substituting this result in Eq. (7.7), we find that $\beta_{2}=\frac{3}{4}$.
Notice that $S_{2}(t)>S_{1}(t)^{2}$. This inequality reflects the fact that
the incremental threat to the lamb from the second lion is less than the
first.

In general, we can map the motion of the lamb and $N$ lions in one dimension
to a single effective particle in $N+1$ dimensions, with absorption when the
effective particle hits any of the $N$ constraint planes. However, the
calculation of the survival probability of the effective particle within the
domain where the effective particle is confined—known as a Weyl
chamber—appears to be intractable.

Figure 8: Space-time trajectories of 6 lions and 1 lamb. Although the motion
of each lion is isotropic, the trajectory of the lion (whose identity may
change) that happens to be closest to the lamb (solid red curve) tends to move
towards the lamb.

While the problem for $N=3,4,\ldots$ lions is difficult, the problem becomes
much simpler for large $N$. To determine the lamb survival probability for
large $N$, we only need to focus on the lion closest to the lamb, because this
last lion ultimately kills the lamb. As shown in Fig. 8, the individual
identity of this last lion can change with time due to the crossing of
different lion trajectories. For large $N$, there is a systematic bias of the
motion of the last lion, $x_{\rm last}(t)$. This bias becomes stronger for
increasing $N$, so that $x_{\rm last}(t)$ becomes smoother as $N$ increases
(Fig. 9). This gradual approach of the last lion to the lamb is the mechanism
which leads to the survival probability of the lamb decaying as
$t^{-\beta_{N}}$, with $\beta_{N}$ a slowly increasing function of $N$.

Figure 9: Position of the last lion versus time for 4, 64, and 1024 lions.

To estimate the location of this last lion when $N\gg 1$ lions are initially
at the origin, we use the extreme statistics condition [15]

$\int_{x_{\rm last}}^{\infty}\frac{N}{\sqrt{4\pi
Dt}}\,e^{-x^{2}/4Dt}\,\,dx=1.$ (8.3)

Equation (8.3) states that one lion out of an initial group of $N$ is in the
range $[x_{\rm last},\infty]$. Although the integral in Eq. (8.3) can be
expressed in terms of the complementary error function, it is instructive to
evaluate it explicitly by writing $x=x_{\rm last}+\epsilon$ and re-expressing
the integrand in terms of $\epsilon$. We thus find

$\int_{0}^{\infty}\frac{N}{\sqrt{4\pi Dt}}\,e^{-x_{\rm
last}^{2}/4Dt}\,e^{-x_{\rm
last}\epsilon/2Dt}\,e^{-\epsilon^{2}/4Dt}\,\,d\epsilon=1.$

Over the range of $\epsilon$ for which the second exponential factor is non-
negligible, the third exponential factor is nearly equal to one. The integral
now reduces to an elementary form, with the result

$\frac{N}{\sqrt{4\pi Dt}}\,e^{-x_{\rm last}^{2}/4Dt}\,\frac{2Dt}{x_{\rm
last}}=1.$ (8.4)

We now define $y=x_{\rm last}/\sqrt{4Dt}$ and $M=N/\sqrt{4\pi}$, so that the
above condition can be simplified to $y\exp(e^{y^{2}})=M$, whose asymptotic
solution is

$y=\sqrt{\ln M\,}\Bigl{(}1-\frac{1}{4}\frac{\ln(\ln M)}{\ln
M}+\ldots\Bigr{)}.$

To lowest order, this gives

$x_{\rm last}(t)\sim\sqrt{4D\ln N\,t}\equiv\sqrt{A_{N}t},$ (8.5)

for $N$ finite. For $N=\infty$, $x_{\rm last}(t)$ would always equal $t$ if an
infinite number of discrete random walking lions were initially at the origin.
A more suitable initial condition therefore is a concentration $c_{0}$ of
lions that are uniformly distributed from $-\infty$ to 0. In this case, only
$N\propto\sqrt{c^{2}_{0}Dt}$ of the lions are “dangerous,” that is, within a
diffusion distance from the edge of the pack and thus potential candidates for
eating the lamb. Consequently, for $N\to\infty$, the leading behavior of
$x_{\rm last}(t)$ becomes

$x_{\rm last}(t)\sim\sqrt{2D\ln(c^{2}_{0}Dt)\,t}\,.$ (8.6)

An important feature of the time dependence of $x_{\rm last}$ is that
fluctuations in this quantity decrease for large $N$ (Fig. 9). Therefore the
lamb and $N$ diffusing lions can be recast as a two-body system of a lamb and
an absorbing boundary which deterministically advances toward the lamb
according to $x_{\rm last}(t)=\sqrt{A_{N}t}$. This determinism is what makes
the problem for large $N$ tractable.

To solve this effective two-body problem, it is convenient to change
coordinates from $(x,t)$ to $(^{\prime}=x-x_{\rm last}(t),t)$ to fix the
absorbing boundary at the origin. By this construction, the diffusion equation
for the lamb probability distribution is transformed to the convection-
diffusion equation

$\frac{\partial p(x^{\prime},t)}{\partial t}-\frac{x_{\rm
last}}{2t}\frac{\partial p(x^{\prime},t)}{\partial
x^{\prime}}=D\frac{\partial^{2}p(x^{\prime},t)}{\partial x^{\prime
2}}\,,\quad(0\leq x^{\prime}<\infty)$ (8.7)

with the absorbing boundary condition $p(x^{\prime}=0,t)=0$. In this reference
frame that is fixed on the average position of the last lion, the second term
in Eq. (8.7) accounts for the bias of the lamb towards the absorber with an
effective speed $-x_{\rm last}/2t$. Because $x_{\rm last}\sim\sqrt{A_{N}t}$
and $x^{\prime}\sim\sqrt{Dt}$ have the same time dependence, the lamb survival
probability acquires a nontrivial dependence on the dimensionless parameter
$A_{N}/D$. This behavior arises whenever there is a coincidence of fundamental
length scales in the system (see, e.g., [16] for other such examples).

Equation (8.7) can be transformed into the parabolic cylinder equation by
first introducing the dimensionless length $\xi=x^{\prime}/x_{\rm last}$ and
making the following scaling ansatz for the lamb probability density,

$p(x^{\prime},t)\sim t^{-\beta_{N}-1/2}\mathcal{C}(\xi).$ (8.8)

The power law prefactor in Eq. (8.8) ensures that the integral of
$p(x^{\prime},t)$ over all space, namely the survival probability, decays as
$t^{-\beta_{N}}$, and $\mathcal{C}(\xi)$ expresses the spatial dependence of
the lamb probability distribution in scaled length units. This ansatz codifies
the fact that the probability density is not a function of $x^{\prime}$ and
$t$ separately, but is a function only of the dimensionless ratio
$x^{\prime}/x_{\rm last}(t)$.

Substituting Eq. (8.8) into Eq. (8.7), we obtain

$\frac{D}{A_{N}}\frac{d^{2}{\mathcal{C}}}{d\xi^{2}}+\frac{1}{2}(\xi+1)\frac{d\mathcal{C}}{d\xi}+\bigl{(}\beta_{N}+\frac{1}{2}\bigr{)}\mathcal{C}=0.$
(8.9)

By introducing $\eta=(\xi+1)\sqrt{A_{N}/2D}$ and
$\mathcal{C}(\xi)=e^{-\eta^{2}/4}\,{\mathcal{D}}(\eta)$ in Eq. (8.9), we are
led to the parabolic cylinder equation of order $2\beta_{N}$ [17]

$\frac{d^{2}{\mathcal{D}}_{2\beta_{N}}}{d\eta^{2}}+\biggl{[}2\beta_{N}+\frac{1}{2}-\frac{\eta^{2}}{4}\biggr{]}{\mathcal{D}}_{2\beta_{N}}=0,$
(8.10)

subject to the boundary condition, ${\mathcal{D}}_{2\beta_{N}}(\eta)=0$ for
both $\eta=\sqrt{A_{N}/2D}$ and $\eta=\infty$. Equation (8.10) has the form of
a Schrödinger equation for a quantum particle of energy
$2\beta_{N}+\frac{1}{2}$ in a harmonic oscillator potential $\eta^{2}/4$ for
$\eta>\sqrt{A_{N}/2D}$, but with an infinite barrier at $\eta=\sqrt{A_{N}/D}$
[18]. For the long-time behavior, we need to find the ground state energy in
this potential. For $N\gg 1$, we may approximate this energy as the potential
at the classical turning point, that is,
$2\beta_{N}+\frac{1}{2}\simeq\eta^{2}/4$. We therefore obtain $\beta_{N}\sim
A_{N}/16D$. Using the value of $A_{N}$ given in Eqs. (8.5) and (8.6) the decay
exponent is

$\beta_{N}\sim\begin{cases}\tfrac{1}{4}\ln N&N~{}\text{finite}\\\\[5.69054pt]
\tfrac{1}{8}\ln t&N=\infty\,.\end{cases}$ (8.11)

The latter dependence of $\beta_{N}$ implies that for $N\to\infty$, the
survival probability has the log-normal form

$S_{\infty}(t)\sim\exp\left(-\tfrac{1}{8}\,\ln^{2}t\right).$ (8.12)

The important feature of the exponent $\beta_{N}$ is its very slow increase
with $N$. That is, each successive lion that is added to the hunt has a
decreasing influence on the survival of the lamb. Indeed, only a small subset
of the lions for large $N$ that actually have a chance to catch the lamb.

##  9 The Expanding Interval

We have seen that the survival probability $S(t)$ of a diffusing particle in a
fixed-length absorbing interval of length $L$ asymptotically decays as
$e^{-\pi^{2}Dt/L^{2}}$. What happens if the interval length grows with time,
$L(t)\sim t^{\alpha}$? This simple question illustrates the relative effects
of diffusion and the motion of the boundary on first-passage properties. This
interplay is a classic problem in the first-passage literature, especially
when the boundary motion matches that of diffusion. Solutions to this problem
have been obtained by a variety of methods (see, e.g., [19, 20, 21, 22]). Here
we give a physics-based approach that is based on [23].

It is easy to infer the survival probability for a slowly expanding interval.
Here, slowly expanding means that the interval length grows slower than
diffusion. Consequently, the probability distribution of the particle spreads
faster than the interval grows and so that the survival probability should
decay rapidly with time. Using an adiabatic approximation that one typically
encounters in basic quantum mechanics, we will show that
$S(t)\sim\exp\big{[}-At^{(1-2\alpha)}\big{]}$, where $A$ is a constant.
Conversely, for the rapidly expanding interval, $\alpha>1/2$, the particle is
unlikely reach the either end of the interval and the probability distribution
is close to that for free diffusion. This is the basis of the _free
approximation_ that leads to a non-zero limiting value for $S(t)$ as
$t\to\infty$.

In the marginal case where the interval expands at the same rate as diffusion,
$L(t)=\sqrt{At}$, a new dimensionless parameter arises—the ratio of the
diffusion length to the interval length. As we shall show, this leads to
$S(t)$ decaying as a non-universal power-law in time, $S(t)\sim t^{-\beta}$,
with $\beta=\b{(}A,D)$ diverging for $A/D\ll 1$ and approaching zero for
$A/D\gg 1$.

### Slowly Expanding Interval

For $L(t)\ll\sqrt{Dt}$, we invoke the adiabatic approximation [18], in which
the spatial dependence of the concentration for an interval of length $L(t)$
is assumed to be identical to that of the static diffusion equation at the
instantaneous value of $L$. This assumption is based on the expectation that
the concentration in a slowly expanding interval is always close to that of a
fixed-size interval. Thus we write

$c(x,t)\simeq f(t)\,\cos\left(\frac{\pi x}{2L(t)}\right)\equiv c_{\rm
ad}(x,t),$ (9.1)

with $f(t)$ to be determined. The corresponding survival probability is

$S(t)\approx\int_{-L(t)}^{L(t)}~{}c_{\rm
ad}(x,t)\,dx=\frac{4}{\pi}~{}f(t)L(t).$ (9.2)

For convenience, we now define the interval boundaries as $[-L(t),L(t)]$. To
obtain $f(t)$, we substitute approximation (9.1) into the diffusion equation,
as in separation of variables, to give

$\frac{df}{dt}=-\left(\frac{D\pi^{2}}{4L^{2}}\right)f-\left(\frac{\pi
x}{2L^{2}}\right)\,\frac{dL}{dt}\,\tan\left(\frac{\pi x}{2L}\right)\,f.$ (9.3)

Notice that variable separation does not strictly hold, since the equation for
$f(t)$ also involves $x$. However, when $L(t)$ increases as $(At)^{\alpha}$
with $\alpha<1/2$, the second term on the right-hand side is negligible. Thus
we drop this second term and solve the simplified form of (9.3). We thereby
find that the controlling factor of $f(t)$ is given by

$\displaystyle
f(t)\sim\exp\left[-\frac{D\pi^{2}}{4}\int_{0}^{t}\,\frac{dt^{\prime}}{L^{2}(t^{\prime})}\right]=\exp\left[-\frac{D\pi^{2}}{4(1-2\alpha)A^{2\alpha}}\,t^{1-2\alpha}\right].$
(9.4)

Notice that $f(t)$ reduces to a pure exponential decay for a fixed-length
interval, while for $\alpha\to 1/2$, Eq. (9.4) suggests a more slowly decaying
functional form for $S(t)$.

### Rapidly Expanding Interval

For a rapidly expanding interval, the escape rate from the system is small and
the absorbing boundaries should eventually become irrelevant. We therefore
expect that the concentration profile should approach the Gaussian
distribution of free diffusion at long times [23]. We may then account for the
slow decay of the survival probability by augmenting the Gaussian with an
overall decaying amplitude. This free approximation is a nice example in which
the existence of widely separated time scales, $\sqrt{Dt}$ and $L(t)$,
suggests the nature of the approximation itself.

According to the free approximation, we write

$\displaystyle c(x,t)\approx\frac{S(t)}{\sqrt{4\pi Dt}}\,e^{-x^{2}/4Dt}\equiv
c_{\rm free}(x,t)\,.$

Although this concentration does not satisfy the absorbing boundary condition,
the inconsistency is negligible at large times, since the density is
exponentially small at the interval boundaries. We may now find the time
dependence of the survival probability by equating the probability flux to the
interval boundaries, $2D|\frac{\partial c}{\partial x}|$, to the loss of
probability within the interval. For $L(t)=(At)^{\alpha}$, this flux is

$\frac{S(t)A^{\alpha}}{\sqrt{4\pi
D}}\;t^{\alpha-3/2}\,\exp\left(-\frac{A^{2\alpha}}{4D}t^{2\alpha-1}\right),$
(9.5)

which rapidly goes to zero for $\alpha>1/2$. Since this flux equals
$-\frac{dS}{dt}$, it follows that the survival probability approaches a non-
zero limiting value for $\alpha>1/2$, and that this limiting value goes to
zero as $\alpha\to 1/2$. Thus a diffusing particle has a non-zero probability
of surviving forever when the interval grows fast enough.

### Marginally Expanding Interval

For the marginal case of $\alpha=1/2$, the adiabatic and the free
approximations are ostensibly no longer appropriate, since $L(t)=\sqrt{At}$
and $\sqrt{Dt}$ have a fixed ratio. However, for $A/D\ll 1$ and $A/D\gg 1$, we
might hope that these methods could still be useful. Thus we continue to apply
these heuristic approximations in their respective domains of validity,
$A/D\ll 1$ and $A/D\gg 1$, and check their accuracies a posteriori. We will
see that the survival probability exponents predicted by these two
approximations are each quite close to the exact result except for $A/D\approx
1$.

When the adiabatic approximation is applied, the second term in Eq. (9.3) is,
in principle, non-negligible for $\alpha=1/2$. However, for $A\ll D$, the
interval still expands more slowly (in amplitude) than free diffusion and the
error made by neglecting the second term in Eq. (9.3) may still be small. The
solution to this crudely truncated equation immediately gives $f(t)$, which,
when substituted into approximately (9.2) leads to $S_{\rm ad}(t)\approx
t^{-\beta_{\rm ad}}$, with

$\beta_{\rm ad}\approx\frac{D\pi^{2}}{4A}-\frac{1}{2}.$ (9.6)

The trailing factor of $-1/2$ should not be taken very seriously, because the
neglected term in Eq. (9.3) leads to additional corrections to $\beta_{\rm
ad}$ that are also of the order of 1.

Figure 10: The survival probability exponent $\beta$ for the marginal case
$L(t)=(At)^{1/2}$. Shown is the numerically exact value of $\beta$ (red,
solid) from solution of Eq. (9.11), together with the predictions from the
adiabatic and the free approximations, Eqs.(9.6) and (9.8) respectively
(dashed curves).

Similarly for $A\gg D$, the free approximation gives

$\displaystyle\frac{dS}{dt}\approx-2D\bigg{|}\frac{\partial c(x,t)}{\partial
x}\bigg{|}_{x=\sqrt{At}}\;=\,-\frac{S(t)}{t}\sqrt{\frac{A}{4\pi
D}}\,e^{-A/4D}.$ (9.7)

This again leads to the non-universal power law for the survival probability,
$S_{\rm free}\sim t^{-\beta_{\rm free}}$, with

$\beta_{\rm free}=\sqrt{\frac{A}{4\pi D}}\,e^{-A/4D}.$ (9.8)

As shown in Fig. 10, these approximations are surprisingly accurate over much
of the range of $A/D$.

To complete our discussion, we outline a first-principles analysis for the
survival probability of a diffusing particle in a marginally expanding
interval [23]. When $L(t)=\sqrt{At}$, a natural scaling hypothesis is to write
the density in terms of the two dimensionless variables

$\displaystyle\xi\equiv\frac{x}{L(t)}\qquad{\rm
and}\qquad\rho=\sqrt{\frac{A}{D}}\,.$

We now seek solutions for the concentration in the form,

$c(x,t)=t^{-\beta-1/2}\;\mathcal{C}_{\rho}(\xi),$ (9.9)

where $\mathcal{C}_{\rho}(\xi)$ is a two-variable scaling function that
encodes the spatial dependence. The power law prefactor ensures that the
survival probability, namely, the spatial integral of $c(x,t)$, decays as
$t^{-\beta}$, as defined at the outset of this section.

After substituting Eq. (9.9) into the diffusion equation, the scaling function
satisfies the ordinary differential equation

$\displaystyle\frac{1}{\rho^{2}}\frac{d^{2}\mathcal{C}}{d\xi^{2}}+\frac{\xi}{2}\frac{d\mathcal{C}}{d\xi}+\left(\beta+\frac{1}{2}\right)\mathcal{C}=0\,.$

Then by introducing $\eta=\xi\,\sqrt{\rho/2}$ and
$\mathcal{C}(\xi)=e^{-\eta^{2}/4}\,{\mathcal{D}}(\eta)$, we transform this
into the parabolic cylinder equation [25]

$\frac{d{\mathcal{D}}}{d\eta^{2}}+\left[2\beta+\frac{1}{2}-\frac{\eta^{2}}{4}\right]{\mathcal{D}}=0.$
(9.10)

When the range of $\eta$ is unbounded, this equation has solutions for
quantized values of the energy eigenvalue $E=2\beta+\frac{1}{2}=\frac{1}{2}$,
$\frac{3}{2}$, $\frac{5}{2},\ldots$ [18].

For our interval problem, the range of $\eta$ is restricted to
$|\eta|\leq\sqrt{A/2D}$. In the equivalent quantum mechanical system, this
corresponds to a particle in a harmonic-oscillator potential for
$|\eta|<\sqrt{A/2D}$ and an infinite potential for $|\eta|>\sqrt{A/2D}$. For
this geometry, a spatially symmetric solution to Eq. (9.10), appropriate for
the long-time limit for an arbitrary starting point, is

$\displaystyle{\mathcal{D}}(\eta)\equiv\tfrac{1}{2}\big{[}{\mathcal{D}}_{2\beta}(\eta)+{\mathcal{D}}_{2\beta}(-\eta)\big{]}\,,$

where ${\mathcal{D}}_{\nu}(\eta)$ is the parabolic cylinder function of order
$\nu$. Finally, the relation between the decay exponent $\beta$ and
$\sqrt{A/D}$ is determined implicitly by the absorbing boundary condition,
namely,

${\mathcal{D}}_{2\beta}(\sqrt{A/2D})+{\mathcal{D}}_{2\beta}(-\sqrt{A/2D})=0.$
(9.11)

This condition for $\beta=\beta(A/D)$ simplifies in the limiting cases $A/D\ll
1$ and $A/D\gg 1$. In the former, the exponent $\beta$ is large and the second
two terms in the brackets in Eq. (9.10) can be neglected. Equivalently, the
physical range of $\eta$ is small, so that the potential plays a negligible
role. The solution to this limiting free-particle equation is just the cosine
function, and the boundary condition immediately gives the limiting expression
of Eq. (9.6), but without the subdominant term of $-1/2$. In the latter case
of $A\gg D$, $\beta\to 0$ and Eq. (9.10) approaches the Schrödinger equation
for the ground state of the harmonic oscillator. In this case, a detailed
analysis of the differential equation reproduces the limiting exponent of Eq.
(9.8) (see [23] for details). These provide rigorous justification for the
limiting values of the decay exponent $\beta$ which we obtained by heuristic
means.

### The Khintchine Iterated Logarithm Law

In the marginal situation of $L(t)=(At)^{1/2}$, we have seen that the survival
probability $S(t)$ decays as a power law $t^{-\beta(A,D)}$, with $\beta\to 0$
as $A/D\to\infty$. This decay becomes progressively slower as $A$ increases.
On the other hand, when $L(t)\propto t^{\alpha}$, with $\alpha$ strictly
greater than 1/2, the survival probability at infinite time is greater than
zero. This leads to the following natural question: what is the nature of the
transition between certain death, defined as $S(t\to\infty)=0$, and a non-zero
survival survival probability, $S(t\to\infty)>0$?

The answer to this question is surprisingly rich. There is an infinite
sequence of transitions, where $L(t)$ acquires additional iterated logarithmic
time dependences, which define regimes where $S(t)$ assumes progressively
slower functional forms. The first term in this series is known as the
Khintchine _iterated logarithm law_ ([24, 3]). While the Khintchine law has
been obtained by rigorous methods, we can also obtain this intriguing result,
as well as the infinite sequence of transitions, with relatively little
computation by the free approximation.

Because we anticipate that the transition between life and death occurs when
$L(t)$ grows slightly faster than $(At)^{1/2}$, we make the hypothesis that
$A\propto u(t)$, with $u(t)$ growing slower than a power law in $t$. Now that
$L(t)$ increases more rapidly than the diffusion length $(Dt)^{1/2}$, the free
approximation should be asymptotically exact, since it already works extremely
well when $L(t)=(At)^{1/2}$ with $A$ large. Within this approximation, we
rewrite Eq. (9.7) as

$\ln S(t)\sim-\int^{t}\frac{dt^{\prime}}{t^{\prime}}\sqrt{\frac{A}{4\pi
D}}\,e^{-A/4D}.$ (9.12)

Here we neglect the lower limit, since the free approximation is valid only as
$t\to\infty$, where the short-time behavior is irrelevant. In this form, it is
clear that for $L=(At)^{1/2}$, $\ln S$ decreases by an infinite amount for
$t\to\infty$ because of the divergence of the integral. Thus $S(t\to\infty)\to
0$. To make the integral converge, the other factors in the integral must
somehow cancel the logarithmic divergence that arises from the factor $dt/t$.
Accordingly, let us substitute $L(t)=\sqrt{4Dt\,u(t)}$ into the approximation
(9.12). This gives

$\ln
S(t)\sim-\frac{1}{\sqrt{\pi}}\int^{t}\frac{dt^{\prime}}{t^{\prime}}\sqrt{u(t^{\prime})}\,e^{-u(t^{\prime})}.$

To simplify this integral, it is helpful to define $x=\ln t$ so that

$S(x)\sim-\frac{1}{\sqrt{\pi}}\int^{\ln t}dx\,\sqrt{u(x)}\,e^{-u(x)}.$ (9.13)

To lowest order, it is clear that if we choose $u(x)=\lambda\ln x$ with
$\lambda>1$, the integral converges as $t\to\infty$. Thus the asymptotic
survival probability is positive. Conversely, for $\lambda\leq 1$, the
integral diverges and the particle surely dies. In this latter case,
evaluation of the integral to lowest order gives

$S(t)\sim\exp\left[-\frac{(\ln t)^{1-\lambda}\sqrt{\lambda\ln\ln
t}}{\sqrt{\pi}(1-\lambda)}\right]\qquad\lambda<1.$ (9.14)

This decay is slower than any power law, but faster than any power of
logarithm, that is, $t^{-\beta}<S(t)<(\ln t)^{-\gamma}$ for $\beta\to 0$ and
$\gamma\to\infty$.

What happens in the marginal case of $\lambda=1$? Here we can refine the
criterion between life and death still further by incorporating into $u(x)$ a
correction that effectively cancels the subdominant factor $\sqrt{u(x)}$ in
Eq. (9.13). We therefore define $u(x)$ such that $e^{-u(x)}=1/x(\ln x)^{\mu}$.
Then in terms of $y=\ln x$, Eq. (9.13) becomes

$S(y)\sim-\frac{1}{\sqrt{\pi}}\int^{\ln\ln
t}\frac{dy}{y^{\mu-1/2}}\left(1+\frac{\mu\ln y}{y}\right).$ (9.15)

This integral now converges for $\mu>3/2$ and diverges for $\mu\leq 3/2$. In
the latter case, the survival probability now lies between the bounds $(\ln
t)^{-\gamma}<S(t)<(\ln\ln t)^{-\delta}$ for $\gamma\to 0$ and
$\delta\to\infty$. At this level of approximation, we conclude that when the
cage length grows faster than

$aL^{*}(t)=\sqrt{4Dt\left(\ln\ln t+\tfrac{3}{2}\ln\ln\ln t+\ldots\right)}$
(9.16a) a diffusing particle has a non-zero asymptotic survival probability,
while for a interval that expands as $L^{*}$, there is an extremely slow decay
of the survival probability.

By incorporated successively finer corrections into $u(x)$ and following the
same logic that led to Eq. (9.15), an infinite series of correction terms can
be generated in the expression for $L^{*}(t)$. By this approach, the ultimate
life-death transition corresponds to an ultra-slow decay in which $S(t)$ has
the form $S(t)\sim\lim_{n\to\infty}1/\ln_{n}t$, where $\ln_{2}t\equiv\ln\ln t$
and $\ln_{n}t\equiv\ln\ln_{n-1}t$. It is remarkable that the physically
motivated and relatively naive free approximation can generate such an
intricate solution. As a final note, P. Erdös sharpened the result of Eq.
(9.16a) considerably and found that $L^{*}(t)$ has the infinite series
representation

$L^{*}(t)=\sqrt{4Dt\left(\ln_{2}t+\tfrac{3}{2}\ln_{3}t+\ln_{4}t+\ln_{5}t+\ldots\right)}\,,$
(9.16b)

in which only the coefficient of the term multiplying $\ln_{3}t$ is different
than 1.

##  10 Birth-Death Dynamics

As our last topic, we determine the kinetics of the _birth-death_ process. We
imagine a collection of identical particles, each of which gives birth to an
identical offspring with rate $\lambda$, and each particle can independently
die with rate $\mu$. The goal is to determine the time dependence of the
population size. As one can easily imagine, this is a classic model for a
variety of biological processes and there is vast literature on this general
topic (see, e.g., [26, 27]).

The most interesting case physically is the symmetric situation of equal birth
and death rates for each particle, $\lambda=\mu$, so that the average
population is static. For $\mu>\lambda$, the population size decreases as
$e^{-(\mu-\lambda)t}$, which quickly goes to zero. In the opposite case, the
population grows exponentially with time and an additional mechanism is need
to cut off this growth. For $\lambda=\mu$, the average population is fixed,
but the time dependence of the distribution of the number of particle exhibits
non-trivial kinetics on the positive infinite line. We can alternatively view
the birth-death process as a continuous-time random walk on the line, but with
birth and death rates for the entire population that are linear functions of
$n$. That is, the overall process is symmetric but moves faster for a larger
population.

Let $n$ denote the number of particles in the population. The time dependence
of the average number of particles obeys the rate equation
$\langle\dot{n}\rangle=(\lambda-\mu)\langle n\rangle=0$, where the overdot
denotes the time derivative. Thus the average number of particles is
conserved, as is clear from the condition $\lambda=\mu$. More meaningful
information is obtained from the full population distribution. For simplicity
in the ensuing formulas, we now set $\lambda=\mu=1$ without loss of
generality. Let $P_{n}(t)$ denote the probability that the population consists
of $n$ particles at time $t$. This probability distribution changes in time
according to

$\displaystyle\dot{P}_{n}=(n-1)P_{n-1}-2nP_{n}+(n+1)P_{n+1}\,,$ (10.1)

where we define $P_{-1}=0$, so that this equation is valid for all $n\geq 0$.
For the standard continuous-time random walk, the corresponding master
equation is $\dot{P}_{n}=P_{n-1}-2P_{n}+P_{n+1}$. We know that this random
walk eventually hits the origin, but that the average time to do so is
infinite. We want to find the behavior of these two first-passage properties
for the birth-death process.

A convenient and powerful way to solve the master equation (10.1) is by the
generating function method. We first define the generating function

$g(z,t)=\sum_{n\geq 0}P_{n}(t)z^{n}\,,$

then take each of the equations for $\dot{P}_{n}$, multiply it by $z^{n}$, and
then sum over all $n$. In doing so, we will encounter terms from the right-
hand side of (10.1), for example, the second term on the right, that looks
like

$\displaystyle\sum_{n\geq 0}2nP_{n}\,z^{n},$

which we can recast as

$\displaystyle 2z\frac{\partial}{\partial z}\sum_{n\geq
0}P_{n}\,z^{n}=2z\frac{\partial g}{\partial z}\,.$

By this device of converting multiplication by $n$ to differentiation for all
three terms on the right-hand side of (10.1), we recast (10.1) as

$g_{t}=(z^{2}-2z+1)g_{z}=(1-z)^{2}g_{z}\,,$ (10.2)

where the subscripts now denote partial differentiation and the arguments of
$g$ are not written for compactness.

This first-order partial differential equation can be simplified further by
defining the variable $y$ via $dy=dz/(1-z)^{2}$, which implies that
$y=1/(1-z)$, or equivalently, $z=1-y^{-1}$. In terms of the variable $y$,
(10.2) is converted to the classic wave equation $g_{t}=g_{y}$. This equation
has the general solution $g=F(t+y)$, where the function $F$ is, in principle,
arbitrary, and whose explicit form is fixed by the initial condition. Let us
specialize to the simple case of the single-particle initial condition,
namely, $P_{n}(t=0)=\delta_{n,1}$. This immediately leads to
$g(z,t\\!=\\!0)=z$. Then at $t=0$ the function $F$ is simply given by
$F(y)=z$. However, we must express the right-hand side in terms of the true
dependent variable $y$, which means that $F(y)=1-y^{-1}$. Thus for any $t\geq
0$ the generating function is

$\displaystyle g(z,t)=F(y+t)=1-\frac{1}{t+y}\,.$ (10.3)

To extract the individual terms in the power-series representation of the
generating function, we now need to re-express this function in terms of $z$:

$\displaystyle g(z,t)$
$\displaystyle=1-\frac{1}{t+1/(1-z)}=1-\frac{1-z}{(1+t)(1-zx)}$
$\displaystyle=1-\frac{1-z}{1+t}\sum_{n\geq 0}(xz)^{n}$
$\displaystyle=1-\frac{1}{1+t}\sum_{n\geq 0}(xz)^{n}+\frac{z}{1+t}\sum_{n\geq
0}(xz)^{n}\,,$ (10.4)

where, for notational simplicity, we introduce $x\equiv t/(1+t)$. From the
last line of the above, we can immediately extract all the $P_{n}(t)$ and
obtain the well-known formulas:

$\displaystyle P_{0}(t)=\frac{t}{1+t}\qquad\quad
P_{n}(t)=\frac{t^{n-1}}{(1+t)^{n+1}}\qquad n\geq 1\,.$ (10.5)

With these results, we now obtain the first-passage properties of the birth-
death process. The quantity $P_{0}(t)$ may be interpreted as the probability
that the population has gone extinct by time $t$, while
$S(t)=1-P_{0}(t)=1/(1+t)$ is the probability that the population survives up
to to time $t$. Thus extinction is sure to occur, but the average extinction
time is finite, just as for isotropic diffusion. The main distinction with
isotropic diffusion is that $S(t)\sim t^{-1/2}$ for diffusion, while $S(t)\sim
t^{-1}$ for the birth-death process. Thus survival is less likely when the
hopping rate is a linearly increasing function of $n$.

## Concluding Comments

These lecture notes have given a whirlwind tour through some basic and some
not-so-basic aspects of first-passage processes. At the level of fundamentals,
I presented some classic results about first passage in the simplest
geometries of the infinite half line and the finite interval, including first-
passage probabilities, first-passage times, and splitting probabilities. I
also discussed the intriguing connection between first passage and
electrostatics. I then presented a number of applications. Some, like the
reaction rate of a cell and the birth-death process are classic and have many
immediate applications. Some, like the survival of a diffusing lamb that is
hunted by $N$ diffusing lions and survival of a diffusing particle in a
growing interval may seem somewhat idiosyncratic. However, the solution
methods are quite generic and may prove useful in many other settings. I hope
that the uninitiated reader will enjoy learning about some of these
applications of first-passage processes and will be inspired to delve further
into this fascinating topic.

Much of the material in Secs. 8 and 9 stems from joint work with Paul
Krapivsky. I thank him for pleasant collaborations on these projects, as well
as pointing out Ref. [28] to me. I also thank the National Science Foundation
for financial support over many years that helped advance some of the topics
discussed in these notes, most recently through NSF grant DMR-1910736.

## References

  * [1] S. Redner, _A Guide to First-Passage Processes_ (Cambridge University Press, Cambridge, UK 2001).
  * [2] A. J. Bray, S. N. Majumdar, and G. Schehr, Adv. Phys. 62 225 (2015).
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