Span observables - "When is a foraging rabbit no longer hungry?"

TL;DR

This paper analyzes the span of a random walk, deriving analytical formulas for the probability of reaching a span of 1, including effects of drift and boundaries, validated by simulations.

Contribution

It provides new analytical results for the span distribution of random walks with boundaries and drift, including joint max-min probabilities.

Findings

Derived formulas for span reaching probability and density.

Extended analysis to include drift and reflecting boundaries.

Validated analytical results with numerical simulations.

Abstract

Be $X_t$ a random walk. We study its span $S$, i.e. the size of the domain visited up to time $t$. We want to know the probability that $S$ reaches $1$ for the first time, as well as the density of the span given $t$. Analytical results are presented, and checked against numerical simulations. We then generalize this to include drift, and one or two reflecting boundaries. We also derive the joint probability of the maximum and minimum of a process. Our results are based on the diffusion propagator with reflecting or absorbing boundaries, for which a set of useful formulas is derived.

Equations

$$⟨X_t⟩=\mu t.$$

$$\displaystyle\left<\big{(}X_{t}-\left<X_{t}\right>\!\big{)}^{\!2}\right>=2t\ .$$

$$P_1(x)=x.$$

$$M_{+}(t)$$

$$M_{-}(t)$$

$$S(t):=M_{+}(t)-M_{-}(t).$$

$${\partial }_{t}X(t)=\mu t+\eta (t),⟨\eta (t)\eta (t^{\prime })⟩=2\delta (t-t^{\prime }).$$

$${\partial }_t{P}_{\mathrm{DD}}^{\mu }(x,y,t)=\frac{{\partial }^2}{\partial y^2}{P}_{\mathrm{DD}}^{\mu }(x,y,t)-\mu \frac{\partial }{\partial y}{P}_{\mathrm{DD}}^{\mu }(x,y,t).$$

$$P_{\mathrm{DD}}^{\mu }(x,y,t)=e^{\frac{\mu (y-x)}{2}-\frac{{\mu }^{2}t}{4}}\left[\mathbb{P}(x-y,t)-\mathbb{P}(x+y,t)\right].$$

$$\mathbb{P}(z,t):=\frac{1}{\sqrt{4\pi t}}\sum _{n=-\infty }^{\infty }{e}^{-(z+2n{)}^2/4t}=\frac{1}{2}{\vartheta }_3\left(\frac{\pi }{2}z,e^{-{\pi }^{2}t}\right).$$

$$\mathbb{P}(z,t)=\frac{1}{2}+\sum _{m=1}^{\infty }{e}^{-m^2{\pi }^{2}t}\cos (m\pi z).$$

$$\underset{t\to 0}{\lim }{P}_{+}^{\mu }(x,y,t)=\delta (x-y).$$

$$\mathbb{P}(z,t)=\mathbb{P}(z+2,t)=\mathbb{P}(-z,t).$$

$${\partial }_z\mathbb{P}(z,t){|}_{z=\mathrm{integer}}=0$$

$$\tilde{F}(s):={\mathcal{L}}_{t\to s}\left[F(t)\right]={\int }_0^{\infty }\mathrm{d}t{e}^{-st}F(t).$$

$$\tilde{\mathbb{P}}(z,s)$$

$$=$$

$$\mathcal{L}_{t\to s}\left[e^{-\frac{\mu^{2}t}{4}}{\mathbb{P}}(z,t)\right]=\tilde{\mathbb{P}}\Big{(}z,s+\frac{\mu^{2}}{4}\Big{)}\ .$$

$$\mathbb{P}(x-y,t)-\mathbb{P}(x+y,t)$$

$${\partial }_t{P}_{\mathrm{DD}}^{\mu }(x,y,t)+{\partial }_y{J}_{\mathrm{DD}}^{\mu }(x,y,t)=0.$$

$$J_{\mathrm{DD}}(x,y,t)=(\mu -{\partial }_y)P_{\mathrm{DD}}^{\mu }(x,y,t)$$

$$\displaystyle\qquad=e^{\frac{\mu(y-x)}{2}-\frac{\mu^{2}t}{4}}\Big{(}\frac{\mu}{2}-\partial_{y}\Big{)}\left[{\mathbb{P}}(x-y,t)-{\mathbb{P}}(x+y,t)\right]\ .$$

$${\int }_0^1\mathrm{d}y{\partial }_t{P}_{\mathrm{DD}}^{\mu }(x,y,t)=J_{\mathrm{DD}}^{\mu }(x,0,t)-J_{\mathrm{DD}}^{\mu }(x,1,t).$$

$$P_{\mathrm{exit}}^{\mathrm{DD}}(x,t)=J_{\mathrm{DD}}^{\mu }(x,1,t)-J_{\mathrm{DD}}^{\mu }(x,0,t)$$

$$=2e^{-\frac{{\mu }^{2}t}{4}}\left[e^{\frac{\mu (1-x)}{2}}{\partial }_x\mathbb{P}(1-x,t)-e^{-\frac{\mu x}{2}}{\partial }_x\mathbb{P}(x,t)\right].$$

$$J_{\mathrm{DD}}^{\mu }(x,1,t)$$

$$-J_{\mathrm{DD}}^{\mu }(x,0,t)$$

$$-{\tilde{J}}_{\mathrm{DD}}^{\mu }(x,0,s)$$

$${\tilde{J}}_{\mathrm{DD}}^{\mu }(x,1,s)$$

$$P_{\mathrm{DD},0}^{\mu }(x)$$

$$P_{\mathrm{DD},1}^{\mu }(x)$$

$$\displaystyle\left<T_{\rm exit}^{\mu}(x)\right>_{0}=-\partial_{s}\left[\tilde{J}_{\rm DD}^{\mu}(x,1,s)-\tilde{J}_{\rm DD}^{\mu}(x,0,s)\right]\Big{|}_{s=0}$$

$$=\frac{e^{\mu }(1-x)-e^{\mu (1-x)}+x}{\left(e^{\mu }-1\right)\mu }$$

$${\int }_0^1\mathrm{d}x{⟨T_{\mathrm{exit}}^{\mu }(x)⟩}_0=\frac{\mu \coth \left(\frac{\mu }{2}\right)-2}{2{\mu }^2}.$$

$$\displaystyle\left<T_{\rm exit}^{\mu}(x)^{2}\right>_{0}=\partial_{s}^{2}\left[\tilde{J}^{\mu}_{\rm DD}(x,1,s)-\tilde{J}_{\rm DD}^{\mu}(x,0,s)\right]\Big{|}_{s=0}$$

$${\int }_0^1\mathrm{d}x{⟨T_{\mathrm{exit}}^{\mu }(x{)}^2⟩}_0=\frac{{\mu }^2+3{\mu }^2{\text{csch}}^2(\frac{\mu }{2})-12}{3{\mu }^4}.$$

Full text

I Introduction

Consider a Brownian motion $X_{t}$, starting at $X_{0}=0$, with drift $\mu$, and variance 2,

[TABLE]

A sample trajectory is sketched on Fig. 1 (for $\mu=0$). A key problem in stochastic processes are the first-passage properties FellerBook ; RednerBook in a finite domain, say the unit interval $[0,1]$. For a Brownian, the probability to exit at the upper boundary $x=1$ without visiting the lower boundary at $x=0$, while starting at $x$ is

[TABLE]

Another key observable is the exit time, starting at $x$, which behaves as $\left<T_{\rm exit}(x)\right>_{0}\sim x(1-x)$.

Here we consider a different set of observables, namely the span of a process: Define the positive and negative records (a.k.a. the running max and min) as

[TABLE]

These observables are drawn on Fig. 1. The span $S(t)$ is their difference, i.e. the size of the (compact) domain visited up to time $t$,

[TABLE]

We study the probability that $S(t)$ becomes 1 for the first time. Curiously, this observable is rarely treated in the literature, and most of the studies we found are concerned with questions of convergence of the first moments, which is non-trivial when the process is more complicated than a random walk: Let us mention the mean first-passage time WeissDiMarzioGaylord1986 , with some discrepancies stated in Ref. PalleschiTorquati1989 . The full distribution as a function of times is derived below. A related but distinct observable is the density of the span at time $t$, considered in the classic references Daniels1941 ; Feller1951 ; WeissRubin1976 ; PalleschiTorquati1989 . A beautiful recent result is the covariance of the span AnnesiMarinariOshanin2019 .

One may ask where span observables actually occur in nature? One example is the Hungry Rabbit Problem. Suppose a hungry and myopic rabbit is released. It will perform a random walk, until its stomach is full, i.e. the span of its trajectory reaches $1$. This is a variant of the myopic rabbit introduced in RagerBhatBenichouRedner2018 . We will give the probability for the time that the rabbit is no longer hungry analytically, including some drift in the rabbit’s motion, when e.g. it prefers to move downhill. One may object that the problem is not realistic, since foraging the rabbit consumes food. We currently have no solution for the latter problem, even though diffusion with moving boundaries can, at least in principle, be treated via a set of integral equations Cannon1984 ; however we do not know of a closed-form solution. A notable exception are expanding boundaries in the limit of large times, where the survival probability can be evaluated analytically BraySmith2007 .

Another example arises in measuring the exit probability from a strip, a problem studied for fractional Brownian motion in Ref. Wiese2018 . The question is how long one has to run a simulation until the process $Y_{t}:=X_{t}+x$ has exited from the unit interval $[0,1]$, for all $x\in[0,1]$. We claim that the simulation can be stopped at time $t_{*}$ when the span first reaches 1. To understand this statement, define $x^{*}:=-M_{-}(t^{*})$. Then consider the process $Y_{t}:=X_{t}+x$. If $x>x^{*}$, then $Y_{t}$ will exit at the upper boundary at time $t<t^{*}$, while for $x<x^{*}$ the process $Y_{t}$ will exit at the lower boundary at time $t<t^{*}$. For $t<t_{*}$ there will be some $x\approx x_{*}$ where no conclusion can be drawn. Thus $x_{*}$ samples $P_{1}^{\prime}(x)$, the derivative of the exit probability at the upper boundary, and $t_{*}$ is the time one can stop the process without loss of information.

A related quantity is the joint density of the running maximum and minimum, given $t$. This question is relevant in the analysis of stock-market data WergenBognerKrug2011 , where it allows one to quantify violations of the Markov property.

The span is also relevant in the search of a protein for its binding site on a DNA molecule. The idea of facilitated diffusion MirnySlutskyWunderlichTafviziLeithKosmrlj2009 is to alternatively diffuse along the DNA molecule or in 3d space, thus optimizing the search.

Finally, the span is not a Markov process, but a process with memory, as it remembers its positive and negative records. This places our study in the larger context of processes with memory, of which fractional Brownian motion may be the most relevant one NourdinBook ; DiekerPhD ; Krug1998 ; SadhuDelormeWiese2017 .

The remainder of this article is organized as follows: We first derive key results for Brownian motion in the unit interval $[0,1]$, with absorbing boundary conditions at both ends, see section II. This is then generalized to one absorbing and one reflecting boundary in section III, and to two reflecting ones in section IV. Most of these results are known. We give analytical results for span observables in section V, and the joint distribution of running maximum and minimum in section VI. A generalization to a random walk with one reflecting boundary is presented in section VII, while two reflecting boundaries are treated in section VIII. We conclude in section IX with open problems.

II Basic formulas for Brownian Motion with two absorbing boundaries

II.1 Solving the Fokker-Planck equation

Consider a random walk $X(t)$ given by its Langevin equation

[TABLE]

There are absorbing (Dirichlet) boundary conditions both at $x=0$, and $x=1$. If the trajectory starts at $X_{0}=x$, and ends at $X_{t}=y$, then the forward Fokker-Planck equation reads FellerBook ; RednerBook

[TABLE]

The index “DD” refers to the two absorbing (Dirichlet) boundary conditions at $x=0$, and $x=1$. The probability to survive at time $t$ is given by $\int_{0}^{1}\mathrm{d}y\,P_{\rm DD}(x,y,t)$. The general solution of the Fokker-Planck equation (8) can be written as

[TABLE]

The key object in this construction is

[TABLE]

$\vartheta$ is the elliptic $\vartheta$-function. Using the Poisson summation formula, an alternative form for ${\mathbb{P}}(z,t)$ is

[TABLE]

To prove the above statements it is enough to remark that Eq. (9) satisfies the Fokker-Planck equation (8), vanishes at $y=0$ and $y=1$, and reduces for $t\to 0$ to a $\delta$-function

[TABLE]

The function $\mathbb{P}(z,t)$ has the following properties

[TABLE]

As a consequence,

[TABLE]

It is useful to consider its Laplace-transformed version. We define the Laplace transform of a function $F(t)$, with $t\geq 0$, and marked with a tilde as

[TABLE]

This yields for $-2<z<2$

[TABLE]

And

[TABLE]

Note that the combination in square brackets in Eq. (9) can also be written as

[TABLE]

The form (19) facilitates its integration over $x$ and $y$, which is useful when concatenating several propagators Wiese2018 .

II.2 Boundary currents and conservation of probability

Conservation of probability reads (the variable $x$ is the initial condition, here a dummy variable)

[TABLE]

$J^{\mu}_{\rm DD}$ is the current, which from Eqs. (8), (9) and (20) can be identified as

[TABLE]

Due to the Dirichlet conditions at $y=0$ and $y=1$, we have

[TABLE]

Thus, the probability to exit at time $t$, when starting in $x$ at time [math] reads

[TABLE]

The outgoing currents at the upper and lower boundary are

[TABLE]

The Laplace transforms of these outgoing currents are

[TABLE]

II.3 Absorption probabilities at $x=0$ and $x=1$

The absorption probabilities at $x=0$ and $x=1$ are

[TABLE]

II.4 Moments of the absorption time, starting at $x$

Moments of the absorption time are extracted from the Laplace-transformed currents as

[TABLE]

III Propagator with one absorbing and one reflecting boundary

The propagator with an absorbing (Dirichlet) boundary at $y=0$ and a reflecting (Neumann) one at $y=1$ reads

[TABLE]

The generalization to include drift is as in Eq. (9). The Laplace transform of Eq. (33) is

[TABLE]

It can also be written as

[TABLE]

Expanding in $s$, we find

[TABLE]

The outgoing current at the lower boundary is

[TABLE]

Taylor expanding in $s$ yields

[TABLE]

The first term indicates that all trajectories exist, while the time it takes and its second moment are

[TABLE]

The propagator with a Dirichlet boundary condition at the upper, and a Neumann boundary condition at the lower end is obtained by replacing $x\to 1-x$ and $y\to 1-y$.

IV Propagator with two reflecting boundaries

With two reflecting (Neumann) boundary conditions the propagator is

[TABLE]

Laplace transforming yields

[TABLE]

V Probabilities for the span

V.1 Definition of the span

The span is a classical problem treated e.g. in Daniels1941 ; Feller1951 ; WeissRubin1976 ; PalleschiTorquati1989 , but the observables we wish to study seem not to have been considered. To properly define the problem, we note the positive and negative records (the running maximum and minimum) as

[TABLE]

The span $S(t)$ is their difference, i.e. the size of the (compact) domain visited up to time $t$,

[TABLE]

(We note capital $S$ for the span, in order to distinguish it from the Laplace variable $s$ conjugate to time $t$.)

V.2 The probability that the span reaches 1 for the first time

We want to know the probability that $S(t)$ becomes 1 for the first time. We note this time by $T_{1}$, and its probability distribution by $P_{T_{1}}(t)$. There are two contributions, depending on whether the process stops while at its minimum or maximum. The probability to stop when the process is at its minimum can be obtained as follows: Consider the outgoing current for the process starting at $X_{0}=x$, with the lower boundary positioned at $m_{1}$, and the upper boundary at $m_{2}$, i.e.

[TABLE]

(The scale factor can be understood from the observation that the current is a density in the starting point times a spatial derivative of a probability.) The probability that the walk reached $m_{2}$ before being absorbed at $m_{1}$ is $\partial_{m_{2}}{\mathbf{J}(x,m_{1},m_{2},t)}$. Finally, the probability to have span 1 at time $t$ is this expression, integrated over $x$ between the two boundaries. There is another term, where the process stops while at its maximum. It is obtained from this first contribution when exchanging the two boundaries, and replacing $\mu$ by $-\mu$. Setting w.l.o.g. $m_{1}=0$ and $m_{2}=m$, the sum of the two terms is

[TABLE]

For $\mu=0$, this simplifies to

[TABLE]

Using Eqs. (21) and (9) allows us to rewrite the integral (for $\mu=0$) as

[TABLE]

Thus

[TABLE]

Inserting the definition (10) of $\mathbb{P}$, we get

[TABLE]

With the help of the Poisson-formula transformed Eq. (11) this can be written as

[TABLE]

This result is compared to a numerical simulation on Fig. 3. Our expansions allow us to give simple formulas for the small and large-$t$ asymptotics,

[TABLE]

These expansions work in a rather large, and overlapping domain, as can be seen on Fig. 3. Its Laplace transform is

[TABLE]

Extracting the moments from the Laplace transform yields

[TABLE]

Let us now return to the case with drift in Eq. (V.2). Since formulas become rather cumbersome, we only give one well-converging series expansion, based on the representation (11),

[TABLE]

V.3 Density of the span

Let us connect to the classical work on the span Daniels1941 ; Feller1951 ; WeissRubin1976 ; PalleschiTorquati1989 . We will show how to reproduce formulas (3.7)-(3.8) in Feller1951 . The latter give the density $\rho_{t}(S)$ for the span $S$ at time $t$. In our formalism, it can be obtained as

[TABLE]

where $\mathbf{P}^{\mu}_{\rm DD}(x,y,m_{1},m_{2},t)$ is the probability to go from $x$ to $y$ in time $t$, without being absorbed by the lower boundary positioned at $m_{1}$, or the upper boundary positioned at $m_{2}$. In terms of the propagator $P_{\rm DD}^{\mu}(x,y,t)$, this can be written as

[TABLE]

We start with $\mu=0$: Using Eq. (9), and the series expansions (10) and (11) yields after integration and simplifications two different representations,

[TABLE]

This is equivalent to Eqs. (3.7)-(3.8) in Feller1951 , if one there replaces $t\to 2t$. (Our variance (2) is $2t$ instead of $t$ as in Feller1951 .) The small and large-$S$ asymptotics are

[TABLE]

Note that in Eq. (62) we have also retained the subleading term for small $S$, which considerably improves the numerical accuracy. A test is presented on Fig. 5.

Let us now turn to the general case with $\mu\neq 0$. There we have using the generating function (11)

[TABLE]

This formula is checked on Fig. 6. The small-$S$ asymptotics can be obtained by retaining only the leading term in $n$. Let us finally note that for large $\mu$, this density tends to

[TABLE]

For $\mu>0$, the first term is the probability density for the max of the endpoint, supposing that the minimum is at [math]. For $\mu<0$, the second term arises, with max and min interchanged.

VI Joint density of maximum and minimum

We can also derive the joint density of the maximum $M_{+}\equiv m_{2}>0$ and minimum $M_{-}\equiv m_{1}<0$, starting at $x=0$. In analogy of Eq. (59), this can be written as

[TABLE]

The equivalent of Eq. (60) then becomes

[TABLE]

Inserting Eq. (9) and one of the two representations (10) or (11) yields two converging series expansions. Since in general these expressions are little enlightening, we continue with $\mu=0$. To simplify our analysis, we rewrite the density (66) in terms of $S:=m_{2}-m_{1}$ and $x:=m_{1}/(m_{1}-m_{2})$:

[TABLE]

Its marginal density coincides with Eq. (61) ,

[TABLE]

The two series expansions in question are

[TABLE]

Interestingly, the latter equation allows us to obtain the marginal distribution of $x$ in closed form. Since this function is independent of $t$, we drop the time index:

[TABLE]

This is the density for the relative location of the starting point w.r.t. the domain given by the maximum and minimum. It is also the distribution of the final position w.r.t. the same domain. This density is larger at the boundaries, as is easily understood: After a new record, the particle diffuses away from the record, but the probability density remains higher close to the last record.

VII The span with a reflecting wall

Now consider diffusion with a reflecting wall at $x=0$. We want to know the probability density for the span to reach $1$ for the first time. For simplicity, we restrict to the drift-free case $\mu=0$. We also assume $x<1$, since for $x>1$ the reflecting boundary can never be reached, and we recover the result of section V.2. Suppose the process starts at $x$, with $0\leq x<1$. There are two possibilities: Either the process first reaches 0, or 1. The probabilities for these two events are $x$ and $1-x$, respectively. If it first reaches 1, then it almost surely also reaches $1+\delta$ with $\delta$ small before its span becomes 1; as a consequence its minimum is bounded by $\delta$. Thus it never reaches the lower boundary at $x=0$.

Consider the two contributions in turn: The first contribution, when the process never reaches $x=0$, is similar to the one obtained in Eq. (V.2). It can itself be decomposed into two sub-contributions, depending on whether, when the span reaches 1, $X_{t}$ equals its maximum (case 1a) or minimum (case 1b). We start with case 1a. Denoting $\mathbf{J}_{\rm DD}(y,m_{2},t|m_{1},m_{2})$ the outgoing current at the upper boundary $m_{2}$, for a particle starting at $y$, with lower boundary $m_{1}$, we have

[TABLE]

Let us first evaluate its normalization, using that the time-integrated current is the exit probability,

[TABLE]

Note that this is smaller than the probability $x$ to exit at the upper boundary. This can be understood from the fact that the trajectory has to go beyond 1,

or more precisely to $1+\rm min$, where ${\rm min}>0$ is the minium of the trajectory. Continuing with Eq. (VII), we obtain

[TABLE]

Integrating this yields

[TABLE]

(The first term on the last line vanishes). To simplify this expression, introduce the function $\mathbb{R}(x,t)$ defined as

[TABLE]

This yields

[TABLE]

This is written s.t. $\mathbb{R}$ can be thought of as the principle function of the integrand in Eq. (74). The second contribution where the process never reaches 0 is obtained when the process has its maximum at $1+\delta$ with $\delta>0$, before going down to $\delta<x$, where the process stops (case 1b). By symmetry, this is the same expression as Eq. (VII), where all positions $x$ are sent to $1-x$, i.e.

[TABLE]

The probability for this process is as in Eq. (73) given by the time-integrated current

[TABLE]

Thus, as expected

[TABLE]

Let us continue with the evaluation of $p_{1b}(x,t)$,

[TABLE]

Integration yields

[TABLE]

In analogy of Eq. (77) this can be written as

[TABLE]

The sum of the two contributions $p_{1a}$ and $p_{1b}$ is

[TABLE]

Note that for $x=\frac{1}{2}$, one gets $p_{1}(x,t)=\frac{1}{2}P_{T_{1}}(t)$.

The second contribution is achieved when the process first reaches the lower boundary. It can be obtained by folding the probability to first reach the lower boundary, i.e. the outgoing current at $x=0$, with an absorbing boundary both at $x=0$ and $x=1$, with the outgoing current at $x=1$ with a reflecting boundary at $x=0$ and an absorbing one at 1, i.e.

[TABLE]

Passing to Laplace variables, this reads

[TABLE]

We had calculated the currents before,

[TABLE]

The inverse Laplace transform of Eq. (86) can be written as

[TABLE]

This is checked by evaluating the Laplace transform of each term in the above sum, and then performing the sum over $n$.

The probability to first reach the lower boundary is

[TABLE]

The probability to reach span 1, starting at $x$, and with a reflecting boundary at $x=0$ is finally obtained as

[TABLE]

The mean time to reach span 1 is

[TABLE]

Thus when starting close to the reflecting wall, it takes on average twice as long to reach span 1, as when starting from far away.

A numerical check for $x=0.25$, $x=0.5$, and $x=0.75$ is presented on Fig. 8.

VIII The span with two reflecting boundaries

Finally, consider two reflecting (Neumann) boundaries at $x=0$ and $x=a\geq 1$, and suppose that $0<x<1$, and $0<a-x<1$, so that both boundaries can be reached before the span attains one and the process terminates. These conditions can be summarized in

[TABLE]

In generalization of Eq. (89), one can write

[TABLE]

The function $p_{3}(x,t|a)$ is a modification of $p_{1a}(x,t)$, defined by

[TABLE]

This integral is analogous to (74), with the difference that the lower boundary may be larger than 0; this domain of integration is restricted s.t. the process never touches the lower boundary. For $a\geq 1+x$, this reproduces the probability $p_{1a}(x,t)$,

[TABLE]

Using our assumptions, $p_{3}(x,t|a)$ can be simplified to

[TABLE]

To get to the last line we used our assumption (93).

Similarly, the last term in Eq. (94) reproduces the function $p_{1b}(x,t)$ used above, when choosing

[TABLE]

Note that Eq. (94) has manifestly the symmetry $x\to a-x$, both for $p_{2}$ and $p_{3}$. Choosing $a=1+x$, the sum of the latter terms becomes $p_{1a}(x,t)+p_{1b}(x,t)$, making manifest the hidden symmetry between these terms.

Finally, one checks that for $x$ satisfying condition (93), $\int_{0}^{\infty}\mathrm{d}t\,P_{T_{1}}^{\rm NN}(x,t)=1$, thus the probability (94) is properly normalized. A numerical test is presented on Fig. 9.

IX Conclusions and Open Problems

Let us come back to the image of a myopic foraging rabbit, and ask when he is no longer hungry. Suppose there is a uniform food distribution. The rabbit starts with an empty stomach, does a random walk and eats everything he can get, until his stomach is full ($S=1$). The probability for this time is the probability that the span reaches one for the first time, as given in Eq. (50), and after. But a real rabbit is burning food, so add a (negative) drift, i.e. stop when $S(t)-\mu t=1$. Curiously, this problem is much more difficult to solve, and we (currently) have no analytical solution. One may be able to calculate the probability that the rabbit dies before having a full stomach, following the approach outlined in Ref. BraySmith2007 .

Another open problem is the generalization of the observables obtained here for correlated processes, as fractional Brownian motion. While the first moments of the span distribution have been obtained in an expansion Wiese2018 around $H=1/2$ (Brownian motion), the full distribution remains to be evaluated.

ACKNOWLEDGMENTS

It is a pleasure to thank Olivier Benichou, Joachim Krug, Satya Majumdar, Tridib Sadhu and Sidney Redner for stimulating discussions.