Distribution of extreme first passage times of diffusion

Abstract

Many events in biology are triggered when a diffusing searcher finds a target, which is called a first passage time (FPT). The overwhelming majority of FPT studies have analyzed the time it takes a single searcher to find a target. However, the more relevant timescale in many biological systems is the time it takes the fastest searcher(s) out of many searchers to find a target, which is called an extreme FPT. In this paper, we apply extreme value theory to find a tractable approximation for the full probability distribution of extreme FPTs of diffusion. This approximation can be easily applied in many diverse scenarios, as it depends on only a few properties of the short time behavior of the survival probability of a single FPT. We find this distribution by proving that a careful rescaling of extreme FPTs converges in distribution as the number of searchers grows. This limiting distribution is a type of Gumbel distribution and involves the LambertW function. This analysis yields new explicit formulas for approximations of statistics of extreme FPTs (mean, variance, moments, etc.) which are highly accurate and are accompanied by rigorous error estimates.

Appendix

In this appendix, we collect the proofs of the propositions and theorems of Sect. 3.

Proof of Proposition 1

This proposition merely collects basic results on Gumbel random variables, all of which follow directly from (11). $\square$

Proof of Proposition 2

Since most results in extreme value theory are formulated in terms of the maximum of a set of random variables, define

$$\begin{aligned} M_{N}:=\max \{-\tau _{1},\ldots ,-\tau _{N}\}=-T_{N}, \end{aligned}$$

and $F(x)=\mathbb {P}(-\tau _{1}<x)=S(-x)$. If there exists normalizing constants $\{a_{N}\}_{N\ge 1}$ and $\{b_{N}\}_{N\ge 1}$ so that $(M_{N}-b_{N})/a_{N}$ converges in distribution as $N\rightarrow \infty$ to a nontrivial random variable, then the distribution of that random variable can only be Frechet, Weibull, or Gumbel (Fisher and Tippett 1928). Since

$$\begin{aligned} x^{*} :=\sup \{x:F(x)<1\}=0<\infty , \end{aligned}$$

Theorem 1.2.1 in the book by De Haan and Ferreira (2007) ensures that the limiting distribution cannot be Frechet.

Furthermore, if the limiting distribution is Weibull, then Theorem 1.2.1 in (De Haan and Ferreira 2007) guarantees that there exists some $\gamma <0$ so that

$$\begin{aligned} \lim _{t\rightarrow 0+}\frac{1-F(-tx)}{1-F(-t)} =\lim _{t\rightarrow 0+}\frac{1-S(tx)}{1-S(t)} =x^{-1/\gamma }\quad \text {for all }x>0. \end{aligned}$$

(33)

Now, it follows directly from (12) that $S(t)=1-e^{-C/t+h(t)}$ for some function h(t) satisfying

$$\begin{aligned} \lim _{t\rightarrow 0+}th(t)=0. \end{aligned}$$

(34)

Therefore, we claim that (33) is violated with, for example, $x=2$. To see this, note that

$$\begin{aligned} \lim _{t\rightarrow 0+}\frac{1-S(2t)}{1-S(t)} =\lim _{t\rightarrow 0+}e^{C/t+h(2t)-h(t)}. \end{aligned}$$

By (34), we are assured that

$$\begin{aligned} -\frac{C}{2t} \le h(t) \le \frac{C}{2t}\quad \text {for sufficiently small }t. \end{aligned}$$

Hence,

$$\begin{aligned} \lim _{t\rightarrow 0+}e^{C/t+h(2t)-h(t)} \ge \lim _{t\rightarrow 0+}e^{3C/(4t)}=+\infty , \end{aligned}$$

which indeed violates (33). Therefore, if the limiting distribution is nondegenerate, it must be Gumbel. $\square$

Proof of Proposition 3

Using the assumptions on h(t) in (13), a direct calculation shows that

$$\begin{aligned} \lim _{t\rightarrow 0+}\frac{\text {d}}{\text {d}t}\left( \frac{1-S_{0}(t)}{S_{0}'(t)}\right) =0. \end{aligned}$$

Therefore, Theorem 2.1.2 in (Falk et al. 2010) ensures that

$$\begin{aligned} \lim _{N\rightarrow \infty }(S_{0}(a_{N}x+b_{N}))^{N} =\exp (-e^{x}),\quad \text {for all }x\in \mathbb {R}, \end{aligned}$$

(35)

for some choice of normalizing constants $\{(a_{N},b_{N})\}_{N\ge 1}$. Remark 1.1.9 in the book by De Haan and Ferreira (2007) implies we can take $a_{N}$ and $b_{N}$ as in (14).

Now, (35) is equivalent to

$$\begin{aligned} \lim _{N\rightarrow \infty }N \ln (S_{0}(a_{N}x+b_{N})) =-e^{x},\quad \text {for all }x\in \mathbb {R}. \end{aligned}$$

Hence, it must be the case that $S_{0}(a_{N}x+b_{N})\rightarrow 1$ as $N\rightarrow \infty$, and thus a straightforward application of L’Hospital’s rule gives

$$\begin{aligned} -\ln (S_{0}(a_{N}x+b_{N})) \sim 1-S_{0}(a_{N}x+b_{N})\quad \text {as }N\rightarrow \infty . \end{aligned}$$

Therefore, (35) is equivalent to

$$\begin{aligned} \lim _{N\rightarrow \infty }N(1-S_{0}(a_{N}x+b_{N})) =e^{x},\quad \text {for all }x\in \mathbb {R}. \end{aligned}$$

(36)

Now, $1-S_{0}(t)\sim 1-S_{0}(t)$ as $t\rightarrow 0+$ by assumption. Hence, (36) holds with $S_{0}$ replaced by S, which then implies that (35) holds with $S_{0}$ replaced by S, which completes the proof. $\square$

Proof of Theorem 1

The theorem follows from Proposition 3 upon calculating $\{a_{N}\}_{N\ge 1}$ and $\{b_{N}\}_{N\ge 1}$ in (16) for $h(t)=\ln (At^{p})$ and using properties of the LambertW function (Corless et al. 1996). $\square$

Proof of Theorem 2

The theorem follows immediately from Theorem 1 and (18)-(19). $\square$

Proof of Theorem 3

By assumption, $\mathbb {E}[T_{N}]<\infty$ for some $N\ge 1$. Hence, if $m\in (0,1)$, then $\mathbb {E}[(T_{N})^{m}]\le 1+\mathbb {E}[T_{N}]<\infty$. If $m\ge 1$, then it is straightforward to check that [see the proof of Proposition 2 in the work by Lawley (2020)]

$$\begin{aligned} \mathbb {E}[(T_{2^{m-1}N})^{m}]<\infty . \end{aligned}$$

Since $\mathbb {E}[X^{m}]<\infty$, applying Theorem 2.1 in (Pickands 1968) completes the proof. $\square$

Proof of Theorem 4

The convergence in distribution in (21) and (23) follows immediately from Theorem 1 above and Theorem 3.5 in the book by Coles (2001). $\square$

Proof of Theorem 5

While convergence in distribution does not necessarily imply convergence of moments, it does imply convergence of moments if the sequence of random variables is uniformly integrable (Billingsley 2013). Hence, it is sufficient to prove that

$$\begin{aligned} \sup _{N}\mathbb {E}\Big [\Big (\frac{T_{k,N}-b_{N}}{a_{N}}\Big )^{2}\Big ] <\infty \end{aligned}$$

(37)

since (37) ensures that $\{\frac{T_{k,N}-b_{N}}{a_{N}}\}_{N\ge 1}$ is uniformly integrable (Billingsley 2013).

By assumption, $1-S(t)\sim At^{p}e^{-C/t}$ as $t\rightarrow 0+$. Hence, there exists a $\delta >0$ so that

$$\begin{aligned} 1-A_{1}t^{p}e^{-C/t} \le 1-S(t) \le 1-A_{0}t^{p}e^{-C/t},\quad \text {if }t\in (0,\delta ], \end{aligned}$$

where $0<A_{0}<A<A_{1}$. Define the survival probability

$$\begin{aligned} S_{+}(t) ={\left\{ \begin{array}{ll} 1 &{}\quad t\le 0,\\ 1-A_{0}t^{p}e^{-C/t} &{}\quad t\in (0,\delta ],\\ S(t) &{}\quad t>\delta . \end{array}\right. } \end{aligned}$$

Define $S_{-}(t)$ similarly with $A_{0}$ replaced by $A_{1}$. Hence, $S_{-}(t)\le S(t)\le S_{+}(t)$ for all $t\in \mathbb {R}$. Let $\{U_{n}\}_{n\ge 1}$ be an iid sequence of random variables, each with a uniform distribution on [0, 1]. Define

$$\begin{aligned} \tau _{n}&:=S^{-1}(U_{n}),\\ \tau _{n}^{-}&:=S_{-}^{-1}(U_{n}),\\ \tau _{n}^{+}&:=S_{+}^{-1}(U_{n}), \end{aligned}$$

and

$$\begin{aligned} T_{k,N}&:=\min \big \{\{\tau _{1},\ldots ,\tau _{N}\}\backslash \cup _{j=1}^{k-1}\{T_{j,N}\}\big \},\quad k\in \{1,\ldots ,N\},\\ T_{k,N}^{\pm }&:=\min \big \{\{\tau _{1}^{\pm },\ldots ,\tau _{N}^{\pm }\}\backslash \cup _{j=1}^{k-1}\{T_{j,N}^{\pm }\}\big \},\quad k\in \{1,\ldots ,N\}, \end{aligned}$$

where $T_{1,N}:=\min \{\tau _{1},\ldots ,\tau _{N}\}$ and $T_{1,N}^{\pm }:=\min \{\tau _{1}^{\pm },\ldots ,\tau _{N}^{\pm }\}$. By construction, we have that

$$\begin{aligned} T_{k,N}^{-} \le T_{k,N} \le T_{k,N}^{+}\quad \text {almost surely}. \end{aligned}$$

Therefore, if $1_{{\mathcal {A}}}$ denotes the indicator function on an event ${\mathcal {A}}$, then

$$\begin{aligned} \mathbb {E}\Big [\Big (\frac{T_{k,N}-b_{N}}{a_{N}}\Big )^{2}\Big ]&=\mathbb {E}\Big [\Big (\frac{T_{k,N}-b_{N}}{a_{N}}\Big )^{2}1_{T_{k,N}>b_{N}}\Big ] +\mathbb {E}\Big [\Big (\frac{T_{k,N}-b_{N}}{a_{N}}\Big )^{2}1_{T_{k,N}<b_{N}}\Big ]\\&\le \mathbb {E}\Big [\Big (\frac{T_{k,N}^{+}-b_{N}}{a_{N}}\Big )^{2}1_{T_{k,N}^{+}>b_{N}}\Big ] +\mathbb {E}\Big [\Big (\frac{T_{k,N}^{-}-b_{N}}{a_{N}}\Big )^{2}1_{T_{k,N}^{-}<b_{N}}\Big ]\\&\le \mathbb {E}\Big [\Big (\frac{T_{k,N}^{+}-b_{N}}{a_{N}}\Big )^{2}\Big ] +\mathbb {E}\Big [\Big (\frac{T_{k,N}^{-}-b_{N}}{a_{N}}\Big )^{2}\Big ]. \end{aligned}$$

Hence, it remains to show that

$$\begin{aligned} \sup _{N}\mathbb {E}\Big [\Big (\frac{T_{k,N}^{\pm }-b_{N}}{a_{N}}\Big )^{2}\Big ] <\infty . \end{aligned}$$

Now,

$$\begin{aligned} \mathbb {E}\Big [\Big (\frac{T_{k,N}^{\pm }-b_{N}}{a_{N}}\Big )^{2}\Big ]&=\int _{0}^{\infty }\mathbb {P}\Big (\Big (\frac{T_{k,N}^{\pm }-b_{N}}{a_{N}}\Big )^{2}>t\Big )\,\text {d}t\\&=\int _{0}^{\infty }\mathbb {P}(T_{k,N}^{\pm }-b_{N}>a_{N}\sqrt{t})\,\text {d}t +\int _{0}^{\infty }\mathbb {P}(b_{N}-T_{k,N}^{\pm }>a_{N}\sqrt{t})\,\text {d}t\\&=:I_{1}+I_{2}. \end{aligned}$$

Since $T_{1,N}^{\pm }\le T_{k,N}^{\pm }$ almost surely for any $k\in \{1,\ldots ,n\}$, we have that

$$\begin{aligned} I_{2} \le \int _{0}^{\infty }\mathbb {P}(b_{N}-T_{1,N}^{\pm }>a_{N}\sqrt{t})\,\text {d}t&\le \int _{0}^{\infty }\mathbb {P}\Big (\Big (\frac{T_{1,N}^{\pm }-b_{N}}{a_{N}}\Big )^{2}>t\Big )\,\text {d}t\\&=\mathbb {E}\Big [\Big (\frac{T_{1,N}^{\pm }-b_{N}}{a_{N}}\Big )^{2}\Big ]. \end{aligned}$$

Now, Theorem 3 implies that

$$\begin{aligned} \mathbb {E}\Big [\Big (\frac{T_{1,N}^{\pm }-b_{N}^{\pm }}{a_{N}^{\pm }}\Big )^{2}\Big ] \rightarrow \mathbb {E}[X^{2}]<\infty \quad \text {as }N\rightarrow \infty , \end{aligned}$$

where $\{a_{N}^{\pm }\}_{N\ge 1}$ and $\{b_{N}^{\pm }\}_{N\ge 1}$ are given by (16) with A replaced by $A_{0}$ or $A_{1}$. Now, it is straightforward to check that there exists $\alpha ^{\pm }>0$ and $\beta ^{\pm }\in \mathbb {R}$ so that

$$\begin{aligned} \frac{a_{N}^{\pm }}{a_{N}}\rightarrow \alpha ^{\pm } \quad \text {and}\quad \frac{b_{N}-b_{N}^{\pm }}{a_{N}}\rightarrow \beta ^{\pm }\quad \text {as }N\rightarrow \infty . \end{aligned}$$

Therefore, Proposition 1.1 and Remark 1 in the work by Peng and Nadarajah (2012) imply that $\mathbb {E}[(\frac{T_{1,N}^{\pm }-b_{N}^{\pm }}{a_{N}^{\pm }})^{2}]$ converges to some finite constant as $N\rightarrow \infty$. Hence,

$$\begin{aligned} \sup _{N}I_{2}<\infty . \end{aligned}$$

Moving to $I_{1}$, note first that

$$\begin{aligned} \mathbb {P}(T_{k,N}^{\pm }>x) =\mathbb {P}(T_{1,N}^{\pm }>x) +\sum _{j=1}^{k-1}\mathbb {P}(T_{j,N}^{\pm }<x<T_{j+1,N}^{\pm }). \end{aligned}$$

Hence,

$$\begin{aligned} I_{1}&=\int _{0}^{\infty }\mathbb {P}(T_{1,N}^{\pm }>a_{N}\sqrt{t}+b_{N})\,\text {d}t +\sum _{j=1}^{k-1}\int _{0}^{\infty }\mathbb {P}(T_{j,N}^{\pm }<a_{N}\sqrt{t}+b_{N}<T_{j+1,N}^{\pm })\,\text {d}t\\&=:I_{3}+I_{4}. \end{aligned}$$

Now, $I_{3}$ can be handled similarly to $I_{2}$ to obtain

$$\begin{aligned} \sup _{N}I_{3}<\infty . \end{aligned}$$

Hence, it remains to show that $\sup _{N}I_{4}<\infty$. Now, since $\{\tau _{n}^{\pm }\}_{n\ge 1}$ are iid, it follows that

$$\begin{aligned} \mathbb {P}(T_{j,N}^{\pm }<x<T_{j+1,N}^{\pm }) ={N \atopwithdelims ()j}(1-S_{\pm }(x))^{j}(S_{\pm }(x))^{N-j},\;\text {if }j\in \{1,\ldots ,k-1\}. \end{aligned}$$

Hence,

$$\begin{aligned} I_{4} =\sum _{j=1}^{k-1}{N \atopwithdelims ()j}\int _{0}^{\infty }(1-S_{\pm }(a_{N}\sqrt{t}+b_{N}))^{j}(S_{\pm }(a_{N}\sqrt{t}+b_{N}))^{N-j}\,\text {d}t. \end{aligned}$$

An application of Laplace’s method with a movable maximum [see, for example, section 6.4 in the book by Bender and Orszag (2013)] shows that each term in this sum is bounded in N, and so the proof is complete. $\square$