A Probabilistic Approach to Extreme Statistics of Brownian Escape Times in Dimensions 1, 2, and 3

Abstract

First passage time (FPT) theory is often used to estimate timescales in cellular and molecular biology. While the overwhelming majority of studies have focused on the time it takes a given single Brownian searcher to reach a target, cellular processes are instead often triggered by the arrival of the first molecule out of many molecules. In these scenarios, the more relevant timescale is the FPT of the first Brownian searcher to reach a target from a large group of independent and identical Brownian searchers. Though the searchers are identically distributed, one searcher will reach the target before the others and will thus have the fastest FPT. This fastest FPT depends on extremely rare events and its mean can be orders of magnitude faster than the mean FPT of a given single searcher. In this paper, we use rigorous probabilistic methods to study this fastest FPT. We determine the asymptotic behavior of all the moments of this fastest FPT in the limit of many searchers in a general class of two- and three-dimensional domains. We establish these results by proving that the fastest searcher takes an almost direct path to the target.

Appendix

In this Appendix, we collect the proofs of all the lemmas.

Proof of Lemma 2

First notice that $R_{n}(t)$ is a 2d Bessel process. In particular, $R_{n}(t)$ has the same law as $\sqrt{2D}\sqrt{W_{1}^{2}(t)+W_{2}^{2}(t)}$, where $W_{1}(t)\in \mathbb {R}$ and $W_{2}(t)\in \mathbb {R}$ are independent standard Brownian motions. Next, define the 3d Bessel process,

$$\begin{aligned} R^{\text {3d}}(t) :=\sqrt{2D}\sqrt{W_{1}^{2}(t)+W_{2}^{2}(t)+W_{3}^{2}(t)}, \end{aligned}$$

where $W_{3}(t)\in \mathbb {R}$ is a third independent standard Brownian motion. If $\tau$ is the first time that $R^{\text {3d}}(t)$ hits radius $a>0$,

$$\begin{aligned} \tau :=\inf \{t>0:R^{\text {3d}}(t)=a\}, \end{aligned}$$

then it is immediate that $\mathbb {P}(\tau _{1,side }\le t)\le \mathbb {P}(\tau \le t)$ for all $t\ge 0$. Since $\mathbb {P}(\tau \le t)$ has the explicit formula (see, for example, Bernoff and Lindsay 2018),

$$\begin{aligned} \mathbb {P}(\tau \le t) =\sqrt{\frac{4a^{2}}{\pi Dt}}\sum _{k=0}^{\infty }\mathrm{e}^{-a^{2}(n+\frac{1}{2})^{2}/(Dt)}, \end{aligned}$$

we may take $C_{1}=4a(\pi D)^{-1/2}$ and $C_{2}=a^{2}/(4D)$ to complete the proof. $\square$

Proof of Lemma 3

Recall that $\tau _{1,z}$ is the first time $Z_{1}(t)$ hits $z=0$, assuming $Z_{1}(0)=z_{0}>0$ and $Z_{1}(t)$ reflects from $z=h\ge z_{0}$. It is immediate that

$$\begin{aligned} \mathbb {P}(\tau _{1,z}>t) \le \mathbb {P}(\tau >t), \end{aligned}$$

where $\tau$ is the first time a standard Brownian motion, $W(t)\in \mathbb {R}$, hits $z_{0}/\sqrt{2D}$,

$$\begin{aligned} \tau :=\inf \{t>0:W(t)=z_{0}/\sqrt{2D}\}. \end{aligned}$$

The reflection principle (Durrett 2019) then gives

$$\begin{aligned} \mathbb {P}(\tau >t) =1-\mathbb {P}(\tau \le t) =1-2\mathbb {P}(W(t)\ge z_{0}/\sqrt{2D}) =\text {erf}(z_{0}/\sqrt{4Dt}), \end{aligned}$$

where $\text {erf}(x)$ denotes the error function, which has the large x behavior,

$$\begin{aligned} \text {erf}(x):=\frac{1}{\sqrt{\pi }}\int _{-x}^{x}\mathrm{e}^{-s^{2}}\,\text {d}s =1-\frac{\mathrm{e}^{-x^{2}}}{x\sqrt{\pi }}\Big (1-\frac{2}{x^{2}}+\mathcal {O}(x^{-4})\Big )\quad \text {as }x\rightarrow \infty . \end{aligned}$$

(5.1)

Hence, we may take $C_{3}=\frac{\sqrt{D}}{\sqrt{\pi }z_{0}}$ and $C_{4}=z_{0}^{2}/(4D)$ to complete the proof. $\square$

Proof of Lemma 6

Define the event

$$\begin{aligned} B&:=\{\tau _{1}<\min \{\tau _{1,side },\tau _{1,top }\}\}, \end{aligned}$$

and let $B^{c}$ denote its complement. It is immediate that

$$\begin{aligned} \mathbb {P}(\tau _{1}>t|B) \le \mathbb {P}(\tau _{1}^{+}>t|B), \end{aligned}$$

since $\tau _{1}\le \tau _{1}^{+}$ if $\tau _{1}<\min \{\tau _{1,side },\tau _{1,top }\}$. Next, by the definition of $\tau _{1}^{+}$, we have that

$$\begin{aligned} \mathbb {P}(\tau _{1}^{+}>t|B^{c})=1\quad \text {for all }t<1. \end{aligned}$$

Therefore

$$\begin{aligned} \mathbb {P}(\tau _{1}>t) \le \mathbb {P}(\tau _{1}^{+}>t|B)\mathbb {P}(B) +\mathbb {P}(B^{c})=\mathbb {P}(\tau _{1}^{+}>t)\quad \text {for all }t<1. \end{aligned}$$

Now, it is an identity that

$$\begin{aligned} \mathbb {E}[T_{3d }^{m}] = \int _{0}^{1}(\mathbb {P}(\tau _{1}>t^{1/m}))^{N}\,\text {d}t +\int _{1}^{\infty }(\mathbb {P}(\tau _{1}>t^{1/m}))^{N}\,\text {d}t. \end{aligned}$$

Therefore, we have that

$$\begin{aligned} \mathbb {E}[T_{3d }^{m}]&= \int _{0}^{1}(\mathbb {P}(\tau _{1}>t^{1/m}))^{N}\,\text {d}t +\int _{1}^{\infty }(\mathbb {P}(\tau _{1}>t^{1/m}))^{N}\,\text {d}t\\&\le \int _{0}^{1}(\mathbb {P}(\tau _{1}^{+}>t^{1/m}))^{N}\,\text {d}t +(\mathbb {P}(\tau _{1}>1))^{N}\int _{1}^{\infty }\frac{\mathbb {P}(\tau _{1}>t^{1/m})}{\mathbb {P}(\tau _{1}>1)}\,\text {d}t. \end{aligned}$$

Therefore, Jensen’s inequality yields

$$\begin{aligned} \frac{\mathbb {E}[T_{3d }^{m}]}{\mathbb {E}[T_{+}^{m}]}&\le \frac{\int _{0}^{1}(\mathbb {P}(\tau _{1}^{+}>t^{1/m}))^{N}\,\text {d}t +(\mathbb {P}(\tau _{1}>1))^{N}\int _{1}^{\infty }\frac{\mathbb {P}(\tau _{1}>t^{1/m})}{\mathbb {P}(\tau _{1}>1)}\,\text {d}t}{(\mathbb {E}[T_{+}])^{1/m}}\\&\le 1+\frac{(\mathbb {P}(\tau _{1}>1))^{N}}{(\mathbb {E}[T_{+}])^{1/m}}\int _{1}^{\infty }\frac{\mathbb {P}(\tau _{1}>t^{1/m})}{\mathbb {P}(\tau _{1}>1)}\,\text {d}t. \end{aligned}$$

To complete the proof, we need only that $\mathbb {E}[T_{+}]$ decays slower than $(\mathbb {P}(\tau _{1}>1))^{N}$ as $N\rightarrow \infty$, which is implied by Lemma 7. $\square$

Proof of Lemma 7

Let $T_{1d }$ be the first time that any ${\widetilde{Z}}_{n}$ hits $z=b$,

$$\begin{aligned} T_{1d } :=\min _{n}\{{\widetilde{\tau }}_{n,bot }\}, \end{aligned}$$

and let $n_{f }\in \{1,\dots ,N\}$ denote the random index of this fastest ${\widetilde{Z}}_{n}$ particle

$$\begin{aligned} n_{f }\in \{1,\dots ,N:{\widetilde{Z}}_{n_{f }}(T_{1d })=b\}. \end{aligned}$$

Define the event that the fastest particle first escapes the cylinder through the bottom

$$\begin{aligned} A :=\{\tau _{n_{f },side }>T_{1d }\}\cap \{{\widetilde{\tau }}_{n_{f },top }>T_{1d }\} =\{T_{+}=T_{1d }\}. \end{aligned}$$

Since

$$\begin{aligned} T_{+}1_{A}=T_{1d }1_{A}\quad \text {almost surely}, \end{aligned}$$

we can apply the same argument as in Sect. 2 if we can manage to show that $\sqrt{\mathbb {P}(A^{c})}(\log N)^{m}\rightarrow 0$ as $N\rightarrow \infty$.

By De Morgan’s laws, we have that

$$\begin{aligned} \mathbb {P}(A^{c})\le \mathbb {P}(\tau _{n_{f },side }\le T_{1d })+\mathbb {P}({\widetilde{\tau }}_{n_{f },top }\le T_{1d }). \end{aligned}$$

The first probability can be handled by the same argument as in Sect. 2.

To handle $\mathbb {P}({\widetilde{\tau }}_{n_{f },top }\le T_{1d })$, note that

$$\begin{aligned} \mathbb {P}({\widetilde{\tau }}_{n_{f },top }\le T_{1d }) \le \mathbb {P}({\widetilde{\tau }}_{n_{f },top }\le \delta )+\mathbb {P}(T_{1d }\ge \delta )\quad \text {for any }\delta >0. \end{aligned}$$

As in the proof of Theorem 1, we may bound $\mathbb {P}(T_{1d }\ge \delta )$ for sufficiently small $\delta >0$ by

$$\begin{aligned} \mathbb {P}(T_{1d }\ge \delta ) =\int _{\delta }^{\infty }N(S_{0}(t))^{N-1}f_{0}(t)\,\text {d}t&\le N(S_{0}(\delta ))^{N-1}\\&\le N\big [1-C_{3}\delta ^{1/2}\exp (-C_{4}/\delta )\big ]^{N-1}. \end{aligned}$$

Furthermore, it is immediate that

$$\begin{aligned} \mathbb {P}({\widetilde{\tau }}_{n_{f },top }\le \delta ) \le \mathbb {P}({\widetilde{\tau }}_{1,top }\le \delta ). \end{aligned}$$

Now, notice that ${\widetilde{\tau }}_{1,top }$ has the same law as the first time a standard Brownian motion, $W(t)\in \mathbb {R}$, hits $(h-z_{0})/\sqrt{2D}$. In particular, the reflection principle (Durrett 2019) gives

$$\begin{aligned} \mathbb {P}({\widetilde{\tau }}_{1,top }\le t) =2\mathbb {P}(W(t)\ge (h-z_{0})/\sqrt{2D}) =1-\text {erf}((h-z_{0})/\sqrt{4Dt}), \end{aligned}$$

where $\text {erf}(x)$ denotes the error function with large x behavior given in (5.1). Hence,

$$\begin{aligned} \mathbb {P}({\widetilde{\tau }}_{n_{f },top }\le \delta ) \le \frac{2}{\sqrt{\pi }}\frac{\sqrt{4D\delta }}{(h-z_{0})} \mathrm{e}^{-(h-z_{0})^{2}/(4D\delta )}\quad \text {for sufficiently small }\delta >0. \end{aligned}$$

Taking $\delta =(\log N)^{-1/2}$ completes the proof. $\square$

Proof of Lemma 8

Define the first time that the nth particle leaves a ball centered at ${{\mathbf {x}}}_{0}$ of radius $z_{0}>0$,

$$\begin{aligned} \tau _{n,ball } :=\inf \{t>0:\Vert \mathbf {X}_{n}(t)-{{\mathbf {x}}}_{0}\Vert \ge z_{0}\}. \end{aligned}$$

It is immediate that the nth particle cannot reach the target $\partial \Omega _{T }$ before time $\tau _{n,ball }$. That is,

$$\begin{aligned} \tau _{n,ball }\le \tau _{n}\quad \text {almost surely for each }n\in \{1,\dots ,N\}. \end{aligned}$$

(5.2)

Define the radial process

$$\begin{aligned} R_{n}(t) :=\Vert \mathbf {X}_{n}(t)-{{\mathbf {x}}}_{0}\Vert \quad \text {for }t\ge 0. \end{aligned}$$

Notice that $R_{n}$ is not a three-dimensional Bessel process, due to the reflecting boundary $\partial \Omega$. However, $R_{n}$ does satisfy the following stochastic differential equation,

$$\begin{aligned} \text {d}R_{n}(t) =\text {d}R_{n}^{0}(t)+{{\mathbf {n}}}_{R}(\mathbf {X}_{n}(t))\,\text {d}L_{n}(t), \end{aligned}$$

(5.3)

where $R_{n}^{0}$ is a 3d Bessel process. In (5.3), ${{\mathbf {n}}}_{R}$ is the radial component of the inner normal field ${{\mathbf {n}}}:\partial \Omega \mapsto \mathbb {R}^{3}$, and $L_{n}(t)$ is the local time of $\mathbf {X}_{n}(t)$ on $\partial \Omega$. More precisely, $L_{n}(t)$ is nondecreasing and increases only when $\mathbf {X}_{n}(t)$ is on $\partial \Omega$. The significance of the local time term in (5.3) is that it forces $\mathbf {X}_{n}(t)$ to reflect from $\partial \Omega$.

By our assumption in (3.7) that S in (3.6) is a star domain, we are assured that ${{\mathbf {n}}}_{R}(\mathbf {X}_{n}(t))\le 0$ for all $t\le \tau _{n,ball }$. Hence, $R_{n}(t)\le R_{n}^{0}(t)$ for all $t\le \tau _{n,ball }$. Therefore, $\tau _{n,ball }^{0}\le \tau _{n,ball }$ almost surely. Hence, $T_{ball }^{0}\le T_{3d }$ almost surely by (5.2). $\square$