On the probability of fast exits and long stays of planar Brownian motion in simply connected domains

The distribution of the exit time of planar Brownian motion from a domain measures in some sense the size of the domain. It can also be used for the study of analytic functions via the conformal invariance of Brownian motion.

Let $Z_{t}=X_{t}+iY_{t}$, $t\geq 0$ denote standard Brownian motion moving in the plane and starting from the origin, (that is, $Z_{0}=0$ almost surely). We denote by ${\bf P}$ and ${\bf E}$ the corresponding probability measure and expectation, respectively. For a domain $D$ containing $0$ in the complex plane ${\mathbb{C}}$, we denote by $T^{D}$ the first exit time of $Z_{t}$ from $D$; that is

$$T^{D}=\inf\{t>0:Z_{t}\notin D\}.$$

Suppose also that $f$ is a univalent function in the well-known class $\mathcal{S}$. Thus $f$ is a function univalent and holomorphic in the unit disk ${\mathbb{D}}$ with $f(0)=0$ and $f^{\prime}(0)=1$. By a classical result of P. Lévy, the image of $Z_{t}$ under $f$ is a Brownian motion with a time change. We describe now a precise version of this conformal invariance of Brownian motion; (see [Dav79, §2]). Let

$$\rho_{f}(s)=\int_{0}^{s}|f^{\prime}(Z_{t})|^{2}\,dt,\;\;\;\;\;0\leq s<T^{\mathbb{D}}.$$

Observe that $\rho_{f}$ is almost surely strictly increasing and set

$$W_{t}=f(Z_{\rho_{f}^{-1}(t)}),\;\;\;0\leq t<\rho_{f}(T^{\mathbb{D}}).$$

Define also $W_{\rho_{f}(T^{\mathbb{D}})}=\lim_{t\to T^{\mathbb{D}}}W_{\rho_{f}(t)}$ and

$$W_{\rho_{f}(T^{\mathbb{D}})+t}=W_{\rho_{f}(T^{\mathbb{D}})}+(Z_{T^{\mathbb{D}}+t}-Z_{T^{\mathbb{D}}}),\;\;\;\;t>0.$$

Note that $f$ being univalent and holomorphic implies that $P(\rho_{f}(T^{\mathbb{D}})<\infty)=1$; this is a nontrivial statement, but is shown for instance in [Bur77, p.198]. Lévy’s theorem now asserts that $\{W_{t},t\geq 0\}$ is standard planar Brownian motion starting from the origin.

We set

$$\nu(f)=\rho_{f}(T^{\mathbb{D}})=\int_{0}^{T^{\mathbb{D}}}|f^{\prime}(Z_{t})|^{2}\,dt$$

and observe that $\nu(f)$ is the first exit time of $W_{t}$ from $f({\mathbb{D}})$; that is, $\nu(f)=T^{f({\mathbb{D}})}$. “The distribution of $\nu(f)$ is an intuitively appealing measure of the size of $f({\mathbb{D}})$” [Dav79]. In several classical extremal problems for functions in the class ${\mathcal{S}}$, the identity function $I(z)=z$ is the “smallest” function while the Koebe function $k(z)=z/(1-z)^{2}$ is the “largest” function in $\mathcal{S}$. B. Davis [Dav79] conjectured that

(1.1)

$${\bf E}[\Phi(\nu(I))]\leq{\bf E}[\Phi(\nu(f))],$$

for all $f\in\mathcal{S}$ and all increasing convex functions $\Phi:[0,\infty)\to{\mathbb{R}}$, and suggested that perhaps

$${\bf P}(\nu(I)>t)\leq{\bf P}(\nu(g)>t),$$

for all $t>0$ and $g\in\mathcal{S}$. Apropos the first conjecture T. McConnell [McC85] proved that

(1.2)

$${\bf E}[\nu(I)^{p}]\leq{\bf E}[\nu(f)^{p}],\;\;\;0<p<\infty,$$

but the full conjecture remains open, as far as we know. McConnell also disproved the second conjecture by finding functions $g\in\mathcal{S}$ such that

(1.3)

$${\bf P}(\nu(I)>t)>{\bf P}(\nu(g)>t),$$

for all sufficiently small $t>0$. Davis also asked in what sense, with regard to $\nu(f)$, the Koebe function is the largest in $\mathcal{S}$.

Motivated by these developments, we have considered the following questions. Given two simply connected planar domains $U,W\neq{\mathbb{C}}$ containing $0$, what sufficient conditions can we place on the domains so that we are more likely to have fast exits (meaning for instance ${\bf P}(T^{U}<t)>{\bf P}(T^{W}<t)$ for $t$ small) from $U$ than from $W$, or long stays (meaning ${\bf P}(T^{U}>t)>{\bf P}(T^{W}>t)$ for $t$ large). We have found that the primary factor influencing the probability of fast exits is the proximity of the boundary to the origin. In order to make this more precise, let us introduce the following notation. For any simply connected domain $V$, let

$$d(V)=\inf\{|z|:z\in\partial V\}.$$

We then have the following theorem, which is the main result of the paper.

We do not know whether $\frac{1}{\sqrt{2}}$ is the optimal constant; it may even be that it may be replaced by 1. If so, this would be a surprising property of planar Brownian motion. Evidence for this possibility can be found in the fact that it is true if $U$ is a half-plane. This is discussed in more detail in Section 3.

Naturally, it would be nice to have an analog for long stays. We believe that the important factor for long stays in domains is the moments of the exit time. To be precise, for domain $V$ let

$${\rm H}(V)=\sup\{p>0:{\bf E}[(T^{V})^{p}]<\infty\};$$

note that ${\rm H}(V)$ is proved in [Bur77] to be exactly equal to half of the Hardy number of $V$, a purely analytic quantity, as defined in [Han70], and is therefore calculable for a number of common domains. Furthermore ${\rm H}(V)\geq\frac{1}{4}$ as long as $V\neq{\mathbb{C}}$. We have the following simple result.

We conjecture that this proposition is true with the $\limsup$ replaced by $\lim$, but have not been able to prove it (except when $W$ is either a half-plane or quarter-plane). This is discussed in detail in the final section.

In this section we prove Theorem 1 and Proposition 1.

As was discussed in the Introduction, Theorem 1 shows that the unit disk is not extremal among Schlicht domains for fast exits. In fact, many Schlicht domains, including the Koebe domain and the half-plane $\{{\rm Re}(z)\geq-\frac{1}{2}\}$, have a higher probability of fast exits. We do not know if the constant $\frac{1}{\sqrt{2}}$ in Theorem 1 is optimal; it would be nice to know what is the best possible. As mentioned in the introduction, there is reason to suspect that the best possible constant is even 1. To see this, let $H_{\alpha}=\{{\rm Re}(z)<\alpha\}$. We then have the following proposition.

Remark: Naturally, this result holds with $H_{\alpha}$ replaced by any domain contained in a rotation of $H_{\alpha}$.

The moments of the exit time have been considered previously by several authors. In [Bur77], it is shown that for the wedge $R_{\theta}=\{|{\rm Arg}(z)|<\theta\}$, that is, the infinite wedge centered at the positive real axis of angular width $2\theta$, we have ${\bf E}[(T^{R_{\theta}})^{p}]<\infty$ if and only if $p<\frac{\pi}{4\theta}$, so

(3.2)

$${\rm H}(R_{\theta})=\frac{\pi}{4\theta}.$$

A domain $W$ is spiral-like of order $\sigma\geq 0$ with center $a$ if, for any $z\in W$, the spiral $\{a+(z-a)\mbox{ exp}(te^{-i\sigma}):t\leq 0\}$ also lies within $W$; $W$ is star-like if it is spiral-like of order $\sigma=0$. The quantity ${\rm H}(W)$ can be determined explicitly if $W$ is star-like or spiral-like, as is shown in [Mar15], with equivalent analytic results appearing in [Han70] and [Han71]. In particular, if we take $a=0$ then, since $W$ is spiral-like, the quantity

(3.3)

$${\cal A}_{r,W}=\max\{m(E):E\mbox{ is a subarc of }W\cap\{|z|=r\}\},$$

is non-increasing in $r$ (here $m$ denotes angular Lebesgue measure on the circle). We may therefore let ${\cal A}_{W}=\lim_{r\nearrow\infty}{\cal A}_{r,W}$, and then [Mar15, Thm. 2] we have $H(W)=\frac{\pi}{2{\cal A}_{W}\cos^{2}\sigma}$.

As mentioned before we suspect that in many cases the $\limsup$ in Proposition 1 is not necessary, and venture the following conjecture.

Note that this includes the case that $W$ is star-like, as well as the wedge $R_{\theta}$. This conjecture would follow from the following, if true.

Our evidence for the truth of Conjecture 1 is at follows. First note that Markov’s inequality, used as in Proposition 1, yields the following fact.

Furthermore the bound required is true in the $\limsup$ sense, as is shown in the proof of Proposition 1. Next we prove Conjecture 1 when $W$ is a half-plane or quarter-plane. Let $W=\{{\rm Re}(z)<1\}$; recall from (3.2) that ${\rm H}(W)=\frac{1}{2}$. Then, using the reflection principle, ${\bf P}(T^{U}>t)$ can be bounded below as follows.

			$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!{\bf P}^{0}(X_{s}<1,\;\;\;\forall s\in[0,t])$
		$\displaystyle=$	$\displaystyle 1-2{\bf P}^{0}(X_{t}\geq 1)=1-2\frac{1}{\sqrt{2\pi t}}\int_{1}^{\infty}e^{-x^{2}/2t}\;dx$
		$\displaystyle=$	$\displaystyle\frac{2}{\sqrt{\pi}}\int_{0}^{\frac{1}{\sqrt{2t}}}e^{-\xi^{2}}\;d\xi\geq\frac{C}{\sqrt{t}}.$

Now suppose $W=\{{\rm Re}(z)>0,{\rm Im}(z)>0\}$; recall from (3.2) that ${\rm H}(W)=1$. Then, using the independence of the one dimensional components of planar Brownian motion, and the calculation for the half-plane, we have

$\displaystyle{\bf P}^{1+i}(T^{W}>t)$

$\displaystyle=$

$\displaystyle{\bf P}^{1}(X_{s}>0,\;\;\;\forall s\in[0,t])^{2}\geq\frac{C}{t}$

for $t$ bounded away from $0$. Finally, we prove an improvement of Lemma 5, as we promised in Section 2.

Proof: Let $p^{U}(t,0,w)$ be the transition density function for Brownian motion killed upon hitting $\partial U$. Then (see e.g. [CZ12, Theorem 2.4])

(3.7)

$${\bf P}(T^{U}>t)=\int_{U}p^{U}(t,0,w)\;A(dw),\;\;\;0<t<+\infty,$$

where $A$ denotes the area measure. A similar formula holds for ${\bf P}(T^{K_{\alpha}}>t)$. Therefore, it suffices to prove that

(3.8)

$$\int_{U}p^{U}(t,0,w)\;A(dw)\leq\int_{U}p^{K_{\alpha}}(t,0,w)\;A(dw),\;\;\;0<t<+\infty.$$

We will prove (3.8) using the theory of polarization and symmetrization. We refer to [Hay94], [Dub14], [BS00] for the definitions and basic facts.

The function $p^{U}(t,0,w)$ satisfies the heat equation on $U$. Let $P_{H}U$ denote the polarization of $U$ with respect to a half-plane $H$ with $0\in\partial H$. Then [BS00, Theorem 9.4]

(3.9)

$$p^{U}(t,0,w)+p^{U}(t,0,R_{H}w)\leq p^{P_{H}U}(t,0,w)+p^{P_{H}U}(t,0,R_{H}w),\;\;\;0<t<\infty,$$

where $R_{H}w$ denotes the reflection of $w$ in the line $\partial H$. It follows from (3.9) that for every $r\in(0,+\infty)$,

(3.10)

$$\int_{0}^{2\pi}p^{U}(t,0,re^{i\theta})d\theta\leq\int_{0}^{2\pi}p^{P_{H}U}(t,0,re^{i\theta})d\theta,\;\;\;0<t<\infty.$$