Maximizing the $p$-th moment of exit time of planar Brownian motion from a given domain

Let $Z_{t}:=X_{t}+iY_{t}$ be a planar Brownian motion starting at a point $a$ in a domain $U$. We will let $\tau_{U}=\tau_{U}(a)$ be the first time that $Z_{t}$ exits $U$, and we will use the standard notation $E_{a}$ to denote expectation conditioned on $Z_{0}=a$ a.s. The focus of this paper is the following optimization problem.

For a given domain in the plane and $0<p<\infty$, find the point $a$ which maximizes the quantity $E_{a}[(\tau_{U})^{p}]$.

We will refer to such a point as a $p$-th center of $U$; it is not in general unique, as the easy example of an infinite strip shows. For many domains, even simple ones such as an isosceles triangle, it is difficult to find any of the $p$-th centers, however we will show how elementary coupling arguments and the conformal invariance of Brownian motion in many cases allows us to locate a small region in $U$ which must contain all $p$-th centers. In certain cases in which the domain in question has a high degree of symmetry, it will allow us to locate all $p$-th centers.

Before describing our methods, we present a brief overview of some earlier works related to this problem. The case $p=1$ is commonly referred to as the "torsion problem" due to its connection with mechanics, and is naturally the most tractable. The function $h(a)=E_{a}[\tau_{U}]$ satisfies $\Delta h=-2$, and therefore p.d.e. techniques can be employed to great effect. [1, Ch. 6] contains a good account of this problem and methods of attacking it in special cases, such as when the domain in question is convex. Further results along the same lines, focussing in particular on convex domains, can be found in [2, 3, 4].

Other interesting related problems have been tackled by p.d.e. methods. For example, in the famous paper [5] (see also the related work [6]) eigenvalue techniques are used to demonstrate relationships between $E_{a}[\tau_{U}]$ and geometric qualities of the domain, such as the size of the hyperbolic density and the inradius (the radius of the largest disk contained in the domain). The methods developed there have been extended by other authors in a number of different directions. For example, in [7] a number of related stochastic domination results were proved concerning convex domains in ${\mathbb{R}}^{n}$ and various types of symmetrizations. These results allow conclusions to be reached concerning the comparison of $p$-th moments of the exit times from these domains. One striking consequence of the eigenvalue methods is the fact that over all domains with a given area the disk maximizes the $p$-th moment of the exit time of Brownian motion for all $p$. The recent work [8] contains a discussion and refinements of this result.

Our results differ from those described above in the following ways. We have not employed p.d.e. methods at all, choosing instead to work with an elementary coupling method. Perhaps as a consequence of this, convexity plays little role in our discussion, although a weaker concept called $\Delta$-convexity (defined below) will be important. The type of coupling we will use is not entirely new, and has found a number of uses in related topics, for instance in investigations into the "hot spots" conjecture such as [9, 10, 11]. However, we believe that it has not been applied directly in the manner that we do before. Furthermore, we restrict our attention to two dimensions, which allows conformal mappings to take prominence and to extend the standard notion of coupling. We present several methods for localizing the $p$-centers of a domain, and then consider a number of specific domains, showing in each case how our methods can be used to localize the $p$-th centers of the domain. In what follows we assume $p$ is a fixed positive number; however, in order to reduce the qualifications needed to state our results, for any planar domain $U$ for which we are interested in maximizing the $p$-th moment we will assume that $E_{a}[(\tau_{U})^{p}]<\infty$ for all points $a\in U$; this would follow if $E_{a}[(\tau_{U})^{p}]<\infty$ for any $a\in U$, as is shown in [12].

This theorem allows us in essence to exclude the symmetric side of any partial symmetry axis for $U$ in our search for $p$-th centers. The only exception to this rule is when $\sigma_{\Delta}(S)=U\setminus(S\cup\Delta)$, i.e. when $\Delta$ is a symmetry axis of $U$; however in most cases a symmetry axis will contain all $p$-th centers. To see why this is so, we need another definition.

It is clear that any convex $U$ is $\Delta$-convex for any symmetry axis $\Delta$, and a less trivial example can be given by $\{-f(x)<y<f(x)\}$, where $f$ is a positive continuous function on the real line, which is $\Delta$-convex with $\Delta={\mathbb{R}}$. A domain which is not $\Delta$-convex with respect to a symmetry axis can be given by $W_{\epsilon}=\{-\epsilon<y<\epsilon,|x|<1\}\cup\{|z-1|<\frac{1}{2}\}\cup\{|z+1|<\frac{1}{2}\}$ with $\epsilon<\frac{1}{2}$; this has the real and imaginary axes as symmetry axes but is not $\Delta$-convex with respect to the imaginary axis (though it is with respect to the real line). As will be seen below, this domain also shows why $\Delta$-convexity is required in the following proposition.

Note that this proposition completely solves our problem in the case that our domain is $\Delta$-convex with respect to two or more non-parallel symmetry axes, since all $p$-th centers must lie at their point of intersection, and we have also incidentally proved the purely geometrical fact that all such symmetry axes must coincide at a unique point; more on this in the final section. Thus, for instance, all $p$-centers of any regular polygon, a circle, an ellipse, a rhombus, and any number of other easily constructed examples must lie at their natural centers. To see an example of a domain with intersecting symmetry axes but where the point of intersection is not a $p$-th center, let us return to the domains $W_{\epsilon}$ described immediately before this proposition. Proposition 4 implies that all $p$-th centers lie on the real line, but it is easy to see that if we make $\epsilon$ sufficiently small then 0, the intersection point of the two symmetry axes, will not be a $p$-th center (clearly $\tau_{W_{\epsilon}}(1)\geq\tau_{\{|z-1|<1/2\}}(1)$, so that $E_{1}[\tau_{W_{\epsilon}}^{p}]$ remains always greater than a positive constant, but $\tau_{W_{\epsilon}}(0)$ decreases monotonically to 0 a.s. as $\epsilon\searrow 0$, so that $E_{1}[\tau_{W_{\epsilon}}^{p}]\searrow 0$).

Let us now look at an example that shows the use of the results proved to this point, but also their limitations. Suppose $U$ is the isosceles right triangle with vertices at $-1,1,$ and $i$. The imaginary axis is an axis of symmetry, and $U$ is $\Delta$-convex with respect to this axis, so all $p$-th centers must lie on the imaginary axis. The line $\{y=\frac{1}{2}\}$ is a partial symmetry axis for $U$, with $U\cap\{y>\frac{1}{2}\}$ the symmetric side, so all $p$-centers must lie on $\{x=0,y\leq\frac{1}{2}\}$. Now, common sense tells us that the $p$-th centers can’t be too close to the real axis as well, because this is a boundary component, but there is no good partial symmetry axis to apply to conclude that rigorously. The way out of this difficulty is to extend our method of reflection to curves more general than straight lines. For this, we will need to utilize the conformal invariance of Brownian motion, via the following famous theorem of Lévy (see [13] or [14]).

This allows us to extend Theorem 2, as follows.

We can obtain a corollary which will be useful for the isosceles triangle and in other cases by taking $f(z)=z_{0}+R\Big{(}\frac{z+i}{z-i}\Big{)}$ for $z_{0}\in{\mathbb{C}}$ and $R>0$, which takes the real axis to the circle $C=\{|z-z_{0}|=R\}$, the upper half-plane to the outside of $C$, and the lower half-plane to the inside. We have $|f^{\prime}(z)|=\frac{2R}{|z-i|^{2}}$, and it is easy to check that $|f^{\prime}(z)|>|f^{\prime}(\bar{z})|$ for all $z$ in the upper half-plane. Applying Proposition 6 and working through the implications yields the following.

Note that here $\sigma_{C}$ denotes reflection over the circular arc $C$, that is, $\sigma_{C}(z)=z_{0}+\frac{R^{2}}{\stackrel{{\scriptstyle\line(1,0){18.0}}}{{z-z_{0}}}}$. We remark further that the singularity that $f$ has at $i$ does not cause a problem in this result, because $P_{0}(Z_{t}=i\mbox{ for some }t\geq 0)=0$, so a.s. a Brownian motion starting at $0$ will not hit the singularity, anyway. The compact set $\{B_{t}:0\leq t\leq\tau_{U}\}$ is therefore bounded away from $i$ a.s., and the result goes through.

Let us now apply this corollary to the isosceles right triangle $U$ above. If $C$ is the circle passing through $-1$ and $1$ which intersects the real axis at angles of $\frac{\pi}{8}$, then the reflection of the set ${\cal A}={\cal I}\cap U$ will be the region ${\cal B}$ in the upper half-plane bounded by $C$ and the circle which passes through $-1$ and $1$ and intersects the real axis at angles of $\frac{\pi}{4}$; this can be seen by noting that the transformation $\sigma_{C}$ preserves angles and and also preserves the class of circles on the Riemann sphere (which includes lines, interpreted as circles through $\infty$).

As this region lies within $U$, we conclude that no $p$-th centers lie within ${\cal A}$. A bit of Euclidean geometry shows that $C$ intersects the imaginary axis at $\csc(\frac{\pi}{8})-\cot(\frac{\pi}{8})\approx.20$, and coupled with our observations above we see that all $p$-th centers must lie on the imaginary axis between the points $.2i$ and $.5i$. In fact, the upper bound of $.5i$ can be improved by using the angle bisector of the angles at $1$ or $-1$, see Example 2 in Section 3. The reader may also have observed that the reflected circular domain does not do a good job of filling the triangle, and therefore it stands to reason that the lower bound may also be improved; more on this in the final section.

Finally, the following result can be useful when $U$ is mapped to itself by an antiholomorphic function $\bar{f}$ (this means that the conjugate of $\bar{f}$, $f(z)$, is holomorphic). We will denote the derivative of this function (with respect to $\bar{z}$ by $\overline{f^{\prime}(z)}$). An example of this is when $U$ is an annulus, as will be explored in Section 3.

Remark: Reflection over the circle, obtained above as a corollary of Proposition 6, can just as easily be deduced as a corollary of Proposition 8.

In this section we work through a series of examples that show how our results may be applied.

Remarks:

• •

  Another way to get the same lower bound as above is to note that Proposition 6 extends to non-injective maps in suitable situations. We may use the map

  $$f:\begin{aligned} \left\{\ln r<Re(z)<\ln R\right\}&\longrightarrow\mathcal{A}_{r,R}\\ z&\longmapsto e^{z}\end{aligned}$$

  and apply this extension of Proposition 6 with reflection axis $\{Re(z)=\frac{\ln R+\ln r}{2}\}$ in order to obtain the result.

• •

  It should be mentioned that an explicit formula for the first moment can be obtained by Dynkin’s formula, and it is

  $$E_{z}(\tau_{\mathcal{A}_{r,R}})=\frac{R^{2}\ln\left(\frac{|z|}{r}\right)-r^{2}\ln\left(\frac{|z|}{R}\right)}{\ln\frac{R}{r}}-|z|^{2}.$$

  This can be shown to be maximal at $|z|=\sqrt{\frac{R^{2}-r^{2}}{2\ln\frac{R}{r}}}$. Our estimates are therefore not necessary for the first moments, but as an aside we obtain the inequality

  $$\sqrt{Rr}<\sqrt{\frac{R^{2}-r^{2}}{2\ln\frac{R}{r}}}<\frac{R+r}{2}.$$

  Setting $a=R^{2}$, $b=r^{2}$, and squaring the inequalities gives the following:

  $$\sqrt{ab}<\frac{a-b}{\ln\frac{a}{b}}<\frac{a+b+2\sqrt{ab}}{4}\leq\frac{a+b}{2}.$$

  The quantity $\frac{a-b}{\ln\frac{a}{b}}$ is of great importance in the study of heat flow, and in that context it is known as the logarithmic mean temperature difference, or LMTD (see [15]). We have therefore given a new proof of the fundamental fact that the LMTD lies between the arithmetic and geometric means, and in fact have proved that the upper bound can be lowered to the arithmetic mean of the arithmetic and geometric means.

As remarked earlier, in Figure 2.1 the regions ${\cal A}$ and ${\cal B}$ do not fill all of $U$, and it is natural to search for a a better bound by finding a conformal map that fills the entire domain. Let us consider the Schwarz-Christoffel transformation sending the unit disk to $U(\theta)$ given by (see [17, Ch. 2])

It can be checked then that $|f^{\prime}(z)|>|f^{\prime}(\bar{z})|$ whenever $Re(z)>0$, so Proposition 6 implies that no $p$-th centers can be found in the image of ${\mathbb{D}}\cap\{Re(z)<0\}$. From this point, a numerical method can be employed, and the resulting bound should improve the one we found, if desired.

As was mentioned in connection with $\Delta$-convexity, there are some purely geometrical consequences of our results. In that context, the following may be proved.

As a corollary of this, and of our probabilistic results above, we obtain the following.

The authors would like to thank Paul Jung, Fuchang Gao, and Lance Smith for helpful conversations. We are also grateful to several anonymous referees for helpful comments.