# Brownian symmetrization of planar domains

Kais Hamza Monash University. Maher Boudabra King Fahd University for
Petroleum and Minerals.

###### Abstract

One of aims of this note is to capture the interest of the mathematical
community to a novel transformation, which we shall call “Brownian
symmetrization”. This transformation arises from the solution of the planar
Skorokhod embedding problem, first appeared in [13]. Brownian symmetrization
shares some properties with the famous Steiner symmetrization. However, we
show that these two transformations are not the same and they do not affect
each other.

##  1 Introduction

In [13], the following question has been posed and answered : Given a
distribution $\mu$ with zero mean and finite second moment, is there a simply
connected domain $U$ (containing the origin) such that if
$(Z_{t}=X_{t}+Y_{t}i)_{t\geq 0}$ is a standard planar Brownian motion, then
$X_{\tau_{U}}=\Re(Z_{\tau_{U}})$ samples as $\mu$ and
$\mathbb{E}(\tau_{U})<+\infty$, where $\tau_{U}$ is the exit time from $U$ (We
shall omit the subscript of $\tau$ if the domain is clear from the context) ?
In other words, we seek a domain $U$ so that $Z_{\tau_{U}}\sim\mu$. A domain
that solves the problem shall be called a $\mu$-domain [4]. The answer of the
question is affirmative. The construction process adopted by the author
generates a domain $U$ with the following geometric properties :

  * •

$U$ is symmetric over the real line.

  * •

If $\mu(\\{x\\})>0$ then $\partial U$ contains a vertical line segment
(possibly infinite).

One can notice that if $(a,b)$ is not supported by $\mu$, i.e $(a,b)$ is null
set for $\mu$ 111 It means that the c.d.f of $\mu$ is constant on $(a,b)$.,
then, due to the simple connectivity requirement, any $\mu$-domain would
contain the vertical strip $(a,b)\times(-\infty,+\infty)$. The constraint of
the finiteness of the average of the exit time $\tau$ is the one making the
problem challenging. To see this, consider the Redmacher distribution
$\mu=\frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_{1}$. The real part of the
stopped Brownian motion samples according to $\mu$ for the vertical strip
$\\{-1\leq\Re(z)\leq 1\\}$ and the "ripped" plane $\mathbb{C}-\\{\Re(z)=\pm
1,|\Im(z)|\geq 1\\}$. But, in average, the Brownian motion spends a finite
amount of time to leave the strip while it needs an infinite amount of time to
get out of the "ripped" plane. That is, the vertical strip is the correct
domain for such a distribution. For the sake of clarity, we give an outline of
the key idea of the proof (See [13] for more details). The core tool of Gross
idea is the following theorem.

######  Theorem 1 (LÃ©vy’s theorem).

Let $f:U\rightarrow\mathbb{C}$ be a non constant univalent function and
$Z_{t}$ be a planar Brownian motion running inside $U$. Then there is a planar
Brownian motion $W_{t}$ such that $f(Z_{t})=W_{\sigma(t)}$ with

$\sigma(t)=\int_{0}^{t}|f^{\prime}(Z_{s})|^{2}ds$

and $t\in[0,\tau_{U}]$.

Theorem 1 is commonly known as the conformal invariance principle. That is,
the image of a planar Brownian motion under the action of non constant
analytic function is yet another planar Brownian motion, except it runs in a
different speed. With the same notation, it is equivalent to say that
$f(Z_{\sigma^{-1}(t)})$ is planar Brownian motion. The random time
$\sigma(\tau_{U}^{-})$ is referred to as the projection of the exit time
$\tau_{U}$. It is not always the case that $\sigma(\tau_{U}^{-})$ is the exit
time of $f(U)$ unless $f$ is proper which is the case when $f$ is univalent.
In particular, the law of $f(Z_{\tau_{U}})$ is obtainable from that of
$Z_{\tau_{U}}$(we refer the reader to [17] for more illustrations of this
fact). In order to solve the problem, Gross’s smart idea was to construct a
univalent map acting on the unit disc such that $\Re(f(e^{ti}))$ samples as
$\mu$. By the aforementioned conformal invariance principle, the distribution
of the real part of a stopped planar Brownian motion on the boundary of
$f(\mathbb{D})$ follows exactly $\mu$. If $F(x)$ denotes the c.d.f of $\mu$
then the pseudo inverse of $\mu$ (also referred to as the quantile function)
is defined by $G_{\mu}(u):=\inf\\{x\mid F(x)\geq u\\}$. A well known property
of $G_{\mu}$ is that

$G_{\mu}(\mathrm{Uni}(0,1))\sim\mu$

which follows immediately from the definition of $G_{\mu}$ [9]. Gross then
considered the Fourier series of the function
$\varphi(\theta):=G_{\mu}(\frac{|\theta|}{\pi})$ over the interval
$(-\pi,\pi)$, say

$\sum_{n=1}^{\infty}\widehat{\varphi}(n)\cos(n\theta)$

where $\widehat{\varphi}(n)$ is the $n^{th}$ Fourier coefficient (
$\widehat{\varphi}(0)$ is zero because $\mu$ is centered). The clue result of
Gross is

######  Theorem 2.

The map

$\psi(z):=\sum_{n=1}^{\infty}\widehat{\varphi}(n)z^{n}$

is one to one in $\mathbb{\mathbb{D}}$.

In [5], the authors extended the construction to any distribution with some
finite $p^{th}$ moment with $p>1$. Their extension uses a deep result in
Fourier analysis: the Carleson-Hunt theorem [12, 8]. It states that Fourier
series converges point-wisely a.e when the underlying function is in $L^{p}$
for some $p>1$, which makes the provided solution a strong one. As one can
see, the technique to produce such $\mu$-domains is purely analytic as it
relies on the Fourier expansion of the quantile function $\varphi(\theta)$. In
particular, the geometric properties are encoded by the conformal map
$\psi(z)$. For example, the symmetry with respect to the real axis is due to
the fact that the coefficients of $\psi(z)$ are real. In [5], the two authors
gave a criterion to guarantee the uniqueness of Gross construction. That is,
they proved that if a $\mu$-domain is symmetric over the real line and
$\Delta$-convex then it is unique, provided that
$\mathbb{E}(\tau^{\frac{p}{2}})<\infty$ for some $p>1$. The $\Delta$-convexity
property means that the segment joining any point of the domain and its
symmetric point w.r.t to the real line remains inside the domain [2, 5]. We
shall call such a domain of Gross type. In particular, there is one to one
correspondence between distributions and Gross domains. For example, the unit
disc corresponds to the scaled and centered arc-sine law

$d\mu(x)=\frac{1}{\pi\sqrt{1-x^{2}}}1_{\\{x\in(-1,1)\\}}.$

The area of Gross $\mu$-domain is given by

$\int_{\mathbb{D}}|\psi^{\prime}(z)|^{2}d\sigma(z)=\pi\sum_{n=1}^{\infty}n\widehat{\varphi}(n)^{2}.$
(1)

where $d\sigma(z)$ denotes the infinitesimal area [11]. The identity (1) shows
that the area of Gross domains is at least $\pi\widehat{\varphi}(1)$, which
corresponds to the centered disc of radius $|\widehat{\varphi}(1)|$.

The following algorithm can be coded to generate the $\mu$-domain using Gross
method. Note that the Cartesian equation of the boundary is obtainable using
Hilbert transform [3]

  1. 1.

Run $X=\Re(Z_{\tau})$ where $Z_{\tau}$ is the stopped Brownian motion.

  2. 2.

Compute $F_{X}(x)=\mathbb{P}(X\leq x)$.

  3. 3.

Compute $G(u)=F_{x}^{-1}(u)$.

  4. 4.

Compute
$\widehat{\varphi}(n)=\frac{2}{\pi}\int_{0}^{\pi}G(\frac{\theta}{\pi})\cos(n\theta)d\theta$.

  5. 5.

Plot $\psi(e^{ti})$ for $t\in(-\pi,\pi)$.

Algorithm 1 Algorithm for Gross solution

The planar curve $\psi(e^{ti})$ is the radial limit of $\psi$, i.e
$\psi(e^{ti}):=\lim_{r\rightarrow 1}\psi(re^{ti})$. The existence of the
radial limit follows from the theory of Hardy spaces [10, 19]. More generally,
algorithm (1) applies to $U$ as long as its underlying distribution $\mu$ has
some finite $p^{th}$ moment with $p>1$. This condition is obviously fulfilled
by domains contained in vertical strips since bounded distributions have
finite $p^{th}$ moment for any $p>1$. Although, catching some $p>1$ is not an
easy task for general unbounded domains. Such a $p$ is calculable in the case
of wedges. More precisely, if $W_{2\theta}$ denotes the wedge
$\\{\mathbf{Arg}(z)\in(-\theta,\theta)\\}$ then $p^{th}$ moments of $\mu$ are
finite whenever $p<\frac{\pi}{2\theta}$ [1, 6]. In particular, any wedge
strictly included in the right half plane carries a distribution $\mu$ of a
finite $p^{th}$ moment for some $p>1$. In a striking work of D. Burkholder
[6], the author provided a very strong result connecting planar Brownian
motion and analytic functions. More precisely, he proved the following
equivalences

$\textstyle{H_{p}(f)<+\infty}$
$\textstyle{\mathbf{\mathbf{E}}(\tau^{\frac{p}{2}})<+\infty}$
$\textstyle{\mathbf{\mathbf{E}}(Z_{\tau}^{*p})<+\infty}$
$\scriptstyle{\mathbf{\mathbf{E}}(\ln(\tau))<+\infty}$$\textstyle{\mathbf{\mathbf{E}}(\lvert
Z_{\tau}\rvert^{p})<+\infty}$

where $Z_{\tau}^{*}:=\sup_{0\leq t\leq\tau}|Z_{\tau}|$ and $H_{p}(f)$ is the
$p^{th}$ Hardy norm 222The word ”norm” is an abuse of terminology. Hardy norms
are ”true” norms only when $p\geq 1$ since we lose the convexity property if
$p<1$. of $f:\mathbb{D}\rightarrow U$ defined by

$H_{p}(f)=\sup_{0\leq r<1}\left\\{\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{\theta
i})|^{p}d\theta\right\\}^{\frac{1}{p}}.$

Note that the requirement on $\mathbb{E}(\ln(\tau))$ has been relaxed to the
finiteness of $\mathbb{E}(\ln(\tau+1))$ which was shown to cover all simply
connected domains (See [1] for more details). The supremum of $p$ such that
$H_{p}(f)$ is finite is called Hardy number of $f$, first introduced in [14].

The technique developed to construct $\mu$-domains in [13] triggered us to
consider an underlying geometric transformation that we shall call Brownian
symmetrization (or Gross symmetrization). That is, consider a bounded domain
$U$ and run a planar Brownian motion $(Z_{t})_{t}$ until it hits $\partial U$
and denote by $\mu$ the distribution of $\Re(Z_{\tau})$ with $\tau$ being the
exit time of $Z_{t}$ from $U$. The Brownian symmetrization of $U$ is simply
the $\mu$-domain constructed as in [13]. We shall denote the Brownian
symmetrization by $\mathfrak{B}(U)$

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$extract
$\mu$$\textstyle{\mu\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W=\mathfrak{B}(U)=\text{Gross
domain generated form $\mu$}}$

##  2 Steiner symmetrization and Brownian symmetrization are different

Steiner symmetrization is a geometric transformation technique used in convex
geometry and differential geometry. It involves replacing a given geometric
shape, typically a bounded set in Euclidean space, with its symmetric
counterpart with respect to a chosen axis of symmetry. More precisely, Steiner
symmetrization works as follows : Let $U$ be a planar bounded domain
containing the origin. For every $z$ in $A$ draw the vertical line $x=\Re(z)$,
say $L_{x}$, and let $\ell_{x}$ be the length of $L_{x}\cap U$. The Steiner
symmetrization of $U$, denoted here by $\mathfrak{S}(A)$, is simply the union
of all segments
$\\{\Re(z)+t\ell_{\Re(z)}\mid-\frac{1}{2}<t<\frac{1}{2}\\}_{z\in U}$. A well
known property of Steiner symmetrization is that it preserves the area of $U$
and does not increase the perimeter [15]. Furthermore, one can remark that
$\mathfrak{S}(U)$ is $\Delta$-convex and symmetric over the real line. In
particular, both Brownian symmetrization and Steiner symmetrization share two
common properties : $\Delta$-convexity and the symmetry over the real axis.
Using the uniqueness criterion in [3], the following identities hold

$\mathfrak{S}\circ\mathfrak{B}(U)=\mathfrak{B}(U)$

and

$\mathfrak{B}\circ\mathfrak{S}(U)=\mathfrak{S}(U)$

In [13], the author provided the following example to show the non uniqueness
of the solution of the planar Skorokhod embedding problem. Let $\mu$ be the
probability distribution of $\Re(Z_{\tau})$ where $\tau$ is the exit time from
the domain $B$ limited by the blue boundary. The Brownian symmetrization of
$B$ is the domain $R$ limited by the red boundary. Note that the domain $R$ is
obtained empirically.

Figure 1: The domain $R$ on the right is the Brownian symmetrization of the
domain $B$ on the left.

It is easy to see that the Steiner symmetrization of $B$ is the domain $G$
limited by the green boundary.

Figure 2: The Steiner symmetrization of the domain $B$.

If we believe the simulation then the two domains are not the same, i.e

$\mathfrak{B}(B)=R\neq\mathfrak{S}(B)=G.$

Although $R$ and $B$ are far of being the same, we provide a convincing
argument that proves that the two symmetrizations are different.

######  Theorem 3.

The Steiner symmetrization and the Brownian symmetrization are different.

###### Proof.

Let $U$ be the following domain

$U=\\{-1<x\leq 0,-\frac{1}{2}<y<1\\}\cup\\{0<x\leq
1,\frac{1}{2}<y<-1\\}\cup\\{x=0,|y|<\frac{1}{2}\\}.$

Figure 3: The domain $U$.

The Steiner symmetrization of $U$ is the the rectangle
$S=\\{|x|<0,|y|<\frac{3}{4}\\}$.

Figure 4: The Steiner symmetrization of $U$.

Let $\mu$ be the probability law of $\Re(Z_{\tau_{U}})$. The distribution
$\mu$ has three point masses namely at $x=-1,x=0$ and $x=1$. This fact uses F.
and M. Riesz Theorem that asserts that the harmonic measure of a simply
connected domain with rectifiable boundary is mutually continuous with respect
to the arc length. However, we provide a short explanation articulating on the
conformal invariance of the Brownian motion [18]. Let $f$ be a conformal map
from the unit disc onto $U$ that sends zero to zero. As the boundary of $U$ is
of Jordan type (in addition of being rectifiable) then by CarathÃ©odory’s
theorem [7] the function $f$ extends continuously to the closed unit disc
without losing the injectivity property. For any portion $E$ of the boundary
the harmonic measure of $E$ is simply $\frac{|f^{-1}(E)|}{2\pi}$ where
$|\cdot|$ denotes the Lebesgue measure on the unit circle. This follows from
the fact that the stopped Brownian motion on the unit circle is uniformly
distributed. That is, $E$ has zero harmonic measure if and only its arc length
is zero ($f$ is an homeomorphism). As the vertical parts of the Boundary of
$U$ are of positive lengths then they induce point masses for $\mu$. Therefore
the Brownian symmetrization, say $V$, would preserve these point masses.
According to the construction process, the boundary of $V$ will contain three
vertical segments (and their symmetric images over the real line) at
$x=-1,x=0$ and $x=1$. Therefore $V=\mathfrak{B}(U)\neq S=\mathfrak{S}(U)$. ∎

##  3 Concluding remarks

The Skorokhod embedding problem has another different solution given in [4],
which can be studied as another version of symmetrization. Unlike Gross’s
method, it generates unbounded domains even for bounded distributions. More
precisely, it creates $\mu$-domains that are $\Delta^{\infty}$ convex, i.e
with the property that the upward half ray starting at any point of the domain
remains inside it (More details are in [4]). In addition, domains with such a
property are proved to be unique under some technical requirement on their
exit times. In [16], an interesting optimization problem was suggested : Fix a
probability distribution $\mu$, among all $\mu$-domains, which one has the
highest and lowest rate, where the rate is half of the principal Dirichlet
eigenvalue. The authors answered the question regarding the lowest rate for
$\mu$ being the uniform distribution on $(-1,1)$. It turns out that their
solution is a particular case of the solution appeared in [4]. The problem of
the highest rate is still open. Although, it is guessed that the Brownian
symmetrization maximizes the rate. In the same context, it is known that
Steiner symmetrization minimizes the principal Dirichlet eigenvalue. So a
natural question presents itself : What would be the effect of the Brownian
symmetrization on such an eigenvalue? Another related question is what is the
effect of the Brownian symmetrization on the perimeter?

At the end, we provide this example followed by some conjectures. Let $U$ be
the set $\\{x^{2}+(y+\kappa)^{2}<1\\}$ where $\kappa\in(0,1)$. The law of
$\Re(Z_{\tau})$ is given by

$d\mu(x)={\textstyle\frac{(1-\kappa^{4})}{\pi\left((1-\kappa^{2})^{2}+4\kappa^{2}x^{2}\right)\sqrt{1-x^{2}}}}$

[5]. The c.d.f of $\mu$ is

$\displaystyle F_{\mu}(x)$
$\displaystyle=\frac{1}{\pi}\arctan\left({\textstyle\frac{1+\kappa^{2}}{1-\kappa^{2}}\frac{x}{\sqrt{1-x^{2}}}}\right)+{\textstyle\frac{1}{2}}$
$\displaystyle=\frac{1}{\pi}\arctan\left({\textstyle\eta\frac{x}{\sqrt{1-x^{2}}}}\right)+{\textstyle\frac{1}{2}}$

and so

$\varphi_{\mu}(\theta)=G_{\mu}({\textstyle\frac{|\theta|}{\pi}})=-\sqrt{{\textstyle\frac{\cot^{2}(\theta)}{\eta^{2}+\cot^{2}(\theta)}}}1_{\left\\{|\theta|\in(0,\frac{\pi}{2})\right\\}}+\sqrt{{\textstyle\frac{\cot^{2}(\theta)}{\eta^{2}+\cot^{2}(\theta)}}}1_{\left\\{|\theta|\in(\frac{\pi}{2},\pi)\right\\}}.$

Using Mathematica software (many thanks to Professor Rajai Nacer for his
assistance), we obtained the following $\mu$-domain, or equivalently the
Brownian symmetrization of $U$.

Figure 5: $\mathfrak{B}(U)$ is limited by the blue curve.

Using the area formula (1), we found that the area of $\mathfrak{B}(U)$ is
approximately $0.6$. In particular

$\mathfrak{B}(U)\subsetneq\mathfrak{S}(U)=\mathbb{D}.$

In light of this example, we think the following conjecture might be true :

  1. 1.

Brownian symmetrization increases the principal Dirichlet eigenvalue.

  2. 2.

Brownian symmetrization does not increase the area and the perimeter.

Moreover, we speculate that Brownian symmetrization does better than Steiner
symmetrization.

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