Remarks on Gross' technique for obtaining a conformal Skorohod embedding of planar Brownian motion

In a recent work by Gross, it was proved that, given a distribution $\mu$ with zero mean and finite second moment, we can find a simply connected domain $\Omega$ such that if $Z_{t}$ is a standard planar BM, then $\mathcal{R}e(Z_{\tau_{\Omega}})$ has the distribution $\mu$. In this note, we extend his method to prove that if $\mu$ has a finite $L^{p}$ moment then the exit time $\tau_{\Omega}$ has a finite moment of order $\frac{p}{2}$.


Introduction and statement of results
In what follows, Ø is a standard planar Brownian motion starting at 0, and for any plane domain ª containing 0 we let ª denote the first exit time of Ø from ª. In the elegant recent paper [6] the following theorem was proved.
Theorem 1. Given a probability distribution on R with zero mean and finite nonzero second moment, we can find a simply connected domain ª such that Ê ´ ª µ has the distribution .
We will prove several new results related to Gross' construction. Our first result is the following generalization.
Theorem 2. Given a probability distribution on R with zero mean and finite nonzero Ô-th moment (with ½ Ô ½), we can find a simply connected domain ª such that Ê ´ ª µ has the distribution . Furthermore we have ´ ª µ Ô ¾ ℄ ½.
The proof of this result depends on a number of known properties of the Hilbert transform and of the exit time ª , and is rather short. However the results needed are scattered through a number of different subfields of probability and analysis, and in an attempt to make the paper self-contained we have included a certain amount of exposition on these topics. We will prove the theorem in the next section.
There are several reasons why we feel that our extension is worth noting. The moments of ª have special importance in two dimensions, as they carry a great deal of analytic and geometric information about the domain ª. The first major work in this direction seems to have been by Burkholder in [2], where it was proved among other things that finiteness of the Ô-th Hardy norm of ª is equivalent to finiteness of the Ô ¾ -th moment of ª . To be precise, for any simply connected domain ª let À´ªµ ×ÙÔ Ô ¼ E ´ ª µ Ô ℄ ½ note that À´ªµ is proved in [2] to be exactly equal to half of the Hardy number of ª, as defined in [7], which is defined to be where is a conformal map from the unit disk onto ª. This equivalence was used in [2] to show for instance that À´Ï « µ ¾« , where Ï « ¼ Ö ´Þµ « is an infinite angular wedge with angle «. In fact, coupled with the purely analytic results in [7] this can be used to determine À´ªµ for any starlike domain ª. If we assume that Î is starlike with respect to ¼, then we may define Ö ª Ñ Ü Ñ´ µ is a subarc of ª Þ Ö (1.1) and this quantity is non-increasing in Ö (here Ñ denotes angular Lebesgue measure on the circle).
We may therefore let ª Ð Ñ Ö±½ Ö ª , and then combining the results in [7] and [2] (see also [9]) we have À´ªµ ¾ ª . In this sense, the quantity À´ªµ provides us with some sort of measure of the aperture of the domain at ½. Also in [9], a version of the Phragmén-Lindelöf principle was proved that makes use of the quantity À´ªµ. Furthermore, the quantity E ´ ª µ Ô ℄ provides us with an estimate for the tail probability È´ ª AEµ: by Markov's inequality, È´ ª AEµ E ´ ªµ Ô ℄ AE Ô .
For these reasons, we would argue that Theorem 2 gives a partial answer to the following intriguing question posed by Gross in [6]: given a probability distribution and a corresponding ª such that Ê ´ ª µ has distribution , in what sense are properties of reflected in the geometric properties of ª? We will have more comments on this question in the final section.
Our next result is influenced by Gross' observation that the domain corresponding to a given measure is not unique. Without further conditions this is correct, however we have found that natural conditions can be imposed on the domain so that a uniqueness principle holds. Before stating the result, let us make some definitions. A domain Í is symmetric if Þ ¾ Í whenever Þ ¾ Í. We will call a symmetric domain Í ¡-convex if the vertical line segment connecting Þ and Þ lies entirely within Í for any Þ ¾ Í. It is straightforward to verify that any domain constructed by Gross' technique is both symmetric and ¡-convex (see Section 2), and we have the following result.
Theorem 3. For any distribution satisfying the conditions of Theorem 2, there is a unique simply connected domain ª such that Ê ´ ª µ and which is symmetric, ¡-convex, and satisfies ´ ª µ Ô ¾ ℄ ½.
The importance of this result for our purposes is that it allows us to give a number of examples of domains generated by Gross' method. That is, if ª is a simply connected domain which is symmetric, ¡-convex, and satisfies ´ ª µ Ô ¾ ℄ ½, then it must be the domain generated by Gross' method corresponding to the distribution of Ê ´ ª µ. We will exploit this fact in Section 4.

Preliminaries and proof of Theorem 2
The proof of Theorem 2 is mainly based on the Hilbert transform theory for periodic functions, and we give here a brief summary of this. For further details about the topic, we refer the reader to [3].
In this sense, ÓØ´Ø ¾ µ can be seen as the periodic version of the function ½ Ø . Let us now sketch the ideas for Gross' proof, so that we may see where the Hilbert transform comes in. We assume for now that has finite second moment. Let be the c.d.f of and consider the pseudo-inverse function of defined by ´Ùµ Ò Ü ¾ R ´Üµ Ù Note that is defined for Ù ¾ ¼ ½℄. It is well known that ´ÍÒ ´¼ ½µ µ has as distribution. Now consider the ¾ -periodic function ³ whose restriction to´ µ is ´ µ. The map ³ is even, increasing on´¼ µ and belongs to Ä ¾ , where Ä Ô here and elsewhere in the paper denotes Ä Ô´ ℄µ. Thus its Fourier series is well defined and converges to ³ in Ä ¾ . We obtain hence Note that ÁÑ´ ³´ µµ is given by ³´ µ È ·½ Ò ½ ³´Òµ × Ò´Ò µ, and this is the Hilbert transform of ³. A crucial property of ³, as is shown in [6], is that it is univalent. The image domain ª ³´Dµ is therefore simply connected, and it is also symmetric over the Ü-axis as ³´Þµ ³´Þµ. Using the conformal invariance of Ø , and the fact that D is uniformly distributed on the boundary of D, we conclude by (2.1) that Ê ´ ³´ D µµ has distribution . Furthermore, Parseval's identity and martingale theory implies that [1]), and this sum is finite since Let us now see how we can extend this argument to prove Theorem 2. We will assume now that has a finite Ô-th moment, where Ô ½. It follows as above that ³ ¾ Ä Ô . The Fourier series È ·½ Ò ½ ³´Òµ Ó×´Ò µ is still well defined and converges to ³ in Ä Ô ([5, Thm. 3.5.7]). Parseval's identity is no longer available to us, but the following result allows us to conclude that the Hilbert transform If is in Ä Ô then its periodic Hilbert transform À does exist almost everywhere and we have for some positive constant Ô .
We see that, as its real and imaginary parts are in Ä Ô , the analytic function ³´Þµ È ·½ Ò ½ ³´ÒµÞ Ò lies in the Hardy space À Ô , which is the space of all holomorphic maps on the disk with finite Hardy Ô-norm, defined as ³´Þµ is also injective, by the same argument as was used in [6], and therefore ª ³´Dµ is simply connected. By a theorem of Burkholder in [2] we have that if is a conformal function on the unit disk then the following equivalence holds: We see therefore that ´ ª µ Ô ¾ ℄ ½, and the theorem is proved.

Proof of Theorem 3.
In this section we prove Theorem 3, that the domain ª generated by Gross' technique is the unique symmetric, ¡-convex simply connected domain with ´ ª µ Ô ¾ ℄ ½ such that such that Ê ´ ª µ has the distribution . Before going through the proof, we need the following lemma related to the Riemann mapping theorem. Lemma 1. If Í C is a symmetric simply connected domain containing 0 then there exists a conformal map from D to Í such that ´¼µ ¼ and ´´ ½ ½µµ R.
Proof. The existence of a conformal map, say , from the unit disc to Í and sending zero to itself is guaranteed by the Riemann mapping theorem. It remains to add the constraint that ´´ ½ ½µµ R. Consideration of the power series shows that the map ´Þµ is analytic, and as D and Í are symmetric it is a conformal map from D to Í. Therefore it is related to via a rotation acting on the unit disc, that is ´Þµ ´ Þµ for some ¾ ¼ ¾ µ. In particular, if Þ is real then ´Þµ is as well, which ends the proof.
We will compare the radial limits of and and show that they coincide a.e., but first we need another lemma.

Lemma 2.
Í and Î agree in distribution with ´ µ and ´ µ respectively, where is a r.v. uniformly distributed on Þ ½ .
Proof: Note that in applying and to , we are making use of the radial limits defined above. We will prove the statement for . Let Ö Ò be any sequence in´¼ ½µ which increases to 1 as Ò ½, and let Ò Ò Ø ¼ Ø Ö Ò . By standard martingale theory (see for instance [13]), since ´ Ò µ is a martingale bounded in Ä Ô we are guaranteed the existence of a limiting r.v. Å ½ such that ´ Ò µ Å ½ Ô ℄ ¼. Therefore ´ Ò µ converges to Å ½ in distribution. On the other hand, ´ Ò µ is equal in distribution to ´ Ò µ, where Ò is any r.v. uniformly distributed on Þ Ö Ò . Let us choose Ò and as follows. Let the probability space in question be the interval ¼ ¾ µ, with probability measure given by Lebesgue measure divided by ¾ . For in the probability space, let Ò´ µ Ö Ò , and similarly Ò´ µ . By [12, Thm 17.12], we have Thus, ´ Ò µ ´ µ ℄ ¼, which implies that ´ Ò µ converges to ´ µ in distribution. However, ´ Ò µ and ´ Ò µ have the same distribution, and therefore Å ½ and ´ µ agree in distribution. Now, ´ Ø µ is a time-changed Brownian motion, and therefore ´ Ò µ ´ Òµ , where denotes the time-change and is a Brownian motion. By monotone convergence, ´ Ò µ ± Í , and thus ´ Ò µ converges a.s. to Í . It follows that Í is equal in distribution to ´ µ.
¡-convexity and symmetry imply that Ê ´ ´ µµ and Ê ´ ´ µµ are a.e. even functions on ℄ and non-increasing on ¼ ℄. Since È´Í ¾ ¼ ¼ µ ¼ for ¼ ¾´¼ ℄, it follows that for a.e. we must have Ê ´ ´ µµ Ö, where Ö is such that Ö ·½µ , and the same must hold for Ê ´ ´ µµ. We see that Ê ´ µ and Ê ´ µ agree a.e. on Þ ½ , and since ÁÑ´ µ ÁÑ´ µ are obtained from these by the periodic Hilbert transform (see Section 2) we see that and agree a.e. on Þ ½ . ´Þµ and ´Þµ for Þ ¾ D can be obtained from their boundary values via the Poisson integral formula ([12, Cor. 17.12]), and thus and agree. Theorem 3 is proved.
None of the three conditions in the theorem can be omitted. For example, suppose that Í CÒ Ê ´Þµ ½ ÁÑ´Þµ ½ . Í is symmetric and ¡-convex, but ´ ª µ Ô ¾ ℄ ½ for Ô ½. Since Ê ´ ª µ is a measure of bounded support, it will generate by Gross' method a domain ª such that ´ ª µ Ô ¾ ℄ ½ for all Ô, and this can therefore not be equal to Í. An example which is symmetric and has finite Ô-th moment for all Ô but which lacks ¡-convexity is displayed in Figure 2 of [6], and it is pointed out there that uniqueness fails. It is similarly easy to construct a domain which is ¡-convex and has finite Ô-th moment for all Ô but which is not symmetric, and again uniqueness fails.
Another example that may be worth noting can be found in the next section, as it is shown there that the parabola and infinite strip lead to the same distribution . This does not contradict our result, since the parabola is neither ¡-convex nor symmetric, but it is interesting to note that both of these domains are convex, and that therefore convexity does not seem to be the correct condition for uniqueness.

Examples
In this section, we consider a series of domains and the corresponding distributions of Ê ´ µ. In all cases that we consider the boundary of the domain will be well behaved and we will be able to find a p.d.f. of the distribution of on the boundary. By this we mean that we can find a function, ´Þµ, defined for Þ on Í such that for any interval Á on the boundary of Í, we have È ´ Ø ¾ Áµ Ê ´Þ´×µµ ×, where Þ´×µ is a parameterization of Í with Þ ¼´× µ ½ and Þ´´ µµ Á. We will use analytic functions and the conformal invariance of Brownian motion as our primary tool; finding exit distributions in this manner has previously been considered in [10], and following the convention there we will use the notation ´Þµ × to denote this density, with the × to indicate that the curve Þ´×µ is parameterized by arclength. for some differentiable bijective function ³ Þ . To see how, let Ü be an element of Ê ´Þµ Þ ¾ .
Since a positive infinitesimal element Þ ¾ is expressed as Þ Ô Ü ¾ · Ý ¾ , then Finally we get Notice that that both sets Þ Ê ´Þµ Ü and Þ ÁÑ´Þµ Ý are countable due to (4.1), which justifies the sum symbols in (4.2). This proves the formula for the distribution of , and can of course be obtained similarly. The following diagram, which should be viewed at the infinitesimal level, provides the intuitive justification for the formulas.

´Þµ Þ
Before going through examples, we recall the following lemma which will be used across the rest of the work. A proof in the case that is analytic (which is what we use in this paper) can be found in [10].

Unit disc.
If Ø starts at zero at stopped at D then due to the rotational invariance of the Brownian motion D is uniformly distributed on the circle, i.e D ¼´ µ ½ ¾ Using the unit circle equation Ü ¾ · Ý ¾ ½, we extract the distributions of D and D on´ ½ ½µ: Similarly for D ¼´Ý µ. We remark that D and D follow the Arc-sine law on´ ½ ½µ. If the starting point is Ù · Ú ¼, then the distribution of D is given by [10]). Using the coordinates expressions´Ü Ýµ ´Ó× × Ò µ, we find the distributions of D and D :

Parabola
Let Ë be the horizontal strip Þ ½ Im´Þµ ½ and È ´Ëµ where Þ Þ ¾ .The map is not conformal as it is ¾ to ½, however it maps the Ë to È.

Concluding remarks
We do not know whether Theorem 2 holds for ½ ¾ Ô ½. There are many difficulties to proving the result in this range. One is that the analogue of Theorem 4 does not hold, even for Ô ½; for a counterexample, see [8, p. 212]. Furthermore À Ô and Ä Ô are not as well behaved for Ô ½; their respective norms are not true norms, for instance, as the triangle inequality fails. In any event, regardless of the veracity of the theorem for ½ ¾ Ô ½, one should certainly exercise extreme caution in attempting to extend it to Ô ½ ¾ . This is because for any simply connected domain ª strictly smaller than C itself we have ´ ª µ Ô ¾ ℄ ½ for any Ô ½ ¾ ; this is proved in [2]. Thus a measure with infinite Ô-th moment for some Ô ½ ¾ cannot correspond in this manner to a simply connected domain.
The question posed by Gross in [6] on how properties of are reflected in the geometry of ª is, in our opinion, an interesting one. Gross proposed finding a condition which forced ª to be convex; this appears difficult, especially considering that according to Gross' simulations the domain corresponding to a Gaussian is not convex. We would like therefore to suggest several weaker properties that ª might have, and propose that finding sufficient conditions on for these might be interesting problems.
• ª is starlike with respect to 0. • ×ÙÔ Þ¾ª ÁÑ´Þµ ½. That is, ª is contained in an infinite horizontal strip. Note that this would imply that all moments of are finite, because all moments of the exit time of a strip are finite, but that this is not sufficient: if ª is the parabolic region Ü Ý ¾ ½ , then all moments of ª are finite (proof: ª can fit inside a rotated and translated wedge Ï « with arbitrarily small aperture «, and therefore its exit time is dominated by that of the wedge, which can have finite Ô-th moment for as large Ô as we like) but ×ÙÔ Þ¾ª ÁÑ´Þµ ½. • Ð Ñ ×ÙÔ Ê ´Þµ ½ Þ¾ª ÁÑ´Þµ ¼.

Acknowledgements
We would like to thank Zihua Guo, Paul Jung, and Wooyoung Chin for helpful comments.