Conformal Skorokhod embeddings and related extremal problems *†

The conformal Skorokhod embedding problem (CSEP) is a planar variant of the classical problem where the solution is now a simply connected domain D ⊂ C whose exit time embeds a given probability distribution µ by projecting the stopped Brownian motion onto the real axis. In this paper we explore two new research directions for the CSEP by proving general bounds on the principal Dirichlet eigenvalue of a solution domain in terms of the corresponding µ and by proposing related extremal problems. Moreover, we give a new and nontrivial example of an extremal domain U that attains the lowest possible principal Dirichlet eigenvalue over all domains solving the CSEP for the uniform distribution on [ − 1 , 1] . Remarkably, the boundary of U is related to the Grim Reaper translating solution to the curve shortening ﬂow in the plane. The novel tool used in the proof of the sharp lower bound is a precise relationship between the widths of the orthogonal projections of a simply connected planar domain and the support of its harmonic measure that we develop in the paper. The upper bound relies on spectral bounds for the torsion function which have recently appeared in the literature.


Introduction
Let W = (W t : t ≥ 0) be a complex Brownian motion starting at 0, and for any open set D ⊂ C containing 0, let τ D denote the first exit time of W from D. Moreover, let µ be a probability distribution with zero mean and finite nonzero variance. The conformal Skorokhod embedding problem (CSEP) was introduced by Gross in [12] where the author shows that there exists a simply connected domain D ⊂ C containing 0 such that This result was recently generalized in [9]. There the authors showed that if µ is a probability distribution with zero mean and finite nonzero p-th moment for some 1 < p < ∞, then there exists a simply connected domain D ⊂ C containing 0 such that Re W τ D ∼ µ and E 0 [τ p/2 D ] < ∞. They also give conditions on the domain which ensure that a solution domain is unique.
The original Skorokhod embedding problem (SEP) asks the following: Given a standard Brownian motion (B t : t ≥ 0) and a probability distribution µ with zero mean and finite variance, find a stopping time T such that B T ∼ µ and E 0 [T ] < ∞. It was first posed and solved by Skorokhod in 1961 (see [24] for an English translation of the original paper) and since then a veritable zoo of varied and interesting solutions have appeared, see [18] for a thorough survey. The similarity between these problems is clear, and as Gross points out, his solution resembles Root's barrier hitting solution [23] with additional randomness, that of the other Brownian motion in the second coordinate. Gross's paper is more than just an existence result and his method gives a relatively explicit construction of a domain D which solves (1.1 The limit exists by Fekete's subadditivity lemma and is clearly nonnegative. Since D is an open set containing 0, the process W will exit some small enough ball with positive radius centered at 0 before it exits D so it follows that λ(D) is finite. Additionally, 2 λ(D) is equal to the bottom of the spectrum of the semigroup generated by the Laplacian on D with Dirichlet boundary conditions and is usually referred to as the principal Dirichlet eigenvalue of D, see Section 3.1 of [26]. An example which will prove useful later on is that of an infinite strip of width w containing 0 where a straightforward projection argument shows that the rate is half the principal eigenvalue of an interval of length w, namely π 2 2w 2 . Two questions regarding the rate naturally present themselves: A. Find upper and lower bounds on the rate (or principal Dirichlet eigenvalue) in terms of µ.
B. For a specific µ, find extremal domains that attain the highest and lowest possible rate (or principal Dirichlet eigenvalue).
The main results of the paper are the following: We answer Question A. by giving an upper bound in terms of the variance of µ in Theorem 2.1 and by giving a sharp lower bound in terms of the width of the support of µ in Theorem 2.4. The proof of Theorem 2.1 relies on recent spectral bounds for the torsion function which are described at the beginning of Section 2. In order to prove Theorem 2.4, we need a precise relationship between the width of the orthogonal projection of a simply connected planar domain ECP 25 (2020), paper 42. and the width of the support of the orthogonal projection of its harmonic measure. This result, which is stated in Theorem 3.1 and proven in Section 3, may be of independent interest. As far as the authors know, it hasn't appeared in the literature. Finally, we give a partial answer to Question B. in the case of the uniform distribution on [− About one month after these results first appeared on arXiv, Boudabra and Markowsky posted a preprint [8] that gives an interesting new method to construct minimal rate solution domains to the CSEP. We point out that the minimality of their solution domains is a consequence of our general lower bound given in Theorem 2.4 which itself relies crucially on the most technical result of our paper, that of Theorem 3.1. Additionally, their result shows the importance of Theorem 2.4 by demonstrating that our lower bound is sharp in general and not just for U[−1, 1] which was already known from Theorem 2.6.
Another example of a minimal rate solution domain which is in some sense trivial can be found in [9] where it was shown that the infinite strip {z ∈ C : | Im z| < 1} and the domain bounded below by the parabola y = have their own extremal property. For instance, given a probability distribution µ with zero mean and finite variance, Root's embedding minimizes the variance of T over all stopping times T such that B T ∼ µ and E 0 [T ] < ∞, while Rost's reversed barrier embedding maximizes the variance of T over the same class of stopping times. We refer to [5] where extremal solutions to the classical SEP are studied in detail.
Extremal problems for Brownian motion apart from the SEP have also been a popular topic of study. A common theme is optimizing the principal Dirichlet eigenvalue or the maximum expected exit time of a domain taken over all starting points when various constraints are given, see [1,2,16,3,15,4] and references therein for some examples.
The CSEP also has a connection to Walden and Ward's harmonic measure distributions [28] that is worth mentioning. Indeed, if the domain D solves the CSEP for µ, then the orthogonal projection of the harmonic measure of D with pole at 0 onto the real axis has distribution µ. If instead we are given a distribution function F : (0, ∞) → [0, 1], then one might ask whether there exists a domain D ⊂ C containing 0 such that the circular projection of the harmonic measure of D with pole at 0 onto the positive real axis has distribution function F ? This is a topic of current interest and the reader is directed to the recent survey [25] for more on this question. Perhaps an adaptation of Gross's construction will prove fruitful in this area.

Main results
Our first main result is an upper bound on λ(D) for d-dimensional Brownian motion in terms of the second moments of the components at time τ D . When applied to a domain D ⊂ C which solves the CSEP, this general result provides a partial answer to Question A.
The proof involves a spectral upper bound for the torsion function of Brownian motion.
Recall that the torsion function of a domain D is nothing but the expected exit time of ECP 25 (2020), paper 42.
The best constant in the spectral upper bound for the torsion function depends on the dimension d and is defined by It was shown in [7] that u D ∞ < ∞ if and only if λ(D) > 0, so it follows from (2.1) that the spectral upper bound for the torsion function holds for all domains D. While computing C d exactly for d > 1 is still an open problem, bounds on C d for Brownian motion and related processes have been studied extensively via several techniques under various assumptions on D, see for instance [1,7,11,6,20]. The current best explicit upper bound in the Brownian case was derived by H. Vogt in [27] and states that See [4] for a non-explicit improvement of (2.3) along with new results on the p-torsion analogue of (2.2) which involves the p-th moment of the exit time.
With the spectral upper bound for the torsion function (2.2) at our disposal, we can now state our first main result.
In particular, if D solves the CSEP (1.1) for a probability distribution µ with zero mean and finite variance, then      Remark 2.7. The boundary of U is a translation and scaling of y = − log cos x hence it belongs to the family of Grim Reaper curves which are the only self-similar translating solutions to the curve shortening flow in the plane, see [13].
Remark 2.8. The simply connected domain U is a rotation and scaling of a domain studied in [17,Example 4] where the author points out that its boundary curve has been referred to as the "catenary of equal resistance". In that paper the expected exit time is computed but no mention is made of the distribution of the real or imaginary parts of the stopped Brownian motion.   Then for all > 0 we have ECP 25 (2020), paper 42.
so we can focus on proving that the latter inequality holds. Since W has continuous paths, it will hit ∂D before it hits the interior of D c , hence the latter probability appearing in (3.1) is equal to Next, we can bound (3.2) from below by Define the conformal transformation f (w) = − |y−ξ| y−ξ (w − ξ) which maps ξ to 0 and y to −|y − ξ| ∈ (− , 0). For a set E ⊂ C, let E * denote the circular projection of E. That is, E * = {|w| : w ∈ E}. Now we can use conformal invariance of Brownian motion and the version of Beurling's projection theorem from [19, Theorem 1] to get In order to produce a meaningful estimate, f (K) * must contain a proper interval. We claim f (K) * = [0, ]. To see that this is true, consider the connected component of K that contains ξ, call it E. Clearly E contains some point z such that |z − ξ| = , for otherwise E would be a bounded connected component of D c which contradicts D being simply connected. This implies both 0 and are elements of f (E) * . Since E is connected and both f and circular projection are continuous, we know that f (E) * is connected. Hence  Table 4.1] while noting that f (y)/ is purely imaginary, we can write where the last inequality follows from |f (y)| = |y − ξ| < . In conjunction with (3.1), this proves the lemma.
With Lemma 3.3 in hand, we can now give a proof of Theorem 3.1.
Proof of Theorem 3.1. Suppose b < ∞ and let > 0. Then there exists x ∈ D such that | Re x − b| < . Hence B (x) ∩ D c is nonempty, for otherwise we could increase b. It follows that B (x) must contain some ξ ∈ ∂D and that | Re ξ − b| < 2 . By Lemma 3.3, we know that P 0 W τ D ∈ B (ξ) ∩ ∂D > 0, from which it follows we have already shown that β = b, so assume b = ∞. We will show that β < ∞ leads to a contradiction.
A result of Burkholder [10,Equation 3.13] shows that By hypothesis, there exists x ∈ D such that Re x > β. Note that Re ξ ≤ β for all ξ ∈ ∂D, for otherwise we could use Lemma 3.3 to increase β. Since W τ D ∈ ∂D almost surely, this In light of (3.4), this leads to the desired contradiction. The proof that α = a follows similarly.

Proof of Theorem 2.1
Proof of Theorem 2.1. If λ(D) = 0 there is nothing to prove, so assume that λ(D) > 0.
In this case we can use (2.1) to write Since the components of W are all independent, each W (i) is a Brownian motion in the natural filtration of W , with respect to which τ D is a stopping time. In particular, optional stopping applied to the martingale (W ECP 25 (2020), paper 42.
for each 1 ≤ i ≤ d and n ∈ N. Letting n → ∞ in the above equality while using Fatou's lemma on the left-hand side and monotone convergence on the right-hand side leads to for each 1 ≤ i ≤ d. Combining (4.1) and (4.2) proves the theorem. The numerical estimates follow from [4, Table 1].

Proof of Theorem 2.6
The first step in proving Theorem 2.6 is to calculate the rate of U. This will be essential in establishing that it is indeed a minimal rate solution to the CSEP for U[−1, 1].
We do this in the following lemma.
Letting n → ∞ in the above inequality completes the proof. Now that we know λ(U), it remains to show that U is actually a solution to the CSEP for U[−1, 1]. The finite width of U implies E 0 [τ U ] < ∞ and we can show that Re W τ U ∼ U[−1, 1] by way of an explicit conformal map from U onto the upper half-plane H while exploiting conformal invariance of Brownian motion.
Since z → e z is injective on the infinite strip {z ∈ C : | Im z| < π 2 } and since U is contained in the infinite strip {z ∈ C : | Re z| < 1}, it follows that f is injective on U.
This shows that f (U) = y>0 f (h y ) = H, hence f maps U conformally onto H. Now for 0 ≤ x < 1, the conformal invariance of Brownian motion and the fact that ∂U is given by the graph of the function h imply that In the middle equation we used (4.3) with y = 0 and in the last equation we used the fact that the law of Re W τ H under P i is just the harmonic measure of the upper half-plane with pole at i which is well known to have the standard Cauchy distribution [22,