Electronic Communications in Probability

Conformal Skorokhod embeddings and related extremal problems

Phanuel Mariano and Hugo Panzo

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Abstract

The conformal Skorokhod embedding problem (CSEP) is a planar variant of the classical problem where the solution is now a simply connected domain $D\subset \mathbb {C}$ whose exit time embeds a given probability distribution $\mu $ 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 $\mu $ and by proposing related extremal problems. Moreover, we give a new and nontrivial example of an extremal domain $\mathbb {U}$ 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 $\mathbb {U}$ 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.

Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 42, 11 pp.

Dates
Received: 27 March 2020
Accepted: 21 May 2020
First available in Project Euclid: 18 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1592446014

Digital Object Identifier
doi:10.1214/20-ECP324

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J65: Brownian motion [See also 58J65]
Secondary: 30C20: Conformal mappings of special domains 30C70: Extremal problems for conformal and quasiconformal mappings, variational methods 30C85: Capacity and harmonic measure in the complex plane [See also 31A15]

Keywords
conformal Skorokhod embedding principal eigenvalue harmonic measure torsion function Grim Reaper curve catenary of equal resistance

Rights
Creative Commons Attribution 4.0 International License.

Citation

Mariano, Phanuel; Panzo, Hugo. Conformal Skorokhod embeddings and related extremal problems. Electron. Commun. Probab. 25 (2020), paper no. 42, 11 pp. doi:10.1214/20-ECP324. https://projecteuclid.org/euclid.ecp/1592446014


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