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Skorokhod energy of planar domains

Mrabet Becher, Maher Boudabra, Fathi Haggui

TL;DR

The paper defines the Skorokhod energy Λ(U) of a simply connected domain containing the origin via univalent maps and proves that, among μ‑domains solving the planar Skorokhod embedding problem, the Gross solution minimizes Λ(U). It derives an integral representation Λ(μ) = (1/(2π^2)) ∫_0^1 1/ρ(Q(x))^2 dx for continuous μ with bounded support and provides concrete values for well-known distributions, while highlighting infinite energy for certain unbounded or irregular domains. The proofs blend conformal invariance, symmetric decreasing rearrangement, and Hardy-space techniques, linking planar Brownian motion, complex analysis, and energy minimization. These results establish a principled energy-minimization perspective on μ‑domains and suggest new directions in Brownian symmetrization and extremal shape problems in planar Skorokhod embedding.

Abstract

In this work, we introduce the Skorokhod energy of a simply connected domain. We show that among all domains solving the planar Skorokhod embedding problem, Gross solution generates the domain with the minimal Skorokhod energy.

Skorokhod energy of planar domains

TL;DR

The paper defines the Skorokhod energy Λ(U) of a simply connected domain containing the origin via univalent maps and proves that, among μ‑domains solving the planar Skorokhod embedding problem, the Gross solution minimizes Λ(U). It derives an integral representation Λ(μ) = (1/(2π^2)) ∫_0^1 1/ρ(Q(x))^2 dx for continuous μ with bounded support and provides concrete values for well-known distributions, while highlighting infinite energy for certain unbounded or irregular domains. The proofs blend conformal invariance, symmetric decreasing rearrangement, and Hardy-space techniques, linking planar Brownian motion, complex analysis, and energy minimization. These results establish a principled energy-minimization perspective on μ‑domains and suggest new directions in Brownian symmetrization and extremal shape problems in planar Skorokhod embedding.

Abstract

In this work, we introduce the Skorokhod energy of a simply connected domain. We show that among all domains solving the planar Skorokhod embedding problem, Gross solution generates the domain with the minimal Skorokhod energy.
Paper Structure (6 sections, 12 theorems, 51 equations, 3 figures)

This paper contains 6 sections, 12 theorems, 51 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Tools and proofs
  3. Proof of Proposition \ref{['prop:A-finite-energy']}
  4. Proof of Theorem \ref{['main theorem']}
  5. Proof of Theorem \ref{['integral form']}
  6. Comments and further directions

Key Result

Theorem 1

Let $f:U\rightarrow\mathbb{C}$ be a non-constant analytic 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 and $t\in[0,\tau_{U}]$.

Figures (3)

  • Figure 1.1: For the uniform distribution on $(-1,1)$, the left domain is Boudabra-Markowsky's solution while Gross's solution is on the right.
  • Figure 1.2: The domains generated from the shifted-scaled arcsine distribution $d\mu(x)=\frac{1}{\pi\sqrt{1-x^{2}}}dx$. Gross's method generates the unit disc while Boudabra-Markowsky's method generates the left domain.
  • Figure 1.3: The graph of the c.d.f of the distribution of $\Re(Z_{\tau_{S}})$ where $S=\frac{2e^{-\frac{\pi i}{4}}}{\sqrt{2}}f_{4}(\mathbb{D})$ is the square of vertices $\{\pm1\pm i\}$. The presence of the two jumps at $\pm1$ is reflected by the atoms of the distribution.

Theorems & Definitions (20)

  • Theorem 1: Lévy's theorem
  • Lemma 2
  • proof
  • Definition 3
  • Proposition 4
  • Theorem 5
  • Theorem 6
  • Definition 7
  • Theorem 8
  • Definition 9
  • ...and 10 more