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Brownian symmetrization of planar domains

Maher Boudabra, Kais Hamza

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. 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.

Brownian symmetrization of planar domains

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. 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.
Paper Structure (3 sections, 3 theorems, 15 equations, 5 figures, 1 algorithm)

This paper contains 3 sections, 3 theorems, 15 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Steiner symmetrization and Brownian symmetrization are different
  3. Concluding remarks

Key Result

Theorem 1

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

Figures (5)

  • Figure 1: The domain $R$ on the right is the Brownian symmetrization of the domain $B$ on the left.
  • Figure 2: The Steiner symmetrization of the domain $B$.
  • Figure 3: The domain $U$.
  • Figure 4: The Steiner symmetrization of $U$.
  • Figure 5: $\mathfrak{B}(U)$ is limited by the blue curve.

Theorems & Definitions (4)

  • Theorem 1: Lévy's theorem
  • Theorem 2
  • Theorem 3
  • proof