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Loewner Evolution as Itô Diffusion

Hülya Acar, Alexey L. Lukashov

Abstract

F. Bracci, M.D. Contreras, S. Díaz Madrigal proved that any evolution family of order d is described by a generalized Loewner chain. G. Ivanov and A. Vasil'ev considered randomized version of the chain and found a substitution which transforms it to an Itô diffusion.We generalize their result to vector randomized Loewner chain and prove there are no other possibilities to transform such Loewner chains to Itô diffusions.

Loewner Evolution as Itô Diffusion

Abstract

F. Bracci, M.D. Contreras, S. Díaz Madrigal proved that any evolution family of order d is described by a generalized Loewner chain. G. Ivanov and A. Vasil'ev considered randomized version of the chain and found a substitution which transforms it to an Itô diffusion.We generalize their result to vector randomized Loewner chain and prove there are no other possibilities to transform such Loewner chains to Itô diffusions.

Paper Structure

This paper contains 1 section, 1 theorem, 35 equations.

Table of Contents

  1. Introduction

Key Result

Theorem \oldthetheorem

Consider Loewner random differential equation where $\left\vert \tau _{1}\left( t,\omega \right) \right\vert =1$ for each fixed $w\in \Omega$ ($\Omega$ is a sample space) and $\tilde{p}$ is an arbitrary Herglötz function. Suppose $\psi _{t}=m(\phi _{t},B_{t}^{(1)},B_{t}^{(2)},\ldots,B_{t}^{(n)})$ where $B_{t}^{(i)}$ are independent Brownian mot

Theorems & Definitions (1)

  • Theorem \oldthetheorem