Loewner Evolution as It\^o Diffusion

TL;DR

This paper generalizes the connection between Loewner chains and Itô diffusions, proving that vector randomized Loewner chains can only be transformed into Itô diffusions through a specific substitution.

Contribution

It extends previous work by Ivanov and Vasil'ev to vector chains, establishing the uniqueness of the transformation to Itô diffusions.

Findings

Generalization of the randomized Loewner chain to vector case

Proof of the uniqueness of the transformation to Itô diffusion

No other transformations to Itô diffusions are possible for these chains

Abstract

F. Bracci, M.D. Contreras, S. D\'iaz 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 It\^o diffusion.We generalize their result to vector randomized Loewner chain and prove there are no other possibilities to transform such Loewner chains to It\^o diffusions.