On Loewner chains driven by semimartingales and complex Bessel-type SDEs

TL;DR

This paper establishes the existence of traces for Loewner chains driven by semimartingales, introduces complex Bessel-type SDEs, and applies these results to stochastic Loewner evolutions and related curve generation.

Contribution

It proves the existence and simplicity of traces for general semimartingale-driven Loewner chains and introduces complex Bessel-type SDEs with existence and uniqueness results.

Findings

SKLE$_{\alpha,b}$ are generated by curves

For $\alpha > 3/2$, $|B_t|^\alpha$ generates a simple curve for small $t$

Complex Bessel-type SDEs describe Loewner chain traces

Abstract

We prove existence (and simpleness) of the trace for both forward and backward Loewner chains under fairly general conditions on semimartingale drivers. As an application, we show that stochastic Komatu-Loewner evolutions SKLE$_{\alpha,b}$ are generated by curves. As another application, motivated by a question of A. Sep\'{u}lveda, we show that for $\alpha >3/2$ and Brownian motion $B$, the driving function $|B_t|^\alpha$ generates a simple curve for small $t$. On a related note we also introduce a complex variant of Bessel-type SDEs and prove existence and uniqueness of strong solution. Such SDEs appear naturally while describing the trace of Loewner chains. In particular, we write SLE$_\kappa$, $\kappa <4$, in terms of stochastic flow of such SDEs.