On Loewner chains driven by semimartingales and complex Bessel-type SDEs
Vlad Margarint, Atul Shekhar, Yizheng Yuan
TL;DR
This work investigates Loewner chains driven by continuous semimartingales, establishing existence (and when possible, simplicity) of the trace for both forward and backward Loewner differential equations. The authors develop backward-flow methods for existence and forward-flow methods for simplicity, supported by moment bounds for derivatives and a framework of complex Bessel-type SDEs, which connects to a stochastic flow representation of SLE-like processes. They prove trace existence under fairly general driving conditions and derive sharp simplicity criteria, with applications to stochastic Komatu-Loewner evolutions (SKLEs) and Brownian functionals $|B_t|^{\alpha}$ for $\alpha>3/2$. A key contribution is the introduction of complex square-root-based SDEs that describe the trace and enable strong uniqueness results, illuminating the link between Loewner traces and complex stochastic dynamics. These results have implications for SKLE curve generation, SLE representations in non-Brownian settings, and potential future directions in LDP analyses of Bessel-type drives.
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$_{α,b}$ are generated by curves. As another application, motivated by a question of A. Sepúlveda, we show that for $α>3/2$ and Brownian motion $B$, the driving function $|B_t|^α$ 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$_κ$, $κ<4$, in terms of stochastic flow of such SDEs.