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

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 ${\mathrm{SKLE}}_{\mathit{\alpha },\mathit{b}}$ are generated by curves. As another application, motivated by a question of A. Sepúlveda, we show that for $\mathit{\alpha }>3/2$ and Brownian motion B, the driving function $|{\mathit{B}}_{\mathit{t}}{|}^{\mathit{\alpha }}$ 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 ${\mathrm{SLE}}_{\mathit{\kappa }}$, $\mathit{\kappa }<4$, in terms of stochastic flow of such SDEs.