| Popis: |
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. |
