On the slit motion obeying chordal Komatu–Loewner equation with finite explosion time

Abstract

This paper studies the behavior of solutions near the explosion time to the chordal Komatu-Loewner equation for slits, motivated by the preceding studies by Bauer and Friedrich (2008) and by Chen and Fukushima (2018). The solution to this equation represents moving slits in the upper half-plane. We show that the distance between the slits and driving function converges to zero at its explosion time. We also prove a probabilistic version of this asymptotic behavior for stochastic Komatu-Loewner evolutions under some natural assumptions.