We prove that any disjoint union of nitely many simple curves in the upper half{plane can be generated in a unique way by the chordal multiple{slit Loewner equation with constant weights.

In this paper we ask whether one can take the limit of multiple SLE as the number of slits goes to infinity. In the special case of n slits that connect n points of the boundary to one fixed point, one can take the limit of the Loewner equation that describes the growth of those slits in a simultaneous way. In this case, the limit is a deterministic Loewner equation whose vector field is determined by a complex Burgers equation.

We describe the region V(z0) of values of f(z0) for all normalized bounded univalent functions f in the unit disk double-struckD at a fixed point z0∈D. The proof is based on the radial Loewner differential equation. We also prove an analogous result for the upper half‐plane using the chordal Loewner equation.

In this note we consider a multi-slit Loewner equation with constant coefficients that describes the growth of multiple SLE curves connecting N points on R to infinity within the upper halfplane. For every N ∈ N, this equation provides a measure valued process t → {α N,t }, and we are interested in the limit behaviour as N → ∞. We prove tightness of the sequence {α N,t } N ∈N under certain assumptions and address some further problems.

In [5], O. Bauer interpreted the chordal Loewner equation in terms of non-commutative probability theory. We follow this perspective and identify the chordal Loewner equations as the non-autonomous versions of evolution equations for semigroups in monotone and anti-monotone probability theory. We also look at the corresponding equation for free probability theory.