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Motivated by the study of trace for Schramm-Loewner evolutions, we considerevolutions of planar domains governed by ordinary differential equations withholomorphic vector fields $F$ defined on the upper half plane $\mathbb{H}$. Weshow a smoothing effect of the presence of noise on the boundary behaviour ofassociated conformal maps. More precisely, if $F$ is H\"older, we show thatevolving domains vary continuously in uniform topology and their boundaries arecontinuously differentiable Jordan arcs. This is in contrast with examples fromdeterministic setting where the corner points on the boundary of domain$F(\mathbb{H})$ may give rise to corner points on the boundaries ofcorresponding evolving domains.

Smoothing of Boundary Behaviour in Stochastic Planar Evolutions