Smoothing of Boundary Behaviour in Stochastic Planar Evolutions

TL;DR

This paper demonstrates that adding noise to planar domain evolutions governed by holomorphic vector fields results in smoother boundary behavior, transforming potentially jagged corners into continuously differentiable arcs.

Contribution

It shows that stochastic perturbations induce boundary smoothing effects in planar evolutions, contrasting with deterministic cases where corners persist.

Findings

Evolving domains vary continuously in uniform topology.

Boundaries become continuously differentiable Jordan arcs.

Noise induces smoothing of boundary corners.

Abstract

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