Atul Shekhar: Regularization of Planar Boundaries under Stochastic Evolution

Brownian motion $B$ exhibits a curious property called regularization by noise which can be attributed to its quadratic variation process. It was shown by A.M. Davie that differential equations of form $dX_t = f(X_t)dt + dB_t$ admits a unique solution for almost surely all Brownian sample paths even if $f$ is only a bounded measurable function. We will consider the case when $f$ is a holomorphic map in an open set which is irregular as the boundary is approached. We will show that there is a unique flow $\varphi(z)$ associated to the above equation and the complex derivative $\varphi^{'}(z)$ admits a continuous extension to the boundary. The result is compared to classical results from complex analysis on boundary behaviour of derivative of conformal maps.