On Stochastic Curvature Flows

It is well known, that multiple solutions to the mean curvature flow can arise for some initial data. Then an additional information is needed to select a unique solution. Souganidis and Yip (2004) proposed to study this problem by using the mean curvature flow equation perturbed by Brownian noise. Using the methods developed by Lions and Souganidis, one can show that the resulting stochastic PDE has a unique solution, and in the limit of vanishing noise some special solutions to the deterministic equation are identified. Starting from this seminal work, we will describe more recent results and some open problems.