The author establishes the existence of a continuously time-varying random subset \(K(t)\) of Euclidean space such that its boundary, which is a hypersurface, has normal velocity formally equal to the mean curvature plus a random driving force. This random force is modelled by a stochastic flow of diffeomorphisms, and the sets \(K(t)\) are sets of finite perimeter. The approach of the author extends results of \textit{F. Almgren, J. E. Taylor} and \textit{L. Wang} [SIAM J. Control Optimization 31, No. 2, 387-438 (1993; Zbl 0783.35002)] to a stochastic setting. The evolution is obtained in a time-splitting scheme: In each time interval first the set is changed by minimizing a functional and then the set is transported by the flow of diffeomorphisms. The author then proves that this construction produces, as the time steps go to zero, a tight sequence of probability measures first on an appropriate space of measure-valued processes and then on the Skorokhod space of processes with values in the space of sets of finite perimeter. The limit points are shown to concentrate on the continuous evolutions. This proves the existence of \(\partial K(t)\), but problems about its motion law and regularity remain unresolved. The paper should be of interest to researchers in the calculus of variations and stochastic calculus alike. It comprises an appendix containing the necessary tools from stochastic calculus.