A stochastic selection principle in case of fattening for curvature flow

Abstract

Consider two disjoint circles moving by mean curvature plus a forcing term which makes them touch with zero velocity. It is known that the generalized solution in the viscosity sense ceases to be a curve after the touching (the so-called fattening phenomenon). We show that after adding a small stochastic forcing \(\epsilon {\rm d} W\), in the limit \(\epsilon\to 0\) the measure selects two evolving curves, the upper and lower barrier in the sense of De Giorgi. Further we show partial results for nonzero \(\epsilon\).