Nonlocal diffusion of smooth sets

We consider normal velocity of smooth sets evolving by the $ s- $fractional diffusion. prove that for small time, such is nearly proportional to mean curvature boundary initial set s\in [\frac{1}{2}, 1) while, (0, \frac{1}{2}) $, it fractional set. Our results show motion (fractional) flow can be approximated heat diffusion and a means harmonic extension sets.