Lévy showed that the total time that a standard Brownian motion stays positive up to time 1 obeys the arcsine law. We will discuss the deviation of this law for a Brownian motion on a Riemannian manifold near a smooth hypersurface. The deviation has the order of the square root of the total time and is proportional to the mean curvature of the hypersurface. Its explicit form depends on the local time of the transversal Brownian motion properly scaled.

This is a joint work with Cheng Ouyang at University of Illinois at Chicago.