  Let M be a compact Riemannian manifold with smooth boundary. We obtain the
exact long time asymptotic behaviour of the heat kernel on abelian coverings of
M with mixed Dirichlet and Neumann boundary conditions. As an application, we
study the long time behaviour of the abelianized winding of reflected Brownian
motions in M. In particular, we prove a Gaussian type central limit theorem
showing that when rescaled appropriately, the fluctuations of the abelianized
winding are normally distributed with an explicit covariance matrix.
