we study the long time behaviour of the heat kernel on abelian covers of compact riemannian manifolds. for manifolds without boundary work of lott and kotani-sunada establishes precise long time asymptotics. extending these results to manifolds with boundary reduces to a 'cute' eigenvalue minimization problem, which we resolve for a dirichlet and neumann boundary conditions. we will show how these results can be applied to studying the ``winding'' / ``entanglement'' of brownian trajectories.