Publication 18-CNA-007

Long Time Asymptotics Of Heat Kernels And Brownian Winding Numbers On Manifolds With Boundary

Xi Geng
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
xig@andrew.cmu.edu

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu

Abstract: 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.

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