A homogenization result for Laplacians on tubular neighbourhoods of closed Riemannian submanifolds (Q2901289)

Consider a Riemannian manifold \(M\) and a compact submanifold \(L\). Let \(L(\varepsilon)\) be the \(\varepsilon\)-neighbourhood of \(L\) in \(M\), and let \(\Delta_\varepsilon\) be the Laplace-Beltrami operator \(L(\varepsilon)\) with Dirichlet boundary condition. The main result of this thesis is the convergence of \(\Delta_\varepsilon\), after some rescaling, to \(\Delta_L+W_0\), where \(\Delta_L\) is the Laplace-Beltrami operator on \(L\), and \(W_0\) is a potential on \(L\) which can be written in terms of geometric (intrinsic and extrinsic) quantities.NEWLINENEWLINEThis result can be interpreted in terms of the Brownian motion on \(M\) conditioned to stay in \(L(\varepsilon)\) until some finite terminal time. The law of this process converges to a limit which can be written in terms of a Feynman-Kac formula involving the Brownian motion on \(L\) and the potential \(W_0\).