arXiv subjects
Publications and source records attributed to O. Wittich.
We consider the heat equation with Dirichlet boundary conditions on the tubular neighborhood of a closed Riemannian submanifold. We show that, as the tube radius decreases, the semigroup of a suitably rescaled and renormalized generator can be effectively described by a Hamiltonian on the submanifold with a potential that depends on the geometry of the submanifold and of the embedding.
We consider the heat equation with Dirichlet boundary conditions on the tubular neighborhood of a closed Riemannian submanifold. We show that, as the tube diameter tends to zero, a suitably rescaled and renormalized semigroup converges to a limit semigroup in Sobolev spaces of arbitrarily large Sobolev index.
Let (S(t)) be a one-parameter family S = (S(t)) of positive integral operators on a locally compact space L. For a possibly non-uniform partition of [0,1] define a measure on the path space C([0,1],L) by using a) S(dt) for the transition between cosecutive partition times of distance dt, and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tighness results which together yield convergence in law of such measures as the partition gets finer. In particular let L be a closed smooth submanifold of a Riemannian manifold M. We prove convergence of Brownian motion on M, conditioned to visit L at all partition times, to a process on L whose law has a Radon-Nikodym density with repect to Brownian motion on L which contains scalar, mean and sectional curvature terms. Various approximation schemes for Brownian motion are also given. These results substantially extend earlier work by the authors and by Andersson and Driver.