A Homogenization Result for Laplacians on Tubular Neighbourhoods of Closed Riemannian Submanifolds
|Keywords:||Dirichlet form; Wiener measure; submanifolds; tubular neighbourhoods; semi groups|
|Type of publication:||Professorial dissertation|
|Format:||24,0 x 17,0 cm|
|Price:||39,80 € / 49,75 SFr|
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|Abstract||The investigation of Brownian motion conditioned to stay within a small tube around a submanifold requires a detailed analysis of its generator. It turns out that it is determined by the Laplacian of the ambient manifold subject to Dirichlet boundary conditions on the boundary of the tube. However, as the tube diameter tends to zero, the limit of these boundary problems is not a priori well-defined. The perturbation problem is singular.|
In this thesis, the singularity in the problem is removed in the following way: By rescaling space, the perturbation problem on tubes of decreasing diameter is transformed to a corresponding problem on a tube of fixed diameter. The tube diameter from the initial problem now simply becomes the parameter of the transformed problem. It parametrizes a family of Riemannian metrics on the fixed tube obtained by transporting the induced metric on the embedded tube via the rescaling map. The transformed problem is given by the Dirichlet Laplacians associated to the respective metrics.
The key point is that the leading terms of the (still singular) perturbation expansion of the operator family corresponds to Dirichlet Laplacians of another family of metrics, the canonical variation of the so called Sasaki-metric on the tube. That provides rather detailed knowledge about the initial problem, sufficient to introduce a suitable renormalization which makes the singular problem regular.
The regular problem thus obtained is investigated using epi-convergence of the corresponding Dirichlet forms and elliptic regularity of the corresponding boundary problems. We obtain a description of the limit in terms of a Schrödinger operator on the submanifold with a potential that depends on the submanifold and of the embedding. As an application, we prove convergence of the conditioned Brownian motion to a Feynman-Kac Gibbs measure supported by the path space of the submanifold in finite dimensional distributions.