Surface measures on path spaces of riemannian manifolds

  • Oberflächenmaße auf Pfadräumen Riemannscher Mannigfaltigkeiten

Nobis, Vera; Wittich, Olaf (Thesis advisor); Führ, Hartmut (Thesis advisor)

Aachen (2019, 2020)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2019

Abstract

This thesis discusses an approach to define surface measures on the path spaces of Riemannian submanifolds given by a Brownian motion. These surface measures have been discussed from various points of view in the literature. It has been helpful to study Brownian motion on tubular neighborhoods of submanifolds. In this thesis we deal with the following case: Let $(L,g_L)$ be a closed Riemannian manifold, isometrically embedded into the complete Riemannian manifold $(M,g)$. We study a Brownian motion on $M$, starting in $x\in L$. Conditioning Brownian motion totubular neighborhoods $L(\varepsilon), \varepsilon>0,$ of $L$ is closely related to Brownian motion absorbed at the boundary $\partial L(\varepsilon)$ ofthe tube. It has been shown (Sidorova, Smolyanov, von Weizsäcker and Wittich (2014)) that the conditional measures correspond to Markov processes and that they converge weakly for $\varepsilon\to 0$. Furthermore, it has been proved that the solution of the heat equation on tubes of small, decreasing diameters can be described by the solution of the heat equation on $L$ and an additional potential $W_0 \in C^\infty(L)$. It was revealed that $W_0$ reflects properties of $L$, $M$ and the embedding $\varphi:L\to M$, i.e. intrinsic and extrinsic geometric quantities. The aim of this thesis is to investigate and interpret the potential $W_0$ and its properties. The primary focus is to understand the role $W_0$ plays in the above described limit process. Our main result is that for any given closed Riemannian manifold $L$ and $W\in C^\infty (L)$ we are able to construct an embedding of$L$ into an ambient space with $W$ as the associated potential function. Another main result of this work is given in Chapter 2, where we explain the behavior of the potential in special cases and elucidate the relevance of $W_0$ for them. The study of totally geodesic embeddings is remarkably fruitful. In this case, a detailed geometric discussion demonstrates that the potential is related to the first non-zero correction term, which occurs in a well-known volume formula by Gray and Weyl. Hence, we are able to provide an interpretation in this case, since the potential is here closely related to the finite dimensional volume of the tube. Moreover, we give a coordinate-free version of the volume formula. In Chapter 6, we investigate the behavior of $W_0$ for the triple $L_2\subset L_1 \subset M$ of (isometric) embeddings of Riemannian manifolds and compare the results obtained by directly conditioning from $M$ to $L_2$ with the successively conditioning first to $L_1$, then to $L_2$. We prove that these surface measures differ in general, but we are able to give a cocycle relation between them.

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