Related papers: Approximation Schemes for Path Integration on Riem…

This paper gives a rigorous interpretation of a Feynman path integral on a Riemannian manifold M with non-positive sectional curvature. A $L^2$ Riemannian metric $G_P$ is given on the space of piecewise geodesic paths $H_P(M)$ adapted to…

In prior work \cite{AD} of Lars Andersson and Bruce K. Driver, the path space with finite interval over a compact Riemannian manifold is approximated by finite dimensional manifolds $H_{x,\P} (M)$ consisting of piecewise geodesic paths…

Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the…

Path integrals developed by Richard Feynman have been an important tool in Physics in studying quantum field theory. In mathematics, it has also been widely used in providing formal proofs in the study of Index theorem and asymptotic…

Some parts of stochastic analysis on curved spaces are revisted. A concise proof of the quasi-invariance of the Wiener measure on the path spaces over a Riemannian manifold is presented. The shifts are allowed to be in the Cameron-Martin…

We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve…

We prove several versions of Driver's integration by parts formula for the horizontal Wiener measure on a totally geodesic Riemannian foliation and prove that the horizontal Wiener measure has a quasi-invariance property with respect to…

Using the path integral measure factorization method based on the nonlinear filtering equation from the stochastic process theory, we consider the reduction procedure in Wiener path integrals for a mechanical system with symmetry that…

The analysis of manifold-valued data requires efficient tools from Riemannian geometry to cope with the computational complexity at stake. This complexity arises from the always-increasing dimension of the data, and the absence of…

We discuss measures on spaces of unparametrized paths related to the Wiener measure. These measures arise naturally in the study of one-dimensional gravity coupled to scalar fields. Two kinds of discrete approximations are defined, the…

In Euclidean space, the integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a type of surface measure. According to geometric measure theory, this surface measure is realized by the…

On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular,…

We prove Gronwall-type estimates for the distance of integral curves of smooth vector fields on a Riemannian manifold. Such estimates are of central importance for all methods of solving ODEs in a verified way, i.e., with full control of…

We characterize the relaxation of the perimeter in an infinite dimensional Wiener space, with respect to the weak L^2-topology. We also show that the rescaled Allen-Cahn functionals approximate this relaxed functional in the sense of…

A path integral reduction procedure in Wiener-type path integrals, based on the approach developed in arXiv:1912.13124, is applied to a simple invariant mechanical system defined on a product manifold with a given free, proper and isometric…

This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the $L^2$ Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold. The main body of the…

Approximating a manifold-valued function from samples of input-output pairs consists of modeling the relationship between an input from a vector space and an output on a Riemannian manifold. We propose a function approximation method that…

In this work we consider a suitable generalization of the Feynman path integral on a specific class of Riemannian manifolds consisting of compact Lie groups with bi-invariant Riemannian metrics. The main tools we use are the Cartan…

We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L^2 Riemannian metric - so called…

The purpose of this article is to propose a new method to define and calculate path integrals over metrics on a K\"ahler manifold. The main idea is to use finite dimensional spaces of Bergman metrics, as an approximation to the full space…