Approximation Schemes for Path Integration on Riemannian Manifolds

TL;DR

This paper develops a finite-dimensional approximation scheme for Wiener measure on closed Riemannian manifolds, extending previous methods to $L^{1}$-functionals using a new categorical colimit approach.

Contribution

It introduces a novel approximation scheme for Wiener measure on Riemannian manifolds, generalizing existing approaches to $L^{1}$-functionals with a categorical perspective.

Findings

Established a finite-dimensional approximation for Wiener measure on manifolds.

Extended approximation methods to $L^{1}$-functionals.

Introduced a new categorical colimit-based approach.

Abstract

In this paper, we prove a finite dimensional approximation scheme for the Wiener measure on closed Riemannian manifolds, establishing a generalization for $L^{1}$-functionals, of the approach followed by Andersson and Driver on [1]. We follow a new approach motived by categorical concept of colimit.