# Approximation to Wiener measure on a general noncompact Riemannian manifold

## TL;DR

This paper extends the approximation of Wiener measure from compact to noncompact Riemannian manifolds using cutoff methods, enabling analysis of more general path spaces and deriving integration by parts formulas.

## Contribution

It generalizes previous finite-dimensional approximation techniques to noncompact manifolds and applies these to derive new integration by parts formulas in broader path spaces.

## Key findings

- Extended approximation methods to noncompact manifolds.
- Established integration by parts formulas for broader path spaces.
- Demonstrated applicability to infinite interval path spaces.

## Abstract

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 adapted to partitions $\P$ of $[0,T]$, and the associated Wiener measure is also approximated by a sequence of probability measures on finite dimensional manifolds. In this article, we will extend their results to the general path space(possibly with infinite interval) over a non-compact Riemannian manifold by using the cutoff method of compact Riemannian manifolds. Extension to the free path space. As applications, we obtain integration by parts formulas in the path space $W^T_x(M)$ and the free path space $W^T(M)$ respectively.