Higher order integration by parts formulae for Wiener measures on a path space between two curves

TL;DR

This paper develops higher-order integration by parts formulae on path spaces between two curves for Wiener measures, incorporating boundary terms and employing novel construction methods and notation for clarity.

Contribution

It introduces higher-order integration by parts formulae with boundary terms, using Brownian excursion constructions and a new Symmetrization notation for concise expression.

Findings

Formulated higher-order integration by parts formulae with boundary terms

Introduced Symmetrization notation for boundary term expressions

Provided probabilistic explanations for boundary terms

Abstract

We have formulated higher-order integration by parts formulae on the path space restricted between two curves, with respect to pinned/ordinary Wiener measures. The higher-order integration by parts formulae introduce nontrivial boundary terms, unlike the first-order one. Furthermore, in the process of proving these formulae, it becomes necessary to employ the construction methods of Brownian excursion and Brownian house-moving through random walk approximations. To express the integration by parts formula concisely, we introduced a notation called Symmetrization. This notation enables the rewriting of the intricate expressions of boundary terms associated with higher-order integration by parts into more concise forms. Additionally, we provided a probabilistic explanation for the boundary terms by introducing symbols based on the concept of infinitesimal probability. These efforts are aimed at fostering an intuitive understanding of the integration by parts formulae.