# Integration by parts of some non-adapted vector field from Malliavin's lifting approach

## TL;DR

This paper introduces a new orthogonal lift of vector fields from a Riemannian manifold to the Cameron-Martin space, enabling integration by parts formulas for non-adapted vector fields in stochastic analysis.

## Contribution

It proposes a novel orthogonal lift construction based on a least squares approach that accounts for Ricci curvature effects, extending Malliavin's lift to curved manifolds.

## Key findings

- Established an integration by parts formula for the new vector fields.
- Reduced to Malliavin's lift in Euclidean space.
- Demonstrated the lift's properties on curved manifolds.

## Abstract

In this paper we propose a lift of vector field $X$ on a Riemannian manifold $M$ to a vector field $\tilde{X}$ on the curved Cameron-Martin space $H\left(M\right)$ named orthogonal lift. The construction of this lift is based on a least square spirit with respect to a metric on $H(M)$ reflecting the damping effect of Ricci curvature. Its stochastic extension gives rise to a non-adapted Cameron-Martin vector field on $W_o(M)$. In particular, if $M=\mathbb{R}^d$ with Euclidean metric, then the damp disappears and the lift reduces to the well-known Malliavin's lift. We establish an integration by parts formula for these first order differential operators.