# Surface measures and integration by parts formula on levels sets induced by functionals of the Brownian motion in $\mathbb R^n$

## TL;DR

This paper constructs a surface measure on level sets of the $L^2$-norm for solutions of stochastic gradient systems in infinite-dimensional path spaces and derives an integration by parts formula using Malliavin calculus.

## Contribution

It introduces a novel surface measure on level sets in infinite-dimensional Wiener spaces associated with stochastic gradient systems, extending Malliavin calculus techniques.

## Key findings

- Defined a surface measure on level sets of $L^2$-norms in path space
- Derived an integration by parts formula involving the surface measure
- Extended Malliavin calculus methods to this new setting

## Abstract

On the infinite dimensional space $E$ of continuous paths from $[0,1]$ to $\mathbb R^n$, $n \ge 3$, endowed with the Wiener measure $\mu$, we construct a surface measure defined on level sets of the $L^2$-norm of $n$-dimensional processes that are solutions to a class of stochastic gradient system-type equations, and provide an integration by parts formula involving this surface measure. We follow the approach to surface measures in Gaussian spaces proposed via techniques of Malliavin calculus by Airault and Malliavin in 1988.