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

Abstract

On the infinite dimensional space E of continuous paths from [0, 1] to $${\mathbb {R}}^n$$ , $$n \ge 1$$ , 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 general class of stochastic differential 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 in Airault and Malliavin (Bull Sci Math 112:3–52, 1988).