Integration by parts for a Lie group valued Brownian motion (Q1322499)

Let \(G\) be a Lie group of \(d \times d\)-matrices with Lie algebra \(g\). Let \((D_ 1, \dots, D_ n)\) be an orthonormal basis of \(g\) with respect to some Euclidean norm, and let \((\widetilde D_ 1, \dots, \widetilde D_ n)\) be the corresponding left invariant vector fields on \(G\). If \((Y_ t)\) is a Brownian motion on \(g\), then the Stratonovich exponential \(\varepsilon^*(Y)\) is a Brownian motion on \(G\) associated with the Laplacian \(\sum_ i \widetilde D_ i\). A Girsanov-type formula is used to derive an integration-by-parts formula for \(\varepsilon^*(Y)\).