Integration by parts and quasi-invariance for heat kernel measures on loop groups
Let \(G\) be a connected compact Lie group with \(Ad_G\)-invariant inner product in the algebra. Denote by \(L (G)\) the space of continuous loops in \(G\) based at the identity. The author proves two versions of the integration by parts formula: for the Wiener measure on the path space of \(L (G)\) and for the heat kernel measures on \(L (G)\). The former is an infinite-dimensional version of that proved by the author earlier for compact manifolds, it deals with a certain class of vector fields on the path space. The latter is for left-invariant first order differential operators on \(L (G)\), it is an infinite-dimensional analog of the fact that heat kernels have logarithmic derivatives in finite dimension. Both constructions allow the author to show that the heat kernel measures are quasi-invariant under right and left translations by finite energy loops, and to obtain a description of a certain collection of logarithmic Sobolev inequalities for finite-dimensional approximation as a truly infinite-dimensional inequality on the loop group.
scientific article; zbMATH DE number 1078822
scientific article