On the equivalence of measures on loop space
One can construct a left invariant Brownian motion on the loop space \(L(K)\) of a compact Lie group \(K\) (the Lie algebra of which is endowed with an \(Ad_K\)-invariant inner product). The aim of this article is to study the equivalence of the heat kernel (the distribution of the Brownian motion on \(L(K)\) at some time \(t\)) and the pinned Wiener measure (the distribution of the Brownian motion on \(K\) conditioned to end at its starting point). The author proves that this equivalence holds true if \(K\) is simply connected (if \(K\) is connected but not simply connected, the equivalence trivially fails). The proof is based on the two-parameter stochastic calculus developed by J. R. Norris. A motivation of this work is the study of logarithmic Sobolev inequalities on loop groups.
scientific article; zbMATH DE number 1563074