Dilatation vector fields on the loop group (Q1304655)

Suppose \(G\) is a compact connected Lie group. Let \(\mathbb{L}(G)\) be the loop space of \(G\), consisting of all continuous maps \(\gamma:[0,1]\to G\) with \(\gamma(0)= \gamma(1)\). Denoting by \(\mu^{\mathbb{L}}_t\) the Wiener measure of variance \(t\) defined on \(\mathbb{L}(G)\), the dilation vector field -- which is the first-order differential operator \({\mathcal G}_t\) such that the heat equation holds with respect to \(\mu^{\mathbb{L}}_t\) -- is explicitly determined.