Logarithmic Sobolev Inequalities for Pinned Loop Groups

LetGbe a connected compact type Lie group equipped with anAdG-invariant inner product on the Lie algebra g ofG. Given this data there is a well known left invariant “H1-Riemannian structure” on L=L(G)—the infinite dimensional group of continuous based loops inG. Using this Riemannian structure, we define and construct a “heat kernel”νT(g0, ·) associated to the Laplace–Beltrami operator on L(G). HereT>0,g0∈L(G), andνT(g0,·) is a certain probability measure on L(G). For fixedg0∈L(G) andT>0, we use the measureνT(g0,·) and the Riemannian structure on L(G) to construct a “classical” pre-Dirichlet form. The main theorem of this paper asserts that this pre-Dirichlet form admits a logarithmic Sobolev inequality.