Logarithmic Sobolev inequalities for pinned loop groups
This is a detailed paper devoted to various geometric features of the infinite dimensional group \(L(G)\) of continuous based loops in a connected Lie group \(G\) of compact type. An \(\text{Ad}_G\)-invariant inner product in the algebra of \(G\) defines the so-called \(H^t\)-Riemannian structure on \(L(G)\). For this structure the authors construct a heat kernel associated to the Laplace-Beltrami operator and to a certain probability measure. A classical pre-Dirichlet form is constructed. It is shown that this form admits a logarithmic Sobolev inequality.
scientific article; zbMATH DE number 934262