Logarithmic Sobolev Inequalities for Pinned Loop Groups (Joint with T. Lohrenz)(UCSD Preprint, September 15, 1995. To appear in J. of Funct. Anal. )Let $G$ be a connected compact type Lie group equipped with an $Ad_G$invariant inner product on the Lie algebra ${\frak g}$ of $ G.$ Given this data there is a well known left invariant ``$H^1$Riemannian structure'' on ${\cal L}={\cal L}(G)$the infinite dimensional group of continuous based loops in $G.$ Using this Riemannian structure, we define and construct a ``heat kernel'' $\nu_T(g_0,\cdot )$ associated to the LaplaceBeltrami operator on ${\cal L}(G).$ Here $T>0,$ $g_0\in {\cal L}(G),$ and $\nu_T(g_0,\cdot )$ is a certain probability measure on ${\cal L}(G).$ For fixed $g_0\in {\cal L}(G)$ and $T>0 ,$ we use the measure $\nu_T(g_0,\cdot )$ and the Riemannian structure on ${\cal L}(G)$ to construct a ``classical'' preDirichlet form. The main theorem of this paper asserts that this preDirichlet form admits a logarithmic Sobolev inequality. The manuscript is available as a LaTeX file (157K) and as a DVI file (234K). August 8, 1996

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