### 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 Laplace-Beltrami 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'' pre-Dirichlet form. The main theorem of this paper asserts that this pre-Dirichlet form admits a logarithmic Sobolev inequality.

The manuscript is available as a LaTeX file (157K) and as a DVI file (234K).

August 8, 1996

 08/15/2016 02:53 PM