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
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