Log-Sob on Pinned Loop Groups
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Logarithmic Sobolev Inequalities for Pinned Loop Groups (Joint with T. Lohrenz)

(UCSD Preprint, September 15, 1995. To appear in J. of Funct. Anal. )

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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.

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

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August 8, 1996


11/26/2018 10:37 AM