Logarithmic Sobolev inequalities on Lie groups (Q1203510)

Let \(G\) be a connected finite dimensional Lie group and \(\xi_ 1,\dots,\xi_ d\) a basis of its Lie algebra. Put \(\Delta=\sum_{j=1}^ d \xi_ j^ 2\) and define the associated heat kernel \(p_ t\) by the equation \(e^{t\Delta}\)= convolution by \(p_ t\). The \(G\) valued Brownian motion with transition semigroup \(e^{t\Delta}\) is constructed as a limit of \(G\) valued random walks. This is used to derive a logarithmic Sobolev inequality for the energy form in \(L^ 2(G,p_ t(x)dx)\). Although the resulting inequality is known to be best possible in case \(G=R^ n\) it is not best possible if \(G\) is compact. To gain some understanding of why the Gaussian constant fails to be best possible the ``limit of random walk'' representation of the heat kernel measure on \(S^ 1\) is used to justify a calculator computation of the best constants for a three point approximation of \(S^ 1\).