Logarithmic Sobolev inequalities on loop groups. (English) Zbl 0742.22003

Gross, Leonard

J. Funct. Anal. 102, No. 2, 268-313 (1991).

Let \(G\) be a compact connected Lie group. Let \(W_ G\) be the space of paths \([0,1]\buildrel g\over \rightarrow G\) such that \(g(0)=\) identity \(=e\). There is a natural probability measure \(P\) on \(W_ G\) induced by the \(G\)-valued Brownian motion over \([0,1]\) which starts at \(e\). For a point \(x\in G\), the conditional measure \(P(.\mid g(1)=x)=\mu_ x\) is the Brownian measure on the bridge space \(W_ x=\{\gamma \in W_ G: \gamma(1)=x\}\). The purpose of this paper is to prove an inequality for the infinite dimensional loop group \(W_ e\) with respect to the Brownian bridge measure \(\mu_ e\) of the following form, for \(f\) real: \[ \int_{W_ e}f^ 2\log f\mu_ e(d\gamma)\leq \hbox{Const. }\int_{W_ e}\{|\hbox{grad }f(\gamma)|^ 2+V(\gamma)f(\gamma)^ 2\}\mu_ e(d\gamma)+\| f\| ^ 2_{L^ 2(\mu_ e)}\log \| f\|_{L^ 2(\mu_ e)}. \] (Here grad requires a definition.) This is achieved by developing an analogy with the corresponding inequality in finite dimensional manifolds, which is first established. The actual result as stated above relies upon 16 intricate technical lemmas. The final section applies the inequality to certain positivity-preserving hypercontractive semi-groups and their spectra.

Reviewer: E.J.Akutowicz (Montpellier)

MSC:

22E30	Analysis on real and complex Lie groups
22E67	Loop groups and related constructions, group-theoretic treatment
60J65	Brownian motion
43A05	Measures on groups and semigroups, etc.

Keywords:

logarithmic Sobolev inequality; compact connected Lie group; space of paths; Brownian motion; infinite dimensional loop group; Brownian bridge measure; positivity-preserving hypercontractive semi-groups

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