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

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.


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.
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[1] Airault, H., Projection of the infinitesimal generator of a diffusion, J. Funct. Anal., 85, 353-391 (1989) · Zbl 0683.60055
[2] Airault, H.; Malliavin, P., Integration géometrique sur l’espace de Wiener, Bull. Sci. Math. (2), 112, 3-52 (1988) · Zbl 0656.60046
[3] Airault, H.; Van Biesen, J., Le processus d’Ornstein-Uhlenbeck sur une sous variété, Bull. Sci. Math. (2), 115, 185-210 (1991) · Zbl 0716.60091
[4] Albeverio, S.; Hoegh-Krohn, R., The energy representation of Sobolev Lie group̄s, Compositio Math., 36, 37-52 (1978) · Zbl 0393.22013
[5] Bakry, D.; Emery, M., Hypercontractivité de semi-groupes de diffusion, Compte-Rendus, 299, 775-778 (1984) · Zbl 0563.60068
[6] Davies, E. B., Heat Kernels and Spectral Theory (1989), Cambridge Univ. Press: Cambridge Univ. Press New York · Zbl 0699.35006
[8] Deuschel, J.-D; Stroock, D., Large Deviations (1989), Academic Press: Academic Press San Diego · Zbl 0682.60018
[9] Doob, J. L., Stochastic Processes (1953), Wiley: Wiley New York · Zbl 0053.26802
[10] Federer, H., Geometric Measure Theory (1969), Springer: Springer New York · Zbl 0176.00801
[11] Frenkel, I. B., Orbital theory for affine Lie algebras, Invent. Math., 77, 301-352 (1984) · Zbl 0548.17007
[12] Fukushima, M., Dirichlet Forms and Markov Processes (1980), North-Holland: North-Holland New York · Zbl 0422.31007
[13] Gelfand, I. M.; Graev, M. I.; Vershik, A. M., Representations of the group of smooth mappings of a manifold \(X\) into a compact Lie group, Compositio Math., 35, 299-334 (1977) · Zbl 0368.53034
[14] Getzler, E., Dirichlet forms on loop space, Bull. Sci. Math. (2), 113, 151-174 (1989) · Zbl 0683.31002
[15] Gross, L., Logarithmic Sobolev inequalities, Amer. J. Math., 97, 1061-1083 (1975) · Zbl 0318.46049
[16] Gross, L., Existence and uniqueness of physical ground states, J. Funct. Anal., 10, 52-109 (1972) · Zbl 0237.47012
[18] Gross, L., Logarithmic Sobolev inequalities for the heat kernel on a Lie group, (Hida, T.; Kuo, H. H., Bielefeld Conf. on White Noise Analysis. Bielefeld Conf. on White Noise Analysis, 1989 (1990), World Scientific: World Scientific New Jersey) · Zbl 0817.60007
[19] Helgason, S., Differential Geometry and Symmetric Spaces (1962), Academic Press: Academic Press San Diego · Zbl 0122.39901
[20] Holley, R.; Stroock, D., Logarithmic Sobolev inequalities and stochastic Ising models, J. Statist. Phys., 46, 1159-1194 (1987) · Zbl 0682.60109
[21] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1981), North-Holland: North-Holland New York · Zbl 0495.60005
[22] Kuo, H. H., Gaussian measures in Banach spaces, (Lecture Notes in Mathematics, Vol. 463 (1975), Springer-Verlag: Springer-Verlag New York/Berlin)
[23] Malliavin, P., Hypoellipticity in infinite dimension, (Pinsky, M., Diffusion Processes and Related Problems in Analysis (1990), Birkhäuser: Birkhäuser Boston) · Zbl 0723.60059
[24] Malliavin, M.-P; Malliavin, P., Integration on loop groups. I. Quasi invariant measures, J. Funct. Anal., 93, 207-237 (1990) · Zbl 0715.22024
[25] Mueller, C. E.; Weissler, F., Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the \(n\)-sphere, J. Funct. Anal., 48, 252-283 (1982) · Zbl 0506.46022
[26] Pressley, A.; Segal, G., Loop Groups (1986), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 0618.22011
[27] Shigekawa, I., Absolute continuity of probability laws of Wiener functional, (Proc. Japan Acad., 54 (1978)), 230-233 · Zbl 0422.60032
[28] Shigekawa, I., Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ., 20, 263-289 (1980) · Zbl 0476.28008
[29] Stroock, D., An Introduction to the Theory of Large Deviations (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0552.60022
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