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Integration on loop groups. II: Heat equation for the Wiener measure. (English) Zbl 0787.22021

[For Parts I and III, cf. ibid. 93, 207-237 (1990; Zbl 0715.22024) and ibid. 108, 13-46 (1992; Zbl 0762.22019).]
Let \(G\) denote a compact semisimple Lie group, and let \(L(G)\) denote the group of “free” loops over \(G\); i.e. the group of continuous maps \(\gamma\) from \([0,1]\) into \(G\) satisfying \(\gamma(0)=\gamma(1)\). The purpose of the authors is to construct an elliptic operator \(\Delta_ L\) on \(L(G)\) such that the family of quasi-invariant Wiener measures defined on \(L(G)\) is generated by the process associated to \(\Delta_ L\) modified by a Feynman-Kac density. The construction uses a sort of pullback to the space \(P(G)\) of continuous maps from \([0,1]\) into \(G\), the space of “free” paths over \(G\), and a tubular chart of \(P(G)\) along \(L(G)\). In this tubular chart functions on \(L(G)\) are extended to functions on \(P(G)\) in a natural way, the Wiener measure on \(P(G)\) satisfies a heat equation with an appropriate Laplace operator \(\Delta_ P\) acting on smooth cylindrical functions. This gives the possibility to obtain the desired operator on \(L(G)\).
The main theorem is corollary 2 of the theorem stated in point 6 of the paper proving a form of heat equation with a Feynman-Kac density.
The last sentence of the introducing summary states: “This work has two aspects, one is the development of a homotopy operator given by the heat diffusion on loop groups in view of their harmonic analysis…; the other is relative to gaussian geometry in infinite dimension…”.

MSC:

22E67 Loop groups and related constructions, group-theoretic treatment
58D20 Measures (Gaussian, cylindrical, etc.) on manifolds of maps
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
43A85 Harmonic analysis on homogeneous spaces
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References:

[1] Airault, H., Differential calculus on finite condimensional submanifolds of the Wiener space: The divergence operator, J. Funct. Anal., 100, 291-316 (1991) · Zbl 0731.60059
[2] Airault, H.; Malliavin, P., Intégration géométrique sur l’espace de Wiener, Bull. Sci. Math., 112, 3-52 (1988) · Zbl 0656.60046
[3] Baker, H. F., Abel’s Theorem and the Allied Theory Including the Theory of the Theta Functions (1897), Cambridge · JFM 28.0331.01
[4] Cameron, P. H.; Martin, W. T., Transformation of Wiener integrals under translations, Ann. of Math., 45, 389-396 (1944) · Zbl 0063.00696
[5] Cruzeiro, A. B., Equations différentielles sur l’espace de Wiener et formules de Cameron-Martin non linéaires, J. Funct. Anal., 54, 206-227 (1983) · Zbl 0524.47028
[6] Frenkel, I. B., Orbital theory for affine Lie algebras, Invent. Math., 77, 301-352 (1984) · Zbl 0548.17007
[7] Getzler, E., Dirichlet forms on loop space, Bull. Sci. Math. (2), 113, 151-174 (1989) · Zbl 0683.31002
[8] Gross, L., Logarithmic Sobolev inequalities on loop groups, J. Funct. Anal., 102, 268-313 (1991) · Zbl 0742.22003
[9] Ito, K., Brownian motions on a Lie group, (Proc. Japan Acad., 26 (1950)), 4-10 · Zbl 0041.45703
[10] Malliavin, M. P.; Malliavin, P., Integration on loop groups. I. Quasi invariant measures, J. Funct. Anal., 93, 207-237 (1990) · Zbl 0715.22024
[11] M. P. Malliavin and P. MalliavinJ. Funct. Anal.; M. P. Malliavin and P. MalliavinJ. Funct. Anal. · Zbl 0762.22019
[12] Malliavin, P., Naturality of quasi invariance of some measures, (Cruzeiro, A. B., Proceedings of the Lisbonne Conference (1991), Birkhäuser) · Zbl 0767.28013
[13] Perrin, F., Etude mathématique du mouvement brownien de rotation, Ann. Ecole Norm. Sup., 45, 1-51 (1928) · JFM 54.0993.05
[14] Whittaker, E. T.; Watson, G. N., Modern Analysis (1927), Cambridge Univ. Press: Cambridge Univ. Press London/New York · Zbl 0108.26903
[15] Kusuoka, S., Analysis on Wiener spaces. II. Differential forms, J. Funct. Anal., 103 (1992) · Zbl 0772.58003
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