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…”.


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
Full Text: DOI


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