## 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

### Citations:

Zbl 0715.22024; Zbl 0762.22019
Full Text:

### References:

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