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Théorème des résidus asymptotique pour le mouvement brownien sur une surface riemannienne compacte. (Asymptotic residue theorem for the Brownian motion on a compact Riemannian surface). (French) Zbl 0746.60059

The aim of this paper is to give a general, autonomous and geometric version of the asymptotic stochastic residue theorem, whose first version appeared in [J. Pitman and M. Yor, Ann. Probab. 17, No. 3, 965-1011 (1989; Zbl 0686.60085)], with the Euclidean plane as frame; this theorem already was a generalisation of the asymptotic studies of winding numbers. The frame of the present paper is a Riemannian compact surface of volume \(S\), endowed with its Brownian motion \(X\) and with a closed 1- form \(\omega\) having a finite number of singularities; the result is that the Stratonovich integral \({1\over t}\int_ 0^ t \omega (X_ s)\) converges in law towards a Cauchy variable of parameter \(\pi/S \times\Sigma\) (residues of \(\omega\)).
The method is: first to compare the given integral with the numbers of little windings around the singularities of \(\omega\); second to study locally such a number in conformal coordinates and excursion by excursion; third to control the amount of excursions; and fourth to shrink the areas of the excursions until obtaining at the limit an annihilation of the fluctuations of the metric.
Reviewer: J.Franchi

MSC:

60H05 Stochastic integrals
60J65 Brownian motion
60F05 Central limit and other weak theorems
58J65 Diffusion processes and stochastic analysis on manifolds

Citations:

Zbl 0686.60085
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