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) (Q1185272)

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 [\textit{J. Pitman} and \textit{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.