Résumé
On donne une version simplifiée de la démonstration probabiliste du théoreme de l'indice pour l'opérateur de Dirac, donnée par J.M. Bismut. Au lieu d'utiliser la construction du mouvement brownien au moyen du fibré des reperes, on utilise la construction de Schwartz en plongeant la variété ambiante dans un espace vectoriel de dimension plus grande. On évite aussi l'utilisation de la décomposition de l'espace de Wiener en deux utilisée par J.M. Bismut, le calcul des variations stochastiques étant notre outil principal. De ce fait, on perd la relation avec la cohomologie de l'espace des lacets.
Summary
We give a simplified version of Bismut's probabilistic proof of the index theorem for the Dirac operator. By using Schwartz's construction of a Brownian motion over a manifold, we expect to give a simpler approach to the computations of stochastic geometry. Our main tool is the calculus of stochastic variations, rather than the splitting of the Wiener space into two pieces. For that reason, we loose the relation with the cohomology of the loop space.
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Bibliographie
Atiyah, M., Singer, I.: Index of elliptic, operators I. Ann. Math. 87, 484–530 (1968)
Atiyah, M., Singer, I.: Index of elliptic operators II. Ann. Math. 87, 546–604 (1968)
Azencott, R.: Grandes déviations et applications. In: Hennequin, P.L. (ed.) Cours de Probabilité de Saint-Flour. (Lect. Notes Math., vol. 774, pp. 1–176, Berlin Heidelberg New York: Springer 1978
Azencott, R.: Une approche probabiliste du théorème de l'indice. Séminaire Bourbaki. Exposé 633.
Bismut, J.M.: The Atiyah-Singer theorem: a probabilistic approach. I.J.F. Anal. 57, 56–98 (1984)
Bismut, J.M.: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander condition. Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 469–505 (1981)
Bismut, J.M.: Mécanique aléatoire. In: Dugué, D., Lukacs, E., Rohatgi, V.K. (eds.) Proceedings, Oberwolfach 1980. Lect. Notes Math. vol. 861, Berlin Heidelberg New York: Springer 1981
Bismut, J.M.: The Witten complex and the degenerate Morse Inequalities. J. Differ. Geom. 23, 207–240 (1986)
Bismut, J.M.: The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs. Invent. Math. 88, 91–151 (1986)
Bismut, J.M.: Index theorem and equivariant cohomology on the loop space. Comm. Math. Phys. 98, 213–237 (1985)
Bismut, J.M.: Localization formulas, superconnections and the index theorem for families. Comm. Math. Phys. 103, 127–166 (1986)
Berline, N., Vergne, M.: A computation of the equivariant index of the Dirac operator. Bull. Soc. Math. Fr. 113, 305–340 (1985)
Doubrovine, D., Fomenko, A., Novikov, S.: La géométrie comtemporaine, tome II. Moscow: Editions de Moscow 1979
Duncan, T.E.: The heat equation, the Kac-Formula and some index formula in partial differential equation and geometry. In: Byrnes, I. (ed.) (Lect. Notes Pure Appl. Math. Vol. 48. pp. 57–76, New York Basel: Dekker 1979
Getzler, E.: A short proof of the Atiyah-Singer index theorem. Topology 25, 111–117 (1988)
Gilkey, P.B.: Invariance theory, the heat equation and the Atiyah-Singer theorem. Boston: Publish or Perish 1984
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North Holland 1981
Ikeda, N., Watanabe, S.: Malliavin calculus for Wiener's functional and it's application. In: Elworthy, D. (ed.) From local times to global geometry. Montreal: Pitman 1986
Kobayashi, S., Nomizu, S.: Foundations of differential geometry, tome II. New York: Interscience 1969
Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. In: K. Ito (ed.) Part I. Stochastic Analysis. Proceedings Tanaguchi Symposium. Kinokuniyol North Holland 1989
Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. Part. II. J. Fac. Sci. Uni. Tokyo. Sect. 1 A Math. 32, 1–76 (1985)
Leandre, R.: Renormalisation et calcul des variations stochastiques. C. R. Acad. Sci., Paris, Ser. I. 302, 135–138 (1986)
Leandre, R.: Sur le théorème de l'indice des familles. Séminaire de Strasbourg no XXII. In: Azema, I., Meyer, P.A., Yor, M. (eds.) Seminaire de probabilites. (Lect. Notes Math., vol. 1321, pp. 348–414) Berlin Heidelberg New York: Springer 1988
Schwarz, L.: Construction directe d'une diffusion sur une variété. In: Azema, J. Vor, M. (eds.) Séminaire de probabilités, no. XIX. Lect. Notes Math., vol. 1123, pp. 91–113. Berlin Heidelberg New York: Springer 1985
Yor, M.: Remarques sur une formule de Paul Levy. In: Azema, J., Yor, M. (eds.) Séminaire de probabilités, No. XIV, (Lect. Notes Math., vol. 784, pp. 343–346) Berlin Heidelberg New York: Springer 1980
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Léandre, R. Sur le théoreme d'atiyah-singer. Probab. Th. Rel. Fields 80, 119–137 (1988). https://doi.org/10.1007/BF00348755
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DOI: https://doi.org/10.1007/BF00348755