Random Holonomy for Yang–Mills Fields: Long-Time Asymptotics
Tóm tắt
We study weak and strong convergence of the stochastic parallel transport for time t→∞ on Euclidean space. We show that the asymptotic behavior can be controlled by the Yang–Mills action and the Yang–Mills equations. For open paths we show that under appropriate curvature conditions there exits a gauge in which the stochastic parallel transport converges almost surely. For closed paths we show that there exists a gauge invariant notion of a weak limit of the random holonomy and we give conditions that insure the existence of such a limit. Finally, we study the asymptotic behavior of the average of the random holonomy in the case of t'Hooft's 1-instanton.
Tài liệu tham khảo
Atiyah, M.: Geometry of Yang—Mills fields, Scuola Normale Superiore, Pisa, 1979.
Bauer, R.O.: 'Characterizing Yang—Mills fields by stochastic parallel transport, J. Funct. Anal. 155 (1998), 536–549.
Bauer, R.O.: 'Yang—Mills fields and the stochastic parallel transport in small geodesic balls', Stochastic Process. Appl. 89(2) (2000), 213–226.
Bauer, R.O. and Carlen, E.A.: 'Random holonomy for Hopf-fibrations', J. Funct. Anal. (2001), to appear.
Bismut, J.—M.: 'The Atiyah—Singer theorems: A probabilistic approach. I: The index theorem. II: The Lefschetz fixed point formulas, J. Funct. Anal. 57 (1984), 56–99 and 329–348.
Bismut, J.-M.: Large Deviations and the Malliavin Calculus, Prog. Math. 45, Birkhäuser, Boston, 1984.
Elworthy, K.D.: Stochastic Differential Equations on Manifolds, London Math. Soc. Lecture Note Ser. 70, Cambridge University Press, 1982.
Hsu, E.: 'Stochastic local Gauss—Bonnet—Chern theorem', J. Theoret. Probab. 10(4) (1997), 819–834.
Ikeda, N. and Watanabe, S.: 'Malliavin calculus of Wiener functionals and its applications', in K.D. Elworthy (ed.), From Local Times to Global Geometry, Control and Physics, Vol. 150, Longman Scientific and Technical, Harlow, 1987, pp. 132–178.
Ikeda, N. and Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn, North-Holland, Kodansha Ltd., Amsterdam, Tokyo, 1989.
ItÔ, K.: 'The Brownian motion and tensor fields on Riemannian manifold', in Proc. Intern. Congr. Math. (Stockholm), 1963, pp. 536–539.
ItÔ, K.: 'Stochastic parallel displacement', in Probabilistic Methods in Differential Equations, Lecture Notes in Math. 451, Springer-Verlag, Berlin, 1975, pp. 1–7.
Blaine Lawson, H.: The Theory of Gauge Fields in Four Dimensions, Regional Conference Series in Math. 58, Amer. Math. Soc., 1987.
Malliavin, P.: Géométrie différentielle stochastique, Les Presses de l'Université de Montréal, Montréal, 1978.
Malliavin, P.: 'Sur certaines intégrales stochastique oscillantes', C.R. Acad. Sci. Paris 295 (1980), 295–300.
Stafford, S.: 'A stochastic criterion for Yang—Mills connections', in M.A. Pinsky (ed.), Diffusion Processes and Related Problems in Analysis, Vol. 1: Diffusions in Analysis and Geometry, Birkhäuser, Boston, 1990, pp. 313–322.
Ikeda, N., Shigekawa, I. and Taniguchi, S.: 'The Malliavin calculus and long time asymptotics of certain Wiener integrals', in Miniconference on Linear Analysis and Function Spaces (Canberra 1984), Proc. Centre Math. Anal. Austral. Nat. Univ. 9, Austral. Nat. Univ., 1985, pp. 46–113.
Uhlenbeck, K.: 'Connections with L p bounds on curvature', Comm. Math. Phys. 83 (1982), 31–42.
Uhlenbeck, K.: 'Removable singularities in Yang—Mills fields', Comm. Math. Phys. 83 (1982), 11–29.
Ward, R.S. and Wells, R.O.: Twistor Geometry and Field Theory, Cambridge University Press, Cambridge, 1990.