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Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 293))

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Abstract

In this section, we will specialize the notions of Sect. 5 Chap. 0 to the Wiener space W d. This space is a Polish space when endowed with the topology of uniform convergence on compact subsets of ℝ+.

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Notes and Comments

  1. Billingsley, P. Convergence of probability measures. Wiley and Sons, New York 1979. Second edition (1999).

    Google Scholar 

  2. Parthasarathy, K.R. Probability measures on metric spaces. Academic Press, New York 1967.

    MATH  Google Scholar 

  3. Jacod, J. and Shiryaev, A.N. Limit theorems for stochastic processes. Springer, Berlin Heidelberg New York 1987.

    MATH  Google Scholar 

  4. Donsker, M.D. An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc. 6(1951) 1–12.

    Google Scholar 

  5. Pitman, J.W. One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. Appl. Prob. 7(1975) 511–526.

    Article  MathSciNet  MATH  Google Scholar 

  6. Le Gall, J.F. Une approche élémentaire des théorèmes de décomposition de Williams. Sém. Prob. XX, Lect. Notes in Mathematics, vol. 1204. Springer, Berlin Heidelberg New York 1986, pp. 447–464.

    Google Scholar 

  7. Le Gall, J.F. Propriétés d’intersection des marches aléatoires, I. Convergence vers le temps local d’intersection, II. Etude des cas critiques. Comm. Math. Phys. 104 (1986) 471–507 and 509–528.

    Google Scholar 

  8. Papanicolaou, G., Stroock, D.W., and Varadhan, S.R.S. Martingale approach to some limit theorems. Proc. 1976. Duke Conf. On Turbulence. Duke Univ. Math. Series III, 1977.

    Google Scholar 

  9. Nualart, D. Weak convergence to the law of two-parameter continuous processes. Z.W. 55(1981) 255–269.

    Article  MathSciNet  MATH  Google Scholar 

  10. Yor, M. Le drap brownien comme limite en loi de temps locaux linéaires. Sém. Prob. XVII, Lect. Notes in Mathematics, vol. 986. Springer, Berlin Heidelberg New York 1983, pp. 89–105.

    Google Scholar 

  11. Pagès, H. Personal communication.

    Google Scholar 

  12. Rebolledo, R. La méthode des martingales appliquée à l’étude de la convergence en loi des processus. Mémoire de la S.M.F. 62 (1979)

    Google Scholar 

  13. Rebolledo, R. Central limit theorems for local martingales. Z.W. 51(1980) 269–286.

    Article  MathSciNet  MATH  Google Scholar 

  14. Jeulin, T. Semi-martingales et grossissement d’une filtration. Lect. Notes in Mathematics, vol. 833. Springer, Berlin Heidelberg New York 1980.

    Google Scholar 

  15. Papanicolaou, G., Stroock, D.W., and Varadhan, S.R.S. Martingale approach to some limit theorems. Proc. 1976. Duke Conf. On Turbulence. Duke Univ. Math. Series III, 1977.

    Google Scholar 

  16. Pitman, J.W. Cyclically stationary Brownian local time processes. Prob. Th. Rel. Fields 106(1996) 299–329.

    Article  MathSciNet  MATH  Google Scholar 

  17. Le Gall. J.F., and Yor, M. Etude asymptotique de certains mouvements browniens complexes avec drift. Z.W. 71(1986) 183–229.

    MATH  Google Scholar 

  18. Kasahara, Y., and Kotani, S. On limit processes for a class of additive functionals of recurrent diffusion processes. Z.W. 49(1979) 133–143.

    Article  MathSciNet  MATH  Google Scholar 

  19. Kasahara, Y. On Lévy’s downcrossing theorem. Proc. Japan Acad. 56A (10) (1980) 455–458.

    Article  MathSciNet  MATH  Google Scholar 

  20. Biane, P. Comportement asymptotique de certaines fonctionnelles additives de plusieurs mouvements browniens. Sém. Prob. XXIII, Lect. Notes in Mathematics, vol. 1372. Springer, Berlin Heidelberg New York 1989, pp. 198–233.

    Google Scholar 

  21. Bernard, A. and Maisonneuve, B. Décomposition atomique de martingales de la classe H 1. Sém. Prob. XI. Lect. Notes in Mathematics, vol. 581. Springer, Berlin Heidelberg New York 1977, pp. 303–323.

    Google Scholar 

  22. Lewis, J.T. Brownian motion on a submanifold of Euclidean space. Bull. London Math. Soc. (1986) 616–620.

    Google Scholar 

  23. Van den Berg, M. and Lewis, J.T. Brownian motion on a hypersurface. Bull. London Math. Soc. 17(1985) 144–150.

    Article  MATH  Google Scholar 

  24. Rogers, L.C.G., and Walsh, J.B. The intrinsic local time sheet of Brownian motion. Prob. Th. Rel. F. 88(1991) 363–379.

    Article  MathSciNet  MATH  Google Scholar 

  25. Elworthy, K.D. Stochastic differential equations on manifolds. London Math. Soc. Lecture Notes Series 70. Cambridge University Press (1982).

    Google Scholar 

  26. Yor, M. Sur la continuité des temps locaux associés à certaines semi-martingales. Astérisque 5253, Temps locaux (1978) 23–36.

    Google Scholar 

  27. Yor, M. Un exemple de processus qui n’est pas une semi-martingale. Astérisque 523(1978) 219–221.

    Google Scholar 

  28. Yor, M. Les filtrations de certaines martingales du mouvement brownien dans R”. Sém. Prob. XIII, Lect. Notes in Mathematics, vol. 721. Springer, Berlin Heidelberg New York 1979, pp. 427–440.

    Google Scholar 

  29. Yor, M. Etude asymptotique des nombres de tours de plusieurs mouvements browniens complexes corrélés. In: Durrett, R. and Kesten, H. (eds.), Random walks, Brownian motion and interacting particle systems. Birkhäuser, Basel 1992, pp. 441–455.

    Google Scholar 

  30. Le Gall. J.F., and Yor, M. Etude asymptotique des enlacements du mouvement brownien autour des droites de l’espace. Prob. Th. Rel. Fields 74(1987) 617–635.

    MATH  Google Scholar 

  31. Le Gall. J.F., and Yor, M. Enlacements du mouvement brownien autour des courbes de l’espace. Trans. Amer. Math. Soc. 317(1990) 687–722.

    MATH  Google Scholar 

  32. Le Gall. J.F., and Yor, M. Excursions browniennes et carrés de processus de Bessel. C.R. Acad. Sci. Paris, Série I 303(1986) 73–76.

    Google Scholar 

  33. Knight, F.B. Some remarks on mutual windings. Sem. Prob. XXVII, Lect. Notes in Mathematics, vol. 1557. Springer, Berlin Heidelberg New York 1993, pp. 36–43.

    Google Scholar 

  34. Yamazaki, Y. On limit theorems related to a class of “winding-type” additive functionals of complex Brownian motion. J. Math. Kyoto Univ. 32(4) (1992) 809–841.

    MathSciNet  MATH  Google Scholar 

  35. Pitman, J.W., and Yor, M. Bessel processes and infinitely divisible laws. In: D. Williams (ed.) Stochastic integrals. Lect. Notes in Mathematics, vol. 851. Springer, Berlin Heidelberg New York 1981.

    Google Scholar 

  36. Watanabe, S. Asymptotic windings of Brownian motion paths on Riemann surfaces. Acta Appl. Math. 63(2000), n° 1–3, 441–464

    Google Scholar 

  37. Lyons, T.J., and McKean, H.P., Jr. Windings of the plane Brownian motion. Adv. Math. 51(1984) 212–225.

    Article  MathSciNet  MATH  Google Scholar 

  38. Gruet, J.C. Sur la transience et la récurrence de certaines martingales locales continues à valeurs dans le plan. Stochastics 28(1989) 189–207.

    MathSciNet  MATH  Google Scholar 

  39. Gruet, J.C., and Mountford, T.S. The rate of escape for pairs of windings on the Riemann sphere. J. London Math. Soc. 48(2) (1993) 552–564.

    Article  MathSciNet  MATH  Google Scholar 

  40. Mountford, T.S. Transience of a pair of local martingales. Proc. A.M.S. 103(3) (1988) 933–938.

    Article  MathSciNet  MATH  Google Scholar 

  41. Warren, J. and Yor, M. The Brownian burglar: conditioning Brownian motion by its local time process. Sém. Prob. XXXII, Lect. Notes in Mathematics, vol. 1686. Springer, Berlin Heidelberg New York 1998, pp. 328–342.

    Google Scholar 

  42. Hu, Y., Shi, Z., and Yor, M. Rates of convergence of diffusions with drifted Brownian potentials. Trans. Amer. Math. Soc. 351(1999) 3915–3934.

    Article  MathSciNet  MATH  Google Scholar 

  43. Kawazu, K., and Tanaka, H. A diffusion process in a Brownian environment with drift. J. Math. Soc. Japan 49(1997) 189–211.

    MathSciNet  MATH  Google Scholar 

  44. Tanaka, H. Note on continuous additive functionals of the 1-dimensional Brownian path. Z.W. 1(1963) 251–257.

    Article  MATH  Google Scholar 

  45. Le Gall, J.F. Some properties of planar Brownian motion. Ecole d’été de Saint-Flour XX, 1990, Lect. Notes in Mathematics, vol. 1527. Springer, Berlin Heidelberg New York 1992, pp. 112–234.

    Google Scholar 

  46. Duplantier, B., Lawler, G.F., LeGall, J.F., and Lyons, T.J. The geometry of the Brownian curve. Bull. Sci. Math. 117(1993) 91–106.

    MathSciNet  MATH  Google Scholar 

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Revuz, D., Yor, M. (1999). Limit Theorems in Distribution. In: Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol 293. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06400-9_14

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  • DOI: https://doi.org/10.1007/978-3-662-06400-9_14

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