Géométrie de la courbe brownienne plane

Ben Arous, Gérard

Séminaire Bourbaki : volume 1990/91, exposés 730-744, Astérisque, no. 201-202-203 (1991), Exposé no. 730, 36 p.

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Ben Arous, Gérard. Géométrie de la courbe brownienne plane, dans Séminaire Bourbaki : volume 1990/91, exposés 730-744, Astérisque, no. 201-202-203 (1991), Exposé no. 730, 36 p. http://www.numdam.org/item/SB_1990-1991__33__7_0/

Bibliographie
Cité par

[AD] O. Adelman, A. Dvoretzky, Plane Brownian motion has strictly n-multiple points. Israel J. Math. 52, 361-364 (1985). | MR | Zbl

[B1] K. Burdzy, Brownian paths and cones. Ann. Probab. 13, 1006-1010 (1985). | MR | Zbl

[B2] K. Burdzy, Multidimensional Brownian Excursions and Potential Theory. Longman, New-York, 1987. | MR | Zbl

[B3] K. Burdzy, Geometric properties of two-dimensional Brownian paths. Probab. Th. Rel. Fields 81, 485-505 (1989). | MR | Zbl

[B4] K. Burdzy, Cut points on Brownian paths. Ann. Probab. 17, 1012- 1036. | MR | Zbl

[BSM] K. Burdzy, J. San Martin, Curvature of the convex hull of planar Brownian motion near its minimum point. Stoch. Process. Appl. 33, 89-103 (1989). | MR | Zbl

[CT] Z. Ciesielski, S. J. Taylor, First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103, 434-450 (1963). | MR | Zbl

[CHM] M. Cranston, P. Hsu, P. March, Smoothness of the convex hull of planar Brownian motion. Ann. Probab. 17, 144-150 (1989). | MR | Zbl

[Da] B. Davis, Brownian motion and analytic functions. Ann. Probab. 7, 913-932 (1979). | MR | Zbl

[DV] M. D. Donsker, S. R. S. Varadhan, Asymptotics for the Wiener sausage. Comm. Pure Appl. Math. 28, 525-565 (1975). | MR | Zbl

[Du1] R. Durrett, A new proof of Spitzer's result on the winding of two-dimensional Brownian motion. Ann. Probab. 10, 244-246 (1982). | MR | Zbl

[Du2] R. Durett, Brownian Motion and Martingales in Analysis. Wadsworth, Belmont Ca. 1984. | Zbl

[DE] A. Dvoretzky, P. Erdös, Some problems on random walk in space. Proc. Second Berkeley Symposium on Math, Statistics and Probability. 353-367. University of California Press, Berkeley, 1951. | MR | Zbl

[DK1] A. Dvoretzky, P. Erdös, S. Kakutani, Double points of paths of Brownian motion in n-space. Acta Sci. Math. (Szeged) 12, 74-81 (1950). | MR | Zbl

[DK2] A. Dvoretzky, P. Erdös, S. Kakutani, Multiple points of Brownian motion in the plane. Bull. Res. Council Israel Sect. F3, 364-371 (1954). | MR

[DK3] A. Dvoretzky, P. Erdös, S. Kakutani, Points of multiplicity of plane Brownian paths. Bull. Res. Council Israel Sect. F7, 175-180 (1958). | MR

[DKT] A. Dvoretzky, P. Erdös, S. Kakutani, S. J. Taylor, Triple points of Brownian motion in 3-space. Proc. Cambridge Philos. Soc. 53, 856-862 (1957). | MR | Zbl

[Dy1] E. B. Dynkin, Additive functionals of several time-reversible Markov processes. J. Funct. Anal. 42, 64-101 (1981). | MR | Zbl

[Dy2] E. B. Dynkin, Local times and quantum fields, Seminar on Stochastic Processes 1983, pp.69-84, Birkhäuser, Boston, 1984. | MR | Zbl

[Dy3] E. B. Dynkin, Polynomials of the occupation field and related random fields. J. Funct. Anal. 42, 64-101 (1984). | MR | Zbl

[Dy4] E. B. Dynkin, Random fields associated with multiple points of the Brownian motion J. Funct. Anal. 62, 397-434 (1985). | MR | Zbl

[Dy5] E. B. Dynkin, Functionals associated with self-intersections of the Brownian motion. Séminaire de Probabilités XX, Lecture Notes in Math. 1204, 553-571, Springer, Berlin, 1986. | Numdam | MR | Zbl

[Dy6] E. B. Dynkin, Self-intersection gauge for random walks and for Brownian motion. Ann. Probab. 16, 1-57 (1988). | MR | Zbl

[Dy7] E. B. Dynkin, Regularized self-intersection local times of the planar Brownian motion. Ann. Probab. 16, 58-74 (1988). | MR | Zbl

[E] S. F. Edwards, The statistical mechanics of polymers with excluded volume. Proc. Phys. Sci. 85, 613-624 (1965). | MR | Zbl

[EB] M. El Bachir, L'enveloppe convexe du mouvement brownien. Thèse de troisième cycle, Université Paul Sabatier, Toulouse, 1983.

[Ev] S. N. Evans, On the Hausdorff dimension of Brownian cone points. Math. Proc. Camb. Phil. Soc. 98, 343-353 (1985). | MR | Zbl

[GHR] D. Geman, J. Horowitz, J. Rosen, A local time analysis of intersections of Brownian paths in the plane. Ann. probab. 12, 86-107 (1984). | MR | Zbl

[H] J. Hawkes, Some geometric aspects of potentials theory. Stochastic Analysis and Applications. Lecture Notes in Math. 1095, 130-154, Springer, New-York, 1984. | MR | Zbl

[IMK] K. Itô, H.P. Mckean, Diffusion Processes and their Sample Paths. Springer, New-York, 1965. | MR | Zbl

[Kn] F. B. Knight, Essentials of Brownian Motion and Diffusion. A.M.S. Providence, 1981. | MR | Zbl

[KR] G. Kallianpur, H. Robbins, Ergodic property of the Brownian Motion process. Proc. Nat. Acad. Sci. U.S.A. 39, 525-533. | MR | Zbl

[L1] J. F. Le Gall, Sur la mesure de Hausdorff de la courbe brownienne. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123, 297- 313. Springer, New-York, 1985. | Numdam | MR | Zbl

[L2] J. F. Le Gall, Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan. Séminaire de Probabilités XIX. Lecture Notes in Math 1123, 314-331, Springer, New-York 1985. | Numdam | MR | Zbl

[L3] J. F. Le Gall, Sur la saucisse de Wiener et les points multiples du mouvement brownien Ann. Probab. 14, 1219-1244 (1986). | MR | Zbl

[L4] J. F. Le Gall, Le comportement du mouvement brownien entre les deux instants où il passe par un point double. J. Funct. Anal. 71, 246-262 (1987). | MR | Zbl

[L5] J. F. Le Gall, Mouvement brownien, cônes et processus stables. Probab. Th. Rel. Fields 76, 587-627 (1987). | MR | Zbl

[L6] J. F. Le Gall, Temps locaux d'intersection et points multiples des processus de Lévy. Séminaire de probabilités XXI, Lecture Notes in Math. 1247, 341-375. Springer, New-York, 1987. | Numdam | MR | Zbl

[L7] J. F. Le Gall, The exact Hausdorff measure of Brownian multiple points, Seminar on Stochastic Processes 1986, pp.107-137, Birkhäuser, Boston, 1987. | MR | Zbl

[L8] J. F. Le Gall, Fluctuation results for the Wiener sausage. Ann. Probab. 16, 991-1018 (1988). | MR | Zbl

[L9] J. F. Le Gall, Sur une conjecture de M. Kac. Probab. Th. Rel. Fields 78, 389-402 (1988). | MR | Zbl

[L10] J. F. Le Gall, Wiener sausage and self-interaction local times. J. Funct. Anal. 88, 299-341 (1990). | MR | Zbl

[L11] J. F. Le Gall, On the connected components of the complement of a two-dimensional Brownian path. To appear. | MR | Zbl

[L12] J. F. Le Gall, Cours à l'Ecole d'été de St Flour 1990. A paraître en Lecture Notes Springer Verlag.

[LY] J. F. Le Gall, M. Yor, Etude asymptotique de certains mouvements browniens complexes avec drift. Probab. Th. Rel. Fields 71, 183-229 (1986). | MR | Zbl

[LJ] Y. Le Jan, On the Fock space representation of functionals of the occupation field and their renormalization. J. Funct. Anal. 80, 88-108 (1988). | MR | Zbl

[Lé1] P. Lévy, Le mouvement brownien plan. Amer. J. Math. 62, 487-550 (1940). | JFM | MR

[Lé2] P. Lévy, La mesure de Hausdorff de la courbe du mouvement brown- ien. Giornale de l'Istituto Italiano degli Attuari 16, 1-37 (1953). | MR | Zbl

[Lé3] P. Lévy, Le caractère universel de la courbe du mouvement brownien et la loi du logarithme itéré. Rendiconti del circolo matematico di Palermo,II, 4, 337-366 (1955). | MR | Zbl

[Lé4] P. Lévy, Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris, 1965. | MR | Zbl

[Ma] N. G. Makarov, On the distorsion of boundary sets under conformal mappings. Proc. London Math. Soc. (3) 51, 369-384 (1985). | MR | Zbl

[M] B. B. Mandelbrot, The Fractal Geometry of Nature. W.H. Freeman and Co. New-York, 1982. | MR | Zbl

[MM] J. E. Mcmillan, Boundary behaviour of a conformal mapping. Acta Math. 123, 43-68 (1969). | MR | Zbl

[Mo] T. S. Mountford, On the asymptotic number of small components created by planar Brownian motion. Stochastics 28, 177-188 (1989). | MR | Zbl

[PY1] J. W. Pitman, M. Yor, Asymptotic laws of planar Brownian motion. Ann. Probab. 14, 733-779 (1986). | MR | Zbl

[PY2] J. W. Pitman, M. Yor, Further asymptotic laws of planar Brownian Motion. Annals of Probability 1989, vol.17, N°3, 965-1011. | MR | Zbl

[Po] G. Pommerenke, Univalent Functions. Vandenhoeck & Ruprecht, Göttingen, 1975. | MR | Zbl

[PS] S. C. Port, C. J. Stone, Brownian Motion and Classical Potential Theory. Academic Press, New-York, 1978. | MR | Zbl

[ReY] D. Revuz, M. Yor, Continuous Martingale Calculus. Springer, 1990 (to appear).

[RoW] L. C. G. Rogers, D. Williams, Diffusions, Markov Processes, and Mar- tingales, Vol.II. Wiley, Chichester, 1987. | MR

[R1] J. Rosen, A local time approach to the self-intersections of Brownian paths in space. Comm. Math. Physics. 88, 327-338 (1983). | MR | Zbl

[R2] J. Rosen, Self-intersections of random fields. Ann. Probab. 12, 108- 119 (1984). | MR | Zbl

[R3] J. Rosen, Tanaka formula and renormalization for intersections of planar Brownian motion. Séminaire de Probabilités XX. Lecture Notes in Math. 1204, 515-531. Springer, Berlin, 1986. | Numdam | MR | Zbl

[RY] J. Rosen, M. Yor, Tanaka formulae and renormalization for triple intersections of Brownian motion in the plane. Ann. Probab. to appear. | MR | Zbl

[Sh1] M. Shimura, A limit theorem for two-dimensional conditioned random walk. Nagoya Math. J. 95, 105-116 (1984). | MR | Zbl

[Sh2] M. Shimura, Excursions in a cone for two-dimensional Brownian motion. J. Math. Kyoto Univ. 25, 433-443 (1985). | MR | Zbl

[Sh3] M. Shimura, Meandering points of two-dimensional Brownian motion. Kodai Math. J. 11, 169-176 (1988). | MR | Zbl

[S1] F. Spitzer, Some theorems concerning two-dimensional Brownian motion. Trans. Amer. Math. Soc. 87, 187-197 (1958). | MR | Zbl

[S2] F. Spitzer, Electrostatic capacity, heat flow and Brownian motion. Z. Wahrsch. verw. Gebiete 3, 110-121 (1964). | MR | Zbl

[Sz] A. S. Sznitman, Some bounds and limiting results for the Wiener sausage of small radius associated with elliptic diffusions. Stochastic Process. Appl. 25, 1-25 (1987). | MR | Zbl

[Ta] L. Takacs, Expected perimeter length, Amer. Math. Monthly 87- 142.

[T1] S. J. Taylor, The exact Hausdorff measure of the sample path for planar Brownian motion. Proc. Cambridge Philos. Soc. 60, 253-258 (1964). | MR | Zbl

[T2] S. J. Taylor, Sample path properties of processes with stationary independent increments. Stochastic Analysis, D. Kendall, E. Harding eds. Wiley, London, 1973. | MR

[T3] S. J. Taylor, The measure theory of random fractals. Math. Proc. Camb. Phil. Soc. 100, 383-406 (1986). | MR | Zbl

[V] S. R. S. Varadhan, Appendix to "Euclidean quantum field theory", by K. Symanzik. In Local Quantum Theory (R. Jost ed.). Academic, New-York, 1969.

[Wo] R. Wolpert, Wiener path intersections and local time. J. Funct. Anal. 30, 329-340 (1978). | MR | Zbl

[Y1] M. Yor, Compléments aux formules de Tanaka-Rosen. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123, 332-349. Springer, Berlin, 1985. | Numdam | MR | Zbl

[Y2] M. Yor, Renormalisation et convergence en loi pour les temps locaux d'intersection du mouvement brownien dans R3. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123, 350-365. Springer, Berlin, 1985. | Numdam | MR | Zbl

[Y3] M. Yor, Précisions sur l'existence et la continuité des temps locaux d'intersection du mouvement brownien dans Rd. Séminaire de Probabilités XX. Lecture Notes in Math. 1204, 532-541. Springer, Berlin, 1986. | Numdam | MR | Zbl

[Y4] M. Yor, Sur la représentation comme intégrales stochastiques des temps d'occupation du mouvement brownien dans Rd. Séminaire de Probabilités XX. Lecture Notes in Math. 1204, 543-552. Springer, Berlin, 1986. | Numdam | MR | Zbl