Abstract
Let B1, B2, ..., Bk be k independent Brownian motions with values in IRd. We study the long time asymptotics of additive functionals of the type:
where f is an integrable function on IRd. The critical cases are d = 2k−1 and d = 2k. We obtain results of the first order and of the second order (corresponding to 
, which generalize classical results of Kallianpur-Robbins, Papanicolaou-Stroock-Varadhan, and Kasahara-Kotani, for k = 1, as well as recent results of Le Gall and Weinryb-Yor, for k = 2.
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References
E.B. Dynkin: Additive functionals of several time-reversible Markov processes. J. Funct. Anal., 42, p 64–101 (1981).
S.N. Evans: Potential theory for a family of several Markov Processes. Ann. de l'Inst. Henri Poincaré, 23, p 499–530, (1987).
P.J. Fitzsimmons, T.S. Salisbury: Capacity and energy for multiparameter Markov processes. preprint.
D. Geman, J. Horowitz, J. Rosen: A local time analysis of intersections of Brownian paths in the plane. Ann. Prob. 12, p 86–107 (1984).
G. Kallianpur, H. Robbins: Ergodic property of the Brownian motion process. Proc Nat. Acad. Sci. U.S.A. 39, p 525–533 (1953).
Y. Kasahara, S. Kotani: On limit processes for a class of additive functionals of recurrent diffusion processes. Z. Wahrsch. verw. Gebiete 49, p 133–153 (1979).
J.F. Le Gall: Propriétés d'intersection des marches aléatoires II. Etude des cas critiques. Comm. Math. Phys. 104, p 509–528 (1986).
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 Mathematics n o 1123, p 314–332 (1985)
J.F. Le Gall: Sur la saucisse de Wiener et les points multiples du Mouvement Brownien. Ann. Prob., 14, p 1219–1244. (1986).
Mountford: An extension of a result of Kahane using Rosen's local time of intersection Stochastics 23, 4, p 449–464 (1988).
G.C. Papanicolaou, D.W. Stroock, S.R.S. Varadhan: Martingale approach to some limit theorems. Duke Univ. Math. Ser. III, Statistical Mechanics and Dynamical Systems. (1977).
D.L. Burkholder: Sharp inequalities for martingales and stochastic integrals. Colloque Paul Lévy, Astérisque n o 157–158, p 75–94. (1988).
J. Rosen: Tanaka's formula for multiple intersections of planar Brownian motion. Stoch. Proc. and Appl., 23, p 131–141 (1986).
S. Weinryb, M. Yor: Le mouvement Brownien de Lévy indexé par IR3 comme limite centrale de temps locaux d'intersection. Séminaire de Probabilités XXII, Lecture notes in Mathematics n o 1321, p 225–248 (1988).
M. Yor: Compléments aux formules de Tanaka-Rosen. Séminaire de Probabilités XIX, Lecture notes in Mathematics n o 1123, p 332–349 (1985).
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Biane, P. (1989). Comportement asymptotique de certaines fonctionnelles additives de plusieurs mouvements browniens. In: Azéma, J., Yor, M., Meyer, P.A. (eds) Séminaire de Probabilités XXIII. Lecture Notes in Mathematics, vol 1372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083974
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DOI: https://doi.org/10.1007/BFb0083974
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