Penalisations of multidimensional brownian motion, VI

Roynette, Bernard; Vallois, Pierre; Yor, Marc

ESAIM: Probability and Statistics, Tome 13 (2009), pp. 152-180.

As in preceding papers in which we studied the limits of penalized 1-dimensional Wiener measures with certain functionals ${\Gamma }_t$, we obtain here the existence of the limit, as $t\to \infty$, of $d$-dimensional Wiener measures penalized by a function of the maximum up to time $t$ of the brownian winding process (for $d=2$), or in $d$ $\ge$ 2 dimensions for brownian motion prevented to exit a cone before time $t$. Various extensions of these multidimensional penalisations are studied, and the limit laws are described. Throughout this paper, the skew-product decomposition of $d$-dimensional brownian motion plays an important role.

Classification : 60F17, 60F99, 60G44, 60H20, 60J60
Mots clés : skew-product decomposition, brownian windings, Dirichlet problem, spectral decomposition

[1] M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété riemannienne. Lect. Notes Math. 194. Springer-Verlag, Berlin (1971). | MR | Zbl

[2] R. Durrett, A new proof of Spitzer's result on the winding of 2-dimensional Brownian motion. Ann. Probab. 10 (1982) 244-246. | MR | Zbl

[3] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition (1991). | MR | Zbl

[4] N.N. Lebedev, Special functions and their applications. Dover Publications Inc., New York (1972). Revised edition, translated from the Russian and edited by Richard A. Silverman, unabridged and corrected republication. | Zbl

[5] P.A. Meyer, Probabilités et potentiel. Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. XIV. Actualités Scientifiques et Industrielles, No. 1318. Hermann, Paris (1966). | MR | Zbl

[6] G. Pap and M. Yor, The accuracy of Cauchy approximation for the windings of planar Brownian motion. Period. Math. Hungar. 41 (2000) 213-226. | MR | Zbl

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

[8] J. Pitman and M. Yor, Further asymptotic laws of planar Brownian motion. Ann. Probab. 17 (1989) 965-1011. | MR | Zbl

[9] D. Revuz and M. Yor, Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition (1999). | MR | Zbl

[10] B. Roynette and M. Yor, Penalising Brownian paths. Lect. Notes Math. 1969. Springer-Verlag, Berlin (2009). | MR

[11] B. Roynette, P. Vallois and M. Yor, Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III. Period. Math. Hungar. 50 (2005) 247-280. | MR | Zbl

[12] B. Roynette, P. Vallois and M. Yor. Limiting laws associated with Brownian motion perturbed by normalized exponential weights I. Studia Sci. Math. Hungar. 43 (2006) 171-246. | MR | Zbl

[13] B. Roynette, P. Vallois and M. Yor, Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II. Studia Sci. Math. Hungar. 43 (2006) 295-360. | MR | Zbl

[14] B. Roynette, P. Vallois and M. Yor, Pénalisations et extensions du théorème de Pitman, relatives au mouvement brownien et à son maximum unilatère. In Séminaire de Probabilités, XXXIX (P.A. Meyer, in memoriam). Lect. Notes Math. 1874. Springer, Berlin (2006) 305-336. | MR | Zbl

[15] B. Roynette, P. Vallois and M. Yor, Some penalisations of the Wiener measure. Japan. J. Math. 1 (2006) 263-290. | MR | Zbl

[16] B. Roynette, P. Vallois and M. Yor, Some extensions of Pitman's and Ray-Knight's theorems for penalized Brownian motions and their local times, IV. Studia Sci. Math. Hungar. 44 (2007) 469-516. | MR | Zbl

[17] B. Roynette, P. Vallois and M. Yor, Penalizing a $Bes\left(d\right)$ process ($0<d<2$) with a function of its local time at $0$, V. Studia Sci. Math. Hungar. 45 (2008) 67-124. | MR | Zbl

[18] B. Roynette, P. Vallois and M. Yor, Penalizing a Brownian motion with a function of the lengths of its excursions, VII. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009) 421-452. | Numdam | MR

[19] F. Spitzer, Some theorems concerning $2$-dimensional Brownian motion. Trans. Am. Math. Soc. 87 (1958) 187-197. | MR | Zbl

[20] D.W. Stroock and S.R.S. Varadhan, Multidimensional diffusion processes. Classics in Mathematics. Springer-Verlag, Berlin, (2006). Reprint of the 1997 edition. | MR | Zbl

[21] S. Watanabe, On time inversion of 1-dimensional diffusion processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75) 115-124. | MR | Zbl