Penalisations of multidimensional Brownian motion, VI

¹ Université Henri Poincaré, Institut Elie Cartan, BP 239, 54506 Vandoeuvre-lès-Nancy Cedex, France; Pierre.Vallois@iecn.u-nancy.fr
² Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI et VII, 4 place Jussieu – Case 188, 75252 Paris Cedex 05, France.
³ Institut Universitaire de France, France.

Received: 9 July 2007
Revised: 28 January 2008

Abstract

As in preceding papers in which we studied the limits of penalized 1-dimensional Wiener measures with certain functionals Γ_t, we obtain here the existence of the limit, as t → ∞, 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} ≥ 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.