The Brownian motion and the canonical stochastic flow on a symmetric space (original) (raw)
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Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2001
We consider the Brownian bridge of length T on a symmetric space of the noncompact type. We prove that this process, properly rescaled, converges when T → +∞ to a process whose generalized radial part is the bridge of the Euclidean Brownian motion in the Weyl chamber killed at the boundary. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Le pont brownien sur les espaces riemanniens symétriques Résumé. On considère le pont brownien de longueur T sur un espace symétrique de type non compact. On montre que ce processus, convenablement renormalisé, converge lorsque T → +∞ vers un processus dont la partie radiale généralisée est le pont du mouvement brownien dans la chambre de Weyl tué à la frontière. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Version française abrégée Considérons un espace riemannien symétrique de type non compact M. Par définition M = G/K où G est un groupe semi-simple connexe de centre fini et K est un sous-groupe compact maximal. Soit g = k + p la décomposition de Cartan de l'algèbre de Lie g de G. On choisit un sous espace abélien maximal a de p et on le munit de la structure euclidienne donnée par la forme de Killing. Soit a + une chambre de Weyl de a et Σ + 0 l'ensemble des racines indivisibles et positives associées. Rappelons la décomposition polaire généralisée de M. On choisit o = K comme origine dans M. Soit A = exp a et soit M le centralisateur de A dans K. Pour x ∈ M, soitk(x) ∈ K/M et C(x) ∈ a + tels que k(x) e C(x) • o = x, où k(x) ∈ K est un représentant dek(x). Nous utilisons aussi la décomposition d'Iwasawa G = KN A. Chaque g ∈ G s'écrit g = K(g)N (g) e H(g) , où K(g) ∈ K, N (g) ∈ N et H(g) ∈ a. Le mouvement brownien B sur M est le processus de Markov de générateur ∆/2. Son semi-groupe admet des densités p t (x, y) strictement positives et symétriques par rapport à la mesure riemannienne m. Le pont de a ∈ M à b ∈ M de longueur T est le processus de Markov B {a,b,T } t , 0 t T non homogène, Note présentée par Marc YOR. S0764-4442(01)02145-0/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés