A remarkable <i>σ</i>-finite measure on $ \mathcal{C}({\mathbf{R}}_{+},\mathbf{R})$ related to many Brownian penalisations

Joseph Najnudel; Bernard Roynette; Marc Yor

doi:10.1016/j.crma.2007.09.015

Probability Theory

A remarkable σ-finite measure on

C (R_{+}, R)

related to many Brownian penalisations

Presented by: Marc Yor

Joseph Najnudel ¹; Bernard Roynette ²; Marc Yor ^1,³

¹ Laboratoire de probabilités et modèles aléatoires, Université Paris VI, 4, place Jussieu, 75252 Paris cedex 05, France
² Institut de mathématiques Elie-Cartan, Université Henri-Poincaré Nancy I, B.P. 239, 54506 Vandoeuvre-lès-Nancy cedex, France
³ Institut universitaire de France, 103, boulevard Saint-Michel, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 459-466.

Abstract
Résumé

In this Note, we study a σ-finite measure $W$ on the space $C (R_{+}, R)$ , strongly related to Wiener measure, and we construct a large class of Brownian martingales from $W$ . Some of these martingales appear naturally in the study of Brownian penalisations made by B. Roynette, P. Vallois and M. Yor.

Dans cette Note, nous étudions une mesure σ-finie $W$ sur l'espace $C (R_{+}, R)$ , fortement liée à la mesure de Wiener, et nous construisons une grande classe de martingales browniennes à partir de $W$ . Certaines de ces martingales apparaissent naturellement dans l'étude des pénalisations browniennes faite par B. Roynette, P. Vallois et M. Yor.

Received: 2007-07-20
Accepted: 2007-09-07
Published online: 2007-10-15

DOI: 10.1016/j.crma.2007.09.015

Author's affiliations:

Joseph Najnudel ¹; Bernard Roynette ²; Marc Yor ^{1, 3}

¹ Laboratoire de probabilités et modèles aléatoires, Université Paris VI, 4, place Jussieu, 75252 Paris cedex 05, France
² Institut de mathématiques Elie-Cartan, Université Henri-Poincaré Nancy I, B.P. 239, 54506 Vandoeuvre-lès-Nancy cedex, France
³ Institut universitaire de France, 103, boulevard Saint-Michel, 75005 Paris, France

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     author = {Joseph Najnudel and Bernard Roynette and Marc Yor},
     title = {A remarkable \protect\emph{\ensuremath{\sigma}}-finite measure on $ \mathcal{C}({\mathbf{R}}_{+},\mathbf{R})$ related to many {Brownian} penalisations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {459--466},
     publisher = {Elsevier},
     volume = {345},
     number = {8},
     year = {2007},
     doi = {10.1016/j.crma.2007.09.015},
     language = {en},
}

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Joseph Najnudel; Bernard Roynette; Marc Yor. A remarkable σ-finite measure on $ \mathcal{C}({\mathbf{R}}_{+},\mathbf{R})$ related to many Brownian penalisations. Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 459-466. doi : 10.1016/j.crma.2007.09.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.09.015/

References
Cited by

[1] Ph. Biane; M. Yor Valeurs principales associées aux temps locaux browniens, Bull. Sci. Math. (2), Volume 111 (1987) no. 1, pp. 23-101

[2] J. Najnudel, Temps locaux et pénalisations browniennes, PhD thesis, Université Paris VI, 2007

[3] J. Najnudel, B. Roynette, M. Yor, On a remarkable σ-finite measure W on path space, which rules penalisations for a linear Brownian motion, 2007, in preparation

[4] D. Revuz; M. Yor Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 293, Springer-Verlag, Berlin, 1999, pp. 493-511

[5] B. Roynette; P. Vallois; M. Yor Limiting laws associated with Brownian motion perturbated by normalized exponential weights, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 337 (2003) no. 10, pp. 667-673

[6] B. Roynette; P. Vallois; M. Yor Limiting laws for long Brownian bridges perturbed by their one-sided maximum. III, Period. Math. Hungar., Volume 50 (2005) no. 1–2, pp. 247-280

[7] B. Roynette; P. Vallois; M. Yor Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time. II, Studia Sci. Math. Hungar., Volume 43 (2006) no. 3, pp. 295-360

[8] B. Roynette; P. Vallois; M. Yor Some penalisations of the Wiener measure, Japan. J. Math., Volume 1 (2006), pp. 263-290

[9] B. Roynette, M. Yor, Penalising Brownian paths: Rigorous results and meta-theorems, Laboratoire de Probabilités et Modèles Aléatoires, Prépublication, 2007

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