With cedram.org English français Home Overview Search for an article Submit a paper Informations for the authors Subscription Limited access - RUCHE Table of contents for this issue | Previous article | Next article Joseph NajnudelPénalisations de l’araignée brownienne(Penalizations of Walsh Brownian motion)Annales de l'institut Fourier, 57 no. 4 (2007), p. 1063-1093, doi: 10.5802/aif.2287 Article PDF | Reviews MR 2339327 | Zbl 1121.60084 Class. Math.: 60B10, 60J65, 60G17, 60G44, 60J25, 60J55Keywords: penalization, local time, Walsh Brownian motion Résumé - AbstractIn this paper, we penalize a Walsh Brownian motion $(A_t)_{t \ge 0}$ (also called Brownian spider), which takes values in a finite set $E$ of intersecting rays, with a weight equal to $\frac{1}{Z_t} \exp (\alpha _{N_t} X_t + \gamma L_t)$, where $t$ is a positive real, $(\alpha _k)_{k \in E}$ a family of real numbers indexed by $E$, $\gamma$ a real parameter, $X_t$ the distance from $A_t$ to the origin, $N_t$ ($\in E$) the ray on which $A_t$ is to be found, $X_t$ the local time of $(A_s)_{0 \le s \le t}$ at the origin, and $Z_t$ the normalization constant. We show that the family of probability measures obtained by these penalizations converges to a limit probability measure as $t$ tends to infinity, and we study some properties of this limit probability measure. Bibliography[1] M. Barlow, J. Pitman & M. Yor, On Walsh’s Brownian motions, Séminaire de Probabilités, XXIII, Lecture Notes in Math. 1372, Springer, 1989, p. 275–293 Numdam |  Zbl 0747.60072[2] M. Barlow, J. Pitman & M. Yor, Une extension multidimensionnelle de la loi de l’arc sinus, Séminaire de Probabilités, XXIII, Lecture Notes in Math. 1372, Springer, 1989, p. 294–314 Numdam |  Zbl 0738.60072[3] A.-S. Cherny & A.-N. Shiryaev, “Some Distributional Properties of a Brownian Motion with a Drift and an Extension of P. Lévy’s Theorem”, Theory of Probab. and Its Applications 44 (2000) no. 2, p. 412-418 Article |  Zbl 0974.60058[4] Y. Hariya & M. Yor, “Limiting distributions associated with moments of exponential Brownian functionals”, Studia Sci. Math. Hungar. 41 (2004) no. 2, p. 193-242  MR 2082657 |  Zbl 02186333[5] I. Karatzas & S.-E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics 113, Springer-Verlag, New York, 1988  MR 917065 |  Zbl 0638.60065[6] J. Pitman, The distribution of local times of a Brownian bridge, Séminaire de Probabilités, XXXIII, Lecture Notes in Math. 1709, Springer, 1999, p. 388–394 Numdam |  MR 1768012 |  Zbl 0945.60081[7] B. Roynette, P. Vallois & M. Yor, “Limiting laws associated with Brownian motion perturbated by normalized exponential weights”, C. R. Math. Acad. Sci. Paris 337 (2003) no. 10, p. 667-673  MR 2030109 |  Zbl 1031.60021[8] B. Roynette, P. Vallois & M. Yor, “Limiting laws for long Brownian bridges perturbed by their one-sided maximum. III”, Period. Math. Hungar. 50 (2005) no. 1-2, p. 247-280 Article |  MR 2162812 |  Zbl 02213351[9] B. Roynette, P. Vallois & M. Yor, “Limiting laws associated with Brownian motion perturbed by its maximum, minmum and local time. II”, Studia Sci. Math. Hungar. 43 (2006) no. 3, p. 295-360  MR 2253307 |  Zbl 05082381[10] B. Roynette, P. Vallois & M. Yor, Pénalisations et quelques extensions du théorème de Pitman, relatives au mouvement brownien et à son maximum unilatère, Séminaire de Probabilités, XXXIX, Lecture Notes in Math. 1874, Springer, 2006, p. 305–336  MR 2276902 |  Zbl 1124.60034[11] B. Roynette, P. Vallois & M. Yor, “Some penalisations of the Wiener measure”, Jap. Journal of Math. 1 (2006), p. 263-290 Article |  MR 2261065 |  Zbl 1160.60315[12] J.-B. Walsh, “A diffusion with a discontinuous local time”, Temps Locaux, Astérisque 52–53 (1978), p. 37-45 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310