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Joseph Najnudel
Pénalisations 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, 60J55
Keywords: penalization, local time, Walsh Brownian motion

Résumé - Abstract

In 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

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