Hitting probabilities and potential theory for the brownian path-valued process
Annales de l'Institut Fourier, Volume 44 (1994) no. 1, p. 277-306

We consider the Brownian path-valued process studied in [LG1], [LG2], which is closely related to super Brownian motion. We obtain several potential-theoretic results related to this process. In particular, we give an explicit description of the capacitary distribution of certain subsets of the path space, such as the set of paths that hit a given closed set. These capacitary distributions are characterized as the laws of solutions of certain stochastic differential equations. They solve variational problems in the space of probability measures on the path space. We also investigate some special classes of polar sets for the path-values process. These results are closely related to the polarity questions for super Brownian motion recently investigated by Dynkin and others. They are also related to removable singularities for the nonlinear partial differential equation Δu=u 2 .

Nous considérons le “mouvement brownien à valeurs trajectoires” déjà étudié dans [LG1], et dans [LG2], qui est étroitement lié au super mouvement brownien. Nous obtenons plusieurs résultats de théorie du potentiel probabiliste relatifs à ce processus. En particulier, nous donnons une description explicite des mesures capacitaires de certains sous-ensembles de l’espace des trajectoires, tels que l’ensemble des trajectoires qui rencontrent un sous-ensemble fermé fixé de d . Ces mesures d’équilibre, qui sont les lois des solutions de certaines équations différentielles stochastiques, sont associées à des problèmes variationnels dans l’ensemble des mesures de probabilité sur l’espace des trajectoires. Nous nous intéressons aussi à des classes particulières d’ensembles polaires pour le mouvement brownien à valeurs trajectoires. Ces derniers résultats sont très liés aux questions de polarité pour le super mouvement brownien étudiées récemment par Dynkin et d’autres auteurs, ainsi qu’aux problèmes d’éliminabilité de singularités pour l’équation aux dérivées partielles non linéaire Δu=u 2 .

@article{AIF_1994__44_1_277_0,
     author = {Gall, Jean-Fran\c cois Le},
     title = {Hitting probabilities and potential theory for the brownian path-valued process},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {44},
     number = {1},
     year = {1994},
     pages = {277-306},
     doi = {10.5802/aif.1398},
     zbl = {0794.60077},
     mrnumber = {94m:60155},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1994__44_1_277_0}
}
Gall, Jean-François Le. Hitting probabilities and potential theory for the brownian path-valued process. Annales de l'Institut Fourier, Volume 44 (1994) no. 1, pp. 277-306. doi : 10.5802/aif.1398. http://www.numdam.org/item/AIF_1994__44_1_277_0/

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