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 . 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 .
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 .
Mots-clés : Path-valued process, Super Brownian motion, Probabilistic potential theory, Energy, Capacitary measure, Hitting probability, Polar sets, Stochastic differential equations, Nonlinear partial differential equations
@article{AIF_1994__44_1_277_0, author = {Le Gall, Jean-Fran\c{c}ois}, title = {Hitting probabilities and potential theory for the brownian path-valued process}, journal = {Annales de l'Institut Fourier}, pages = {277--306}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {44}, number = {1}, year = {1994}, doi = {10.5802/aif.1398}, mrnumber = {94m:60155}, zbl = {0794.60077}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1398/} }
TY - JOUR AU - Le Gall, Jean-François TI - Hitting probabilities and potential theory for the brownian path-valued process JO - Annales de l'Institut Fourier PY - 1994 SP - 277 EP - 306 VL - 44 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.1398/ DO - 10.5802/aif.1398 LA - en ID - AIF_1994__44_1_277_0 ER -
%0 Journal Article %A Le Gall, Jean-François %T Hitting probabilities and potential theory for the brownian path-valued process %J Annales de l'Institut Fourier %D 1994 %P 277-306 %V 44 %N 1 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/aif.1398/ %R 10.5802/aif.1398 %G en %F AIF_1994__44_1_277_0
Le Gall, Jean-François. Hitting probabilities and potential theory for the brownian path-valued process. Annales de l'Institut Fourier, Tome 44 (1994) no. 1, pp. 277-306. doi : 10.5802/aif.1398. http://www.numdam.org/articles/10.5802/aif.1398/
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