Hitting probabilities and potential theory for the Brownian path-valued process
We consider the Brownian path-valued process studied in the paper reviewed above and in [``A path-valued Markov process and its connections with partial differential equations'' (to appear in: Proc. 1st Europ. Congr. Math.)] 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-valued 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 \(\Delta u=u^ 2\).
scientific article; zbMATH DE number 566973