Hitting probabilities and potential theory for the Brownian path-valued process
type: Item
instance of: scholarly article
title: Hitting probabilities and potential theory for the Brownian path-valued process
zbMATH Open document ID: 0794.60077
publication date: 1994-08-28T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75059
review text: 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 \(\Delta u=u^ 2\).
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 566973
zbMATH Keywords: path-valued process
zbMATH Keywords: super Brownian motion
zbMATH Keywords: potential-theoretic results
zbMATH Keywords: capacitary distributions
zbMATH Keywords: variational problems
zbMATH Keywords: partial differential equation
author: J.-F. Le Gall
reviewed by: J.-F. Le Gall
MaRDI profile type: Publication
OpenAlex ID: W2025836284
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DOI: 10.5802/AIF.1398
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 566973