Towards an innovation theory of spatial Brownian motion under boundary conditions. (English) Zbl 1007.60033
Summary: Set-parametric Brownian motion \({\mathbf b}\) in a star-shaped set \(G\) is considered when the values of \({\mathbf b}\) on the boundary of \(G\) are given. Under the conditional distribution given these boundary values the process \({\mathbf b}\) becomes some set-parametric Gaussian process and not Brownian motion. We define the transformation of this Gaussian process into another Brownian motion which can be considered as “martingale part” of the conditional Brownian motion \({\mathbf b}\) and the transformation itself can be considered as Doob-Meyer decomposition of \({\mathbf b}\). Some other boundary conditions and, in particular, the case of conditional Brownian motion on the unit square given its values on the whole of its boundary are considered.
MSC:
60G48 | Generalizations of martingales |
60G57 | Random measures |
60H99 | Stochastic analysis |
47N30 | Applications of operator theory in probability theory and statistics |