E. Khmaladze

Towards an Innovation Theory of Spatial Brownian Motion under Boundary Conditions

Set-parametric Brownian motion $\boldsymbol{b}$ in a star-shaped set $G$ is considered when the values of $\boldsymbol{b}$ on the boundary of $G$ are given. Under the conditional distribution given these boundary values the process $\boldsymbol{b}$ becomes some set-parametrics 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 $\boldsymbol{b}$ and the transformation itself can be considered as Doob--Meyer decomposition of $\boldsymbol{b}$. 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.