**E. Khmaladze**

## Towards an Innovation Theory of Spatial Brownian Motion under Boundary Conditions

**abstract:**

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.