  We describe generalized Brownian motion related to parabolic equation systems
from a logical point of view, i.e., as a generalization of Anderson's random
walk. The connection to classical spaces is based on the Loeb measure. It seems
that the construction of Roux in [11] is the only attempt in the literature to
define generalized Brownian motion related to parabolic systems with coupled
second order terms, where Lam\'e's equation of elastic mechanics is considered
as an example. In this paper we provide an exact construction from a logical
point of view in a more general situation. A Feynman-Kac formula for
generalized Brownian motion is derived which is a useful tool in order to
design probabilistic algorithms for Cauchy problems and initial-boundary value
(of a class of) parabolic systems as well as for stationary boundary problems
of (a class of) elliptic equation systems. The article includes a selfcontained
introduction into all tools of nonstandard analysis needed, and which can be
read with a minimum knowledge of logic in order to make the results available
to a wider audience.
