  Factorization of the (formal) path integral measure in a Wiener path
integrals for Yang--Mills diffusion is studied. Using the nonlinear filtering
stochastic differential equation, we perform the transformation of the path
integral defined on a total space of the Yang--Mills principal fiber bundle and
come to the reduced path integral on a Coulomb gauge surface. Integral relation
between the path integral representing the "quantum" evolution given on the
original manifold of Yang--Mills fields and the path integral on the reduced
manifold defined by the Coulomb gauge is obtained.
