Path integral measure factorization in path integrals for diffusion of Yang--Mills fields

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.