  Working in a Hamiltonian formulation with $A_0 = 0$ gauge and also in a path
integral formulation, we show that the vacuum wave functional of
four-dimensional pure Yang-Mills theory has the form of the exponential of a
{\it three}-dimensional Yang-Mills action. This result implies that vacuum
expectation values can be calculated in Yang-Mills theory but one dimension
lower. Our analysis reveals that this dimensional reduction results from a
stochastic nature of the theory.
