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A general equation for the probability distribution of parallel transporterson the gauge group manifold is derived using the cumulant expansion theorem.This equation is shown to have a general form known as the Kramers-Moyallcumulant expansion in the theory of random walks, the coefficients of theexpansion being directly related to nonperturbative cumulants of the shiftedcurvature tensor. In the limit of a gaussian-dominated QCD vacuum the obtainedequation reduces to the well-known heat kernel equation on the group manifold.Comment: 7 page

Kramers-Moyall cumulant expansion for the probability distribution of parallel transporters in quantum gauge fields