Linear and nonlinear Lagrangian equations are derived for stochastic processes that appear as solutions of the averaged hydrodynamic equations, since their moments satisfy the budgets given by these equations. These equations include the potential temperature, so that non-neutral flows can be described. They will be compared with nonlinear and non-Markovian equations that are obtained using concepts of nonequilibrium statistical mechanics. This approach permits the description of turbulent motion and buoyancy, where memory effects and driving forces with arbitrary colored noise may occur. The equations depend on assumptions that concern the dissipation and pressure redistribution. In the approximations of Kolmogorov and Rotta for these terms, the dissipation time scale remains open, which can be determined by the calculation of the production–dissipation ratio of turbulent kinetic energy. The features of these equations are illustrated by the calculation of turbulent states in the space of invariants. ©1997 American Institute of Physics.