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The response of quasi-integrable Hamiltonian systems to lightly linear and/or nonlinear dampings and weakly external and/or parametric excitations of Gaussian white noises can be examined using a stochastic averaging approach. This method considers the stochastic averaging of an energy envelope and the Stratonovich stochastic averaging as two special cases. It is compatible with the quasi-nonintegrable Hamiltonian systems.

A study introduced a stochastic averaging method to roughly predict the response of multi-degree-of-freedom quasi-nonintegrable-Hamiltonian systems. This method was compared with the equivalent nonlinear system method for stochastically excited and dissipated nonintegrable Hamiltonian system and extended to a more general class of system. The two methods yielded consistent solutions for both single- and multi-degree-of-freedom systems.

The integrability and resonant property of a modified Hamiltonian system conditions exact stationary solutions for stochastically excited and dissipated integrable systems. Double summation Wong-Zakai correct terms are used to modify the Hamiltonian system. A nonintegrable system is solved using a Hamiltonian functional. The integrable system is solved by a functional of n independent motion integrals in the modified case.