Consider a perturbed KdV equation:
ut +uxxx −6uux = εf(u(·)), x ∈ T = R/Z,
where the nonlinear perturbation defines analytic operators u(·) → f(u(·)) in sufficiently smooth Sobolev spaces. Assume that the equation has an ε-quasi- invariant measure μ and satisfies some additional mild assumptions. Let uε(t) be a solution. Then on time intervals of order ε−1, as ε → 0, its actions I(uε(t,·)) can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is μ-typical.