Khasminskii--Whitham averaging for randomly perturbed KdV equation

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We consider the damped-driven KdV equation u-νuxx+uxxx-6uuₓ=√νη(t,x), x∈ S¹, ∫ u dx≡ ∫ηdx≡0, where 0<ν≤1 and the random process η is smooth in x and white in t. For any periodic function u(x) let I=(I₁,I₂,...) be the vector, formed by the KdV integrals of motion, calculated for the potential u(x). We prove that if u(t,x) is a solution of the equation above, then for 0≤ tν⁻¹ and ν→0 the vector I(t)=(I₁(u(t,·)),I₂(u(t,·)),...) satisfies the (Whitham) averaged equation.