Khasminskii--Whitham averaging for randomly perturbed KdV equation

We consider the damped-driven KdV equation $$\dot u-\nu{u_{xx}}+u_{xxx}-6uu_x=\sqrt\nu \eta(t,x), x\in S^1, \int u dx\equiv \int\eta dx\equiv0,$$ where $0<\nu\le1$ and the random process $\eta$ is smooth in $x$ and white in $t$. For any periodic function $u(x)$ let $I=(I_1,I_2,...)$ 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 $0\le t\lesssim\nu^{-1}$ and $\nu\to0$ the vector $I(t)=(I_1(u(t,\cdot)),I_2(u(t,\cdot)),...)$ satisfies the (Whitham) averaged equation.