Damped-driven KdV and effective equation for long-time behaviour of its solutions

For the damped-driven KdV equation $$\dot{u}-\nu u_{xx}+u_{xxx}-6u{u}_x=\sqrt{\nu }\eta (t,x),x\in S^1,\int udx\equiv \int \eta dx\equiv 0,$$ with $0<\nu \le 1$ and smooth in $x$ white in $t$ random force $\eta$, we study the limiting long-time behaviour of the KdV integrals of motions $(I_1,I_2,\mathrm{.}\mathrm{.}\mathrm{.})$, evaluated along a solution $u^{\nu }(t,x)$, as $\nu \to 0$. We prove that %if $u=u^{\nu }(t,x)$ is a solution of the equation above, for $0\le \tau :=\nu t\lesssim 1$ the vector $I^{\nu }(\tau )=(I_1(u^{\nu }(\tau ,\cdot )),I_2(u^{\nu }(\tau ,\cdot )),\mathrm{.}\mathrm{.}\mathrm{.}),$ converges in distribution to a limiting process $I^0(\tau )=(I_1^0,I_2^0,\mathrm{.}\mathrm{.}\mathrm{.})$. The $j$-th component $I_j^0$ equals $\text{1}2(v_j(\tau {)}^2+v_{-j}(\tau {)}^2)$, where $v(\tau )=(v_1(\tau ),v_{-1}(\tau ),v_2(\tau ),\mathrm{.}\mathrm{.}\mathrm{.})$ is the vector of Fourier coefficients of a solution of an {\it effective equation} for the dam-ped-driven KdV. This new equation is a quasilinear stochastic heat equation with a non-local nonlinearity, written in the Fourier coefficients. It is well posed.