On The Energy Transfer To High Frequencies In The Damped/Driven Nonlinear Schrodinger Equation (Extended Version)

We consider a damped/driven nonlinear Schrodinger equation in an $n$-cube $K^{n}subsetmathbb{R}^n$, $n$ is arbitrary, under Dirichlet boundary conditions [ u_t- uDelta u+i|u|^2u=sqrt{ u}eta(t,x),quad xin K^{n},quad u|_{partial K^{n}}=0, quad u>0, ] where $eta(t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $ | u(t)|_m^2 le C u^{-m}, $ uniformly in $tge0$ and $ u>0$. In this work we prove that for small $ u>0$ and any initial data, with large probability the Sobolev norms $|u(t,cdot)|_m$ of the solutions with $m>2$ become large at least to the order of $ u^{-kappa_{n,m}}$ with $kappa_{n,m}>0$, on time intervals of order $mathcal{O}(frac{1}{ u})$.