On The Energy Transfer To High Frequencies In The Damped/Driven Nonlinear Schr\"odinger Equation (Extended Version)

We consider a damped/driven nonlinear Schr"odinger equation in an $n$-cube $K^n\subset {\mathbb{R}}^n$, $n$ is arbitrary, under Dirichlet boundary conditions $$u_t-\nu \mathrm{\Delta }u+i\mathrm{\mid }u{\mathrm{\mid }}^{2}u=\sqrt{\nu }\eta (t,x),x\in K^n,u{\mathrm{\mid }}_{\mathrm{\partial }{K}^n}=0,\nu >0,$$ 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 $\mathrm{\parallel }u(t){\mathrm{\parallel }}_m^2\le C{\nu }^{-m},$ uniformly in $t\ge 0$ and $\nu >0$. In this work we prove that for small $\nu >0$ and any initial data, with large probability the Sobolev norms $\mathrm{\parallel }u(t,\cdot ){\mathrm{\parallel }}_m$ of the solutions with $m>2$ become large at least to the order of ${\nu }^{-{\kappa }_{n,m}}$ with ${\kappa }_{n,m}>0$, on time intervals of order $\mathcal{O}(\frac{1}{\nu })$.