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

We consider a damped/driven nonlinear Schrödinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n$ is arbitrary, under Dirichlet boundary conditions \[ u_t-νΔu+i|u|^2u=\sqrtνη(t,x),\quad x\in K^{n},\quad u|_{\partial K^{n}}=0, \quad ν>0, \] where $η(t,x)$ 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 Cν^{-m}, $ uniformly in $t\ge0$ and $ν>0$. In this work we prove that for small $ν>0$ 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 $ν^{-κ_{n,m}}$ with $κ_{n,m}>0$, on time intervals of order $\mathcal{O}(\frac{1}ν)$.