On the energy transfer to high frequencies in the damped/driven nonlinear Schrödinger equation

We consider a damped/driven nonlinear Schrödinger equation in $$\mathbb {R}^n$$ , where n is arbitrary, $${\mathbb {E}}u_t-\nu \Delta u+i|u|^2u=\sqrt{\nu }\eta (t,x), \quad \nu >0,$$ under odd periodic boundary conditions. Here $$\eta (t,x)$$ random force which white time and smooth space. It known that the Sobolev norms of solutions satisfy $$ \Vert u(t)\Vert _m^2 \le C\nu ^{-m}, uniformly $$t\ge 0$$ $$\nu >0$$ . In this work we prove for small any initial data, with large probability $$\Vert u(t,\cdot )\Vert _m$$ $$m>2$$ become at least to order ^{-\kappa _{n,m}}$$ $$\kappa _{n,m}>0$$ on intervals $$\mathcal {O}(\frac{1}{\nu })$$ proves develop short space-scale positive degree, rigorously establishes (direct) cascade energy equation.