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

We consider a damped/driven nonlinear Schrödinger equation in Rⁿ, where n is arbitrary, Eut-νΔu+i|u|2u=νη(t,x),ν>0, under odd periodic boundary conditions. Here η(t, x) is a random force which is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy ‖u(t)‖m2≤Cν-m, uniformly in t≥ 0 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, ·) ‖ _m with m> 2 become large at least to the order of ν-κn,m with κ_n,m> 0 , on time intervals of order O(1ν). It proves that solutions of the equation develop short space-scale of order ν to a positive degree, and rigorously establishes the (direct) cascade of energy for the equation.