On the Well-Posedness for Stochastic Schr(o)dinger Equations with Quadratic Potential

The authors investigate the influence of a harmonic potential and random perturbations on the nonlinear Schr(o)dinger equations.The local and global well-posedness are proved with values in the space ∑(Rn) =｛f ∈ H1(Rn),｜ · ｜f ∈ L2(Rn)}.When the nonlinearity is focusing and L2-supercritical,the authors give sufficient conditions for the solutions to blow up in finite time for both confining and repulsive potential.Especially for the repulsive case,the solution to the deterministic equation with the initial data satisfying the stochastic blow-up condition will also blow up in finite time.Thus,compared with the deterministic equation for the repulsive case,the blow-up condition is stronger on average,and depends on the regularity of the noise.If φ =0,our results coincide with the ones for the deterministic equation.