Almost Sure Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Large Initial Data

We consider the fourth-order nonlinear Schrödinger equation (4NLS) (i∂ _t + εΔ + Δ ^2)u = c_1u^m + c_2(∂ u)u^m - 1 + c_3(∂ u)^2u^m - 2, and establish the conditional almost sure global well-posedness for random initial data in H^s(ℝ^d) for s ∈ ( s c − 1/2, s c ], when d ≥ 3 and m ≥ 5, where s c := d /2 − 2/( m − 1) is the scaling critical regularity of 4NLS with the second order derivative nonlinearities. Our proof relies on the nonlinear estimates in a new M -norm and the stability theory in the probabilistic setting. Similar supercritical global well-posedness results also hold for d = 2, m ≥ 4 and d ≥ 3, 3 ≤ m < 5.