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We consider the Cauchy problem for the nonlinear Schr\"odinger equations(NLS) with non-algebraic nonlinearities on the Euclidean space. In particular,we study the energy-critical NLS on $\mathbb{R}^d$, $d=5,6$, andenergy-critical NLS without gauge invariance and prove that they are almostsurely locally well-posed with respect to randomized initial data below theenergy space. We also study the long time behavior of solutions to theseequations: (i) we prove almost sure global well-posedness of the (standard)energy-critical NLS on $\mathbb{R}^d$, $d = 5, 6$, in the defocusing case, and(ii) we present a probabilistic construction of finite time blowup solutions tothe energy-critical NLS without gauge invariance below the energy space.

On the probabilistic well-posedness of the nonlinear Schr\"{o}dinger equations with non-algebraic nonlinearities