The Cauchy problem for the Hartree equations under random influences

In this paper, the Cauchy problem for the Hartree equations under random influences is studied. In the first part, the randomization of the initial conditions, introduced by N. Burq and N. Tzvetkov, is used to show that the Hartree equations admit strong solutions for data in HσHσ for some σ   below the critical threshold scsc. In the second part, the (global) well-posedness and blow-up results for the Hartree equations with stochastic potential are considered.