Almost Sure Well-Posedness of Fractional Schrödinger Equations with Hartree Nonlinearity

Abstract

We consider a Cauchy problem of an energy-critical fractional Schrödinger equation with Hartree nonlinearity below the energy space. Using randomization of functions on ${\mathbb{R}}^d$ associated with the Wiener decomposition, we prove that the Cauchy problem is almost surely locally well posed. Our result includes the Hartree Schrödinger equation ($\alpha =2$).