Well-posedness for Stochastic Generalized Fractional Benjamin-Ono Equation

This paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation. By using the Bourgain spaces and Fourier restriction method and the assumption that $u_0$ is ${\mathcal{F}}_0$-measurable, we prove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data $u_{0}(x,w)\in L^{2} (\Omega; H^{s}($\mathbb{R}$))$with$s\geq\frac{1}{2}-\frac{\alpha}{4}$,where$0< \alpha \leq 1.$Inparticular,when$u_{0}\in L^{2}(\Omega; H^{\frac{\alpha+1}{2}}($\mathbb{R}$))\cap L^{\frac{2(2+3\alpha)}{\alpha}}(\Omega; L^{2}($\mathbb{R}$))$,weprovethatthereexistsauniqueglobalsolution$u\in L^{2}(\Omega; H^{\frac{\alpha+1}{2}}($\mathbb{R}$))$with$0< \alpha \leq 1.$