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 $0< \alpha \leq 1.$ In particular, when $u_{ 0}\in L^{ 2}(\Omega; H^{ \frac{ \alpha+1}{ 2}}(\mathbb{ R}))\cap L^{ \frac{ 2(2+3\alpha)}{ \alpha}}(\Omega; L^{ 2}(\mathbb{ R}))$, we prove that there exists a unique global solution $u\in L^{ 2}(\Omega; H^{ \frac{ \alpha+1}{ 2}}(\mathbb{ R}))$ with $0< \alpha \leq 1.$
Categories: math.AP